# Detailed results of the comparison of basis functions used in interval-based optimization process

Sébastien Lengagne

August 27, 2024

## 1 Introduction

In this paper, we assess the results of the method proposed in [1] by addressing Optimization Problems (OP) to find the posture of a robot that optimally fits robot constraints and desired end-effector position such as :

$$ 
\begin{array}{ll}
\text{Find} & q \in \mathbb{Q} \subset \mathbb{R}^n \\
\text{such as} & \min_{q} \mathcal{F}(q) \\
\text{with } \forall j \in \{1, \dots, m\} & \mathcal{G}_j(q) \in [\underline{g}_j, \bar{g}_j]
\end{array}
\qquad (1) $$

Where n is called the number of degrees of freedom.

## 2 Detail of the basis functions

The different basis functions presented in [1] are more deeply presented here. Table 1 to 8 represents the coefficient of the matrix B and the coefficient of the matrix B⁻¹. Eventually, Table 9 shows the plot for the different functions from order 2 to 6.

We consider basis functions of the difference types :

* Bernstein,

* MinVariance,

* B-Splines,

* MinNorm,

* MinVolume

* Approximation of MinVolume (ApproxMinVo),

* Recursive,

* Recursive2.

## 3 Bissection mode

The goal of this document is to evaluate the efficiency of the evaluation process in the case of optimization. We do not focus on the bissection process that will be studied in future works. Here, we consider the results of two different bissection process :

* **MinFirst**: Cut the largest interval and store in a FILO (First In Last Out) stack the higher part and then the lower part. So the lower part will be the next to be evaluated.

* **MaxFirst**: Cut the largest interval and store in a FILO (First In Last Out) stack the lower part and then the higher part. So the higher part will be the next to be evaluated.

## 4 Presentation of the Planar robot problem

First, we assess our method on multi degrees-of-freedom 2D-robots, and considering possible constraints on the end effector, the Center of Mass position and maximum torque:
## 4.1 End effector position

We define the end effector by recursively computing:

$$i \in 1, \dots, n \quad T_i = T_{i-1} \times \begin{bmatrix} \cos(q_i) & -\sin(q_i) & l \\ \sin(q_i) & \cos(q_i) & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad (2)$$

With the number of a link $l = 2/n$ and $T_0 = I_{3 \times 3}$ and the end-effector position $\{P_x, P_y\}$ and orientaiton $\theta$ are defined as:

$$T_n = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & P_x \\ \sin(\theta) & \cos(\theta) & P_y \\ 0 & 0 & 1 \end{bmatrix} \qquad (3)$$

## 4.2 Considering several targets

We assess our method on nine different target positions considering shape as a square of size 0.2 by 0.2, as a circle target with a radius of 0.1 and as a ring target with an external radius of 0.15 and an interval radius of 0.05. The targets are localized at three different positions:

* problem 1,4,7: $x = 0.2, y = 1.7$

* problem 2,5,8: $x = 0.4, y = 1.4$

* problem 3,6,9: $x = 0.6, y = 1.1$

We defined the following constraints:

$$\text{square: } \begin{cases} P_x \in [x - 0.1; x + 0.1] \\ P_y \in [y - 0.1; y + 0.1] \end{cases} \qquad (4)$$

$$\text{circle: } (P_x - x)^2 + (P_y - y)^2 \in [0; 0.01] \qquad (5)$$

$$\text{ring: } (P_x - x)^2 + (P_y - y)^2 \in [0.05^2; 0.15^2] \qquad (6)$$

## 4.3 Center of Mass

To assess the performance of the evaluation for more complex functions we consider the Center of Mass (COM) of the robot. We consider that all the links have the same length $l = 2/n$ and same weight 1 and the same Center Of Mass located in the middle of each links. Thus the x-coordinate of the COM of each link $c_i$ can be computed through:

$$c_i = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} T_i \times \begin{bmatrix} l/2 \\ 0 \\ 0 \end{bmatrix} \qquad (7)$$

and the global COM as:

$$COM = \frac{1}{n} \sum_{i=1}^{n} c_i \qquad (8)$$

Finally we consider that the COM of the robot must remain in $[0.1, 0.1]$ on the horizontal plane.

## 4.4 Cost function

We consider two different cost functions to minimize. The first one is the sum of joint position square and the second is the sum or torque square:

$$\text{problem 1 to 9 position: } \sum_{i=1}^{n} q_i^2 \qquad (9)$$

$$\text{problem 11 to 19 torque: } \sum_{i=1}^{n} \Gamma_i^2 \qquad (10)$$

(11)

## 4.5 Results of the Planar robot problem

We assess our contribution with 18 different constraint satisfaction problems, with the two bissection processes (MinFirst and MaxFirst) such as presented in Table 11.
## 4.6 Results of the Bissection process

The results are summarized in the tables 12 to 47 and summarized in table 48. For all the cases, it appears that using basis functions reduce the number of iterations regarding to the state-of-the-art interval bissection process. This phenomenon is all the more important as the number of degrees of freedom increases and as the precision decreases. Regarding the computation time the BSplines basis function is the worst cases (even worse than interval bissection process). Excluding the preparation phase the other basis functions allows the optimization process to be faster than interval bissection. Taking into account the preparation phase, the basis function (except BSplines) are faster for degrees of freedom bigger than 4.

Regarding the global performances the Bernstein function seems to be the best solution, very close to the MinVariance (for computation time and number of iterations). Then the MinNo,the Recursive and the Recurisve2 produce quite good performances and the MinVo and ApproxMinVo are the less effective methods whereas BSplines is the worst one.

# 5 3D robot

## 5.1 Presentation fo the robot

In this section we consider the same kind of problem as presented in Equation 1 but with 3D-robots. To assess the effectiveness of our method on more complex systems, we consider the following problem:

$$
\begin{array}{ll}
\text{Find} & q \in \mathbb{Q} \subset \mathbb{R}^n \\
\text{such as} & \min_q F(q) \\
\text{with} & x \in x_d + [-0.01; 0.01] \\
& y \in y_d + [-0.01; 0.01] \\
& z \in z_d + [-0.01; 0.01] \\
\forall j = \{1, \dots, n\} & \Gamma_j \in [-\alpha \bar{\Gamma}_j : \alpha \bar{\Gamma}_j]
\end{array}
\qquad (12)
$$

We present the results for two different 3D robots (with $n = 4$ and $n = 6$ degrees of freedom) which derive from the KUKA LWR as presented on Figure 1. The 4-dof robot is obtained by considering the joint number 3 and 5 equal to zero.

<figure><img src="image_1.png" /><figcaption>Figure 1: The 3D robot : KUKA LWR</figcaption></figure>

The solution of Problem 12 is the static posture, i.e. the joint position, that minimizes the cost function $F(q)$ and ensures that the end effector position $\{x, y, z\}$ is within a 0.02 meter width box centered on a desired position $\{x_d, y_d, z_d\}$, and checks the joint torque $\Gamma_i$ are within the joint limits $[-\bar{\Gamma}_j : \bar{\Gamma}_j]$ considering different coefficient $\alpha$. These function can be obtained by using the forward kinematic model and the inverse dynamics model as presented in [2].

## 5.2 Constraint evaluation

First, Table 49 and 50 presents the results of the constraint and cost function computation considering several inputs to compare the effectiveness of the different basis function on the evaluation performances.

Figure 2 represents the diameter of the sum of square torque regarding the diameter of the input $[q]$. It shows that for large boxes (diameter of 2), the interval evaluation is quite similar using interval analysis or basis function based evaluation. For smaller intervals, the basis-functions produce tighter evaluation than interval analysis, except for the worst case of B-Splines.
<figure><img src="image_2.png" /><figcaption>Figure 2: Representation of the diameter of the evaluation using basis function regarding interval analysis.</figcaption></figure>

## 5.3 Optimization Results of the 3D robot problem

We assess our contribution with 12 different constraint satisfaction problems, with the two bissection processes (MinFirst and MaxFirst) such as presented in Table 51, with *n* the number of degrees of freedom. Some computation with 6-dof robot and sum of square torques are still in progress with huge computation time (more than 60 days), hence we do not present the results here.

## 5.4 Results of the Bissection process

We can note that the preparation time is quite important: around 100s for 4-dof-robot and around 400s for the 6-dof-robot. This will greatly impact the performance for fast resolution, but can be neglected for more complex problem that are solved in several hours or days.

As for the 2D-cases, we notice that the number of iterations of basis-function based method is lower that state-of-the art interval analysis. This point emphasizes that the evaluation produce tighter boxes that was already showed in Section 5.2.

Regarding the performances of the different basis functions, it is clear that the BSplines are the worst case. The other basis function produce similar results in terms of reducing the number of bissection process or the computation even if we can notice that the Bernstein functions seems too be the best choice.

# 6 Result summary

<figure><img src="image_3.png" /><figcaption>Figure 3: Comparison of the number of iterations</figcaption></figure>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Comparison of the computation time.</figcaption></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Comparison of the total computation time (including preparation phase).</figcaption></figure>

## Acknowledgment

We are grateful to the Mésocentre Clermont-Auvergne of the Université Clermont Auvergne for providing help, computing and storage resources facilities.

## References

[1] Sébastien Lengagne. Comparison of the basis-functions for interval-based optimization. submitted to *International Journal of Approximate Reasoning*, —.

[2] Bruno Siciliano and Oussama Khatib. *Springer Handbook of Robotics*. Springer-Verlag, Berlin, Heidelberg, 2007.
<table><caption>Table 1: Values of the matrices $B$ and $B^{-1}$ for the Bernstein basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="4">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td>1</td><td colspan="4">0.500000 0.500000<br>-0.500000 0.500000</td></tr><tr><td>2</td><td colspan="4">0.250000 0.500000 0.250000<br>-0.500000 0.000000 0.500000<br>0.250000 -0.500000 0.250000</td></tr><tr><td>3</td><td colspan="4">0.125000 0.375000 0.375000 0.125000<br>-0.375000 -0.375000 0.375000 0.375000<br>0.375000 -0.375000 -0.375000 0.375000<br>-0.125000 0.375000 -0.375000 0.125000</td></tr><tr><td>4</td><td colspan="4">0.062500 0.250000 0.375000 0.250000 0.062500<br>-0.250000 -0.500000 0.000000 0.500000 0.250000<br>0.375000 0.000000 -0.750000 0.000000 0.375000<br>-0.250000 0.500000 0.000000 -0.500000 0.250000<br>0.062500 -0.250000 0.375000 -0.250000 0.062500</td></tr><tr><td>5</td><td colspan="4">0.031250 0.156250 0.312500 0.312500 0.156250 0.031250<br>-0.156250 -0.468750 -0.312500 0.312500 0.468750 0.156250<br>0.312500 0.312500 -0.625000 -0.625000 0.312500 0.312500<br>-0.312500 0.312500 0.625000 -0.625000 -0.312500 0.312500<br>0.156250 -0.468750 0.312500 0.312500 -0.468750 0.156250<br>-0.031250 0.156250 -0.312500 0.312500 -0.156250 0.031250</td></tr><tr><td>6</td><td colspan="4">0.015625 0.093750 0.234375 0.312500 0.234375 0.093750 0.015625<br>-0.093750 -0.375000 -0.468750 0.000000 0.468750 0.375000 0.093750<br>0.234375 0.468750 -0.234375 -0.937500 -0.234375 0.468750 0.234375<br>-0.312500 0.000000 0.937500 0.000000 -0.937500 0.000000 0.312500<br>0.234375 -0.468750 -0.234375 0.937500 -0.234375 -0.468750 0.234375<br>-0.093750 0.375000 -0.468750 0.000000 0.468750 -0.375000 0.093750<br>0.015625 -0.093750 0.234375 -0.312500 0.234375 -0.093750 0.015625</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td>1</td><td colspan="4">1.000000 -1.000000<br>1.000000 1.000000</td></tr><tr><td>2</td><td colspan="4">1.000000 -1.000000 1.000000<br>1.000000 0.000000 -1.000000<br>1.000000 1.000000 1.000000</td></tr><tr><td>3</td><td colspan="4">1.000000 -1.000000 1.000000 -1.000000<br>1.000000 -0.333333 -0.333333 1.000000<br>1.000000 0.333333 -0.333333 -1.000000<br>1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>4</td><td colspan="4">1.000000 -1.000000 1.000000 -1.000000 1.000000<br>1.000000 -0.500000 -0.000000 0.500000 -1.000000<br>1.000000 -0.000000 -0.333333 -0.000000 1.000000<br>1.000000 0.500000 0.000000 -0.500000 -1.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>5</td><td colspan="4">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000<br>1.000000 -0.600000 0.200000 0.200000 -0.600000 1.000000<br>1.000000 -0.200000 -0.200000 0.200000 0.200000 -1.000000<br>1.000000 0.200000 -0.200000 -0.200000 0.200000 1.000000<br>1.000000 0.600000 0.200000 -0.200000 -0.600000 -1.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>6</td><td colspan="4">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000 1.000000<br>1.000000 -0.666667 0.333333 -0.000000 -0.333333 0.666667 -1.000000<br>1.000000 -0.333333 -0.066667 0.200000 -0.066667 -0.333333 1.000000<br>1.000000 0.000000 -0.200000 -0.000000 0.200000 0.000000 -1.000000<br>1.000000 0.333333 -0.066667 -0.200000 -0.066667 0.333333 1.000000<br>1.000000 0.666667 0.333333 0.000000 -0.333333 -0.666667 -1.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000</td></tr></tbody></table>
<table><caption>Table 2: Values of the matrices B and B⁻¹ for the MinVariance basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="6">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td rowspan="2">1</td><td colspan="6">0.500000 0.500000</td></tr><tr><td colspan="6">-0.500000 0.500000</td></tr><tr><td rowspan="2">2</td><td colspan="6">0.155869 0.688262 0.155869</td></tr><tr><td colspan="6">-0.463203 0.000000 0.463203</td></tr><tr><td rowspan="2">3</td><td colspan="6">0.344131 -0.688262 0.344131</td></tr><tr><td colspan="6">0.019243 0.480757 0.480757 0.019243</td></tr><tr><td rowspan="2">4</td><td colspan="6">-0.175647 -0.592928 0.592928 0.175647</td></tr><tr><td colspan="6">0.474214 -0.474214 -0.474214 0.474214</td></tr><tr><td rowspan="2">5</td><td colspan="6">-0.317810 0.599471 -0.599471 0.317810</td></tr><tr><td colspan="6">0.000358 0.223129 0.553025 0.223129 0.000358</td></tr><tr><td rowspan="2">6</td><td colspan="6">0.017719 -0.754489 0.000000 0.754489 -0.017719</td></tr><tr><td colspan="6">0.198449 0.414677 -1.226252 0.414677 0.198449</td></tr><tr><td rowspan="2">7</td><td colspan="6">-0.510994 0.754489 0.000000 -0.754489 0.510994</td></tr><tr><td colspan="6">0.297927 -0.637806 0.679758 -0.637806 0.297927</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td rowspan="2">1</td><td colspan="6">0.001914 0.054005 0.444081 0.444081 0.054005 0.001914</td></tr><tr><td colspan="6">0.020330 -0.413584 -0.682843 0.682843 0.413584 -0.020330</td></tr><tr><td rowspan="2">2</td><td colspan="6">-0.000905 0.916831 -0.915926 -0.915926 0.916831 -0.000905</td></tr><tr><td colspan="6">-0.272851 -0.227279 1.473726 -1.473726 0.227279 0.272851</td></tr><tr><td rowspan="2">3</td><td colspan="6">0.496226 -0.970088 0.473862 0.473862 -0.970088 0.496226</td></tr><tr><td colspan="6">-0.244714 0.641611 -0.792900 0.792900 -0.641611 0.244714</td></tr><tr><td rowspan="2">4</td><td colspan="6">0.002525 0.009926 0.212042 0.551016 0.212042 0.009926 0.002525</td></tr><tr><td colspan="6">0.005469 -0.147132 -0.828820 0.000000 0.828820 0.147132 -0.005469</td></tr><tr><td rowspan="2">5</td><td colspan="6">-0.051546 0.691264 0.368931 -2.017299 0.368931 0.691264 -0.051546</td></tr><tr><td colspan="6">-0.013728 -1.008610 1.734068 0.000000 -1.734068 1.008610 0.013728</td></tr><tr><td rowspan="2">6</td><td colspan="6">0.343248 -0.088754 -1.475367 2.441746 -1.475367 -0.088754 0.343248</td></tr><tr><td colspan="6">-0.489013 1.155742 -0.906977 0.000000 0.906977 -1.155742 0.489013</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td rowspan="2">1</td><td colspan="6">1.000000 -1.000000</td></tr><tr><td colspan="6">1.000000 1.000000</td></tr><tr><td rowspan="2">2</td><td colspan="6">1.000000 -1.079440 1.000000</td></tr><tr><td colspan="6">1.000000 0.000000 -0.452934</td></tr><tr><td rowspan="2">3</td><td colspan="6">1.000000 -1.020433 1.013797 -1.009295</td></tr><tr><td colspan="6">1.000000 -0.540983 -0.040579 0.298991</td></tr><tr><td rowspan="2">4</td><td colspan="6">1.000000 0.540983 -0.040579 -0.298991</td></tr><tr><td colspan="6">1.000000 1.020433 1.013797 1.009295</td></tr><tr><td rowspan="2">5</td><td colspan="6">1.000000 -1.013632 1.008328 -1.013632 1.005414</td></tr><tr><td colspan="6">1.000000 -0.686505 0.328883 -0.023804 -0.220271</td></tr><tr><td rowspan="2">6</td><td colspan="6">1.000000 -0.000000 -0.266696 0.000000 0.176443</td></tr><tr><td colspan="6">1.000000 0.686505 0.328883 0.023804 -0.220271</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td rowspan="2">1</td><td colspan="6">1.000000 -1.005471 1.005090 -1.006392 1.005581 -1.004626</td></tr><tr><td colspan="6">1.000000 -0.758010 0.483305 -0.240705 -0.002976 0.205408</td></tr><tr><td rowspan="2">2</td><td colspan="6">1.000000 -0.303057 -0.063107 0.115827 -0.003972 -0.154322</td></tr><tr><td colspan="6">1.000000 0.303057 -0.063107 -0.115827 -0.003972 0.154322</td></tr><tr><td rowspan="2">3</td><td colspan="6">1.000000 0.758010 0.483305 0.240705 -0.002976 -0.205408</td></tr><tr><td colspan="6">1.000000 1.005471 1.005090 1.006392 1.005581 1.004626</td></tr><tr><td rowspan="2">4</td><td colspan="6">1.000000 -1.003855 1.004594 -1.005696 1.005546 -1.005462 1.004382</td></tr><tr><td colspan="6">1.000000 -0.792904 0.578203 -0.378065 0.182506 0.001746 -0.174029</td></tr><tr><td rowspan="2">5</td><td colspan="6">1.000000 -0.469135 0.106350 0.060478 -0.078809 -0.006944 0.147667</td></tr><tr><td colspan="6">1.000000 0.000000 -0.111888 0.000000 0.044865 0.000000 -0.116585</td></tr><tr><td rowspan="2">6</td><td colspan="6">1.000000 0.469135 0.106350 -0.060478 -0.078809 0.006944 0.147667</td></tr><tr><td colspan="6">1.000000 0.792904 0.578203 0.378065 0.182506 -0.001746 -0.174029</td></tr><tr><td rowspan="2">7</td><td colspan="6">1.000000 1.003855 1.004594 1.005696 1.005546 1.005462 1.004382</td></tr><tr><td colspan="6">1.000000 -1.003855 1.004594 -1.005696 1.005546 -1.005462 1.004382</td></tr></tbody></table>
<table><caption>Table 3: Values of the matrices $B$ and $B^{-1}$ for the BSplines basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="2">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td>1</td><td colspan="2">0.500000 0.500000</td></tr><tr><td></td><td colspan="2">-0.500000 0.500000</td></tr><tr><td>2</td><td colspan="2">0.125000 0.750000 0.125000</td></tr><tr><td></td><td colspan="2">-0.250000 0.000000 0.250000</td></tr><tr><td></td><td colspan="2">0.125000 -0.250000 0.125000</td></tr><tr><td>3</td><td colspan="2">0.020833 0.479167 0.479167 0.020833</td></tr><tr><td></td><td colspan="2">-0.062500 -0.312500 0.312500 0.062500</td></tr><tr><td></td><td colspan="2">0.062500 -0.062500 -0.062500 0.062500</td></tr><tr><td></td><td colspan="2">-0.020833 0.062500 -0.062500 0.020833</td></tr><tr><td>4</td><td colspan="2">0.002604 0.197917 0.598958 0.197917 0.002604</td></tr><tr><td></td><td colspan="2">-0.010417 -0.229167 0.000000 0.229167 0.010417</td></tr><tr><td></td><td colspan="2">0.015625 0.062500 -0.156250 0.062500 0.015625</td></tr><tr><td></td><td colspan="2">-0.010417 0.020833 0.000000 -0.020833 0.010417</td></tr><tr><td></td><td colspan="2">0.002604 -0.010417 0.015625 -0.010417 0.002604</td></tr><tr><td>5</td><td colspan="2">0.000260 0.061719 0.438021 0.438021 0.061719 0.000260</td></tr><tr><td></td><td colspan="2">-0.001302 -0.097656 -0.200521 0.200521 0.097656 0.001302</td></tr><tr><td></td><td colspan="2">0.002604 0.054688 -0.057292 -0.057292 0.054688 0.002604</td></tr><tr><td></td><td colspan="2">-0.002604 -0.007812 0.036458 -0.036458 0.007812 0.002604</td></tr><tr><td></td><td colspan="2">0.001302 -0.003906 0.002604 0.002604 -0.003906 0.001302</td></tr><tr><td></td><td colspan="2">-0.000260 0.001302 -0.002604 0.002604 -0.001302 0.000260</td></tr><tr><td>6</td><td colspan="2">0.000022 0.015668 0.228798 0.511024 0.228798 0.015668 0.000022</td></tr><tr><td></td><td colspan="2">-0.000130 -0.030729 -0.188151 0.000000 0.188151 0.030729 0.000130</td></tr><tr><td></td><td colspan="2">0.000326 0.024089 0.025716 -0.100260 0.025716 0.024089 0.000326</td></tr><tr><td></td><td colspan="2">-0.000434 -0.008681 0.018663 0.000000 -0.018663 0.008681 0.000434</td></tr><tr><td></td><td colspan="2">0.000326 0.000651 -0.005534 0.009115 -0.005534 0.000651 0.000326</td></tr><tr><td></td><td colspan="2">-0.000130 0.000521 -0.000651 -0.000000 0.000651 -0.000521 0.000130</td></tr><tr><td></td><td colspan="2">0.000022 -0.000130 0.000326 -0.000434 0.000326 -0.000130 0.000022</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td>1</td><td colspan="2">1.000000 -1.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000</td></tr><tr><td>2</td><td colspan="2">1.000000 -2.000000 3.000000</td></tr><tr><td></td><td colspan="2">1.000000 0.000000 -1.000000</td></tr><tr><td></td><td colspan="2">1.000000 2.000000 3.000000</td></tr><tr><td>3</td><td colspan="2">1.000000 -3.000000 7.666667 -15.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -0.333333 3.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 -0.333333 -3.000000</td></tr><tr><td></td><td colspan="2">1.000000 3.000000 7.666667 15.000000</td></tr><tr><td>4</td><td colspan="2">1.000000 -4.000000 14.333333 -44.000000 105.000000</td></tr><tr><td></td><td colspan="2">1.000000 -2.000000 2.333333 2.000000 -15.000000</td></tr><tr><td></td><td colspan="2">1.000000 0.000000 -1.666667 0.000000 9.000000</td></tr><tr><td></td><td colspan="2">1.000000 2.000000 2.333333 -2.000000 -15.000000</td></tr><tr><td></td><td colspan="2">1.000000 4.000000 14.333333 44.000000 105.000000</td></tr><tr><td>5</td><td colspan="2">1.000000 -5.000000 23.000000 -95.000000 337.800000 -945.000000</td></tr><tr><td></td><td colspan="2">1.000000 -3.000000 7.000000 -9.000000 -14.200000 105.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -1.000000 5.000000 1.800000 -45.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 -1.000000 -5.000000 1.800000 45.000000</td></tr><tr><td></td><td colspan="2">1.000000 3.000000 7.000000 9.000000 -14.200000 -105.000000</td></tr><tr><td></td><td colspan="2">1.000000 5.000000 23.000000 95.000000 337.800000 945.000000</td></tr><tr><td>6</td><td colspan="2">1.000000 -6.000000 33.666667 -174.000000 809.266667 -3254.000000 10395.000000</td></tr><tr><td></td><td colspan="2">1.000000 -4.000000 13.666667 -36.000000 49.266667 124.000000 -945.000000</td></tr><tr><td></td><td colspan="2">1.000000 -2.000000 1.666667 6.000000 -22.733333 -18.000000 315.000000</td></tr><tr><td></td><td colspan="2">1.000000 0.000000 -2.333333 0.000000 17.266667 0.000000 -225.000000</td></tr><tr><td></td><td colspan="2">1.000000 2.000000 1.666667 -6.000000 -22.733333 18.000000 315.000000</td></tr><tr><td></td><td colspan="2">1.000000 4.000000 13.666667 36.000000 49.266667 -124.000000 -945.000000</td></tr><tr><td></td><td colspan="2">1.000000 6.000000 33.666667 174.000000 809.266667 3254.000000 10395.000000</td></tr></tbody></table>
<table><caption>Table 4: Values of the matrices $B$ and $B^{-1}$ for the MinNorm basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="6">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td rowspan="2">1</td><td colspan="6">0.500000 0.500000</td></tr><tr><td colspan="6">-0.500000 0.500000</td></tr><tr><td rowspan="3">2</td><td colspan="6">0.178806 0.642387 0.178806</td></tr><tr><td colspan="6">-0.479297 0.000000 0.479297</td></tr><tr><td colspan="6">0.321194 -0.642387 0.321194</td></tr><tr><td rowspan="4">3</td><td colspan="6">0.028323 0.471677 0.471677 0.028323</td></tr><tr><td colspan="6">-0.208384 -0.572950 0.572950 0.208384</td></tr><tr><td colspan="6">0.466241 -0.466241 -0.466241 0.466241</td></tr><tr><td colspan="6">-0.286180 0.578386 -0.578386 0.286180</td></tr><tr><td rowspan="4">4</td><td colspan="6">0.001808 0.232346 0.531691 0.232346 0.001808</td></tr><tr><td colspan="6">-0.040769 -0.719547 0.000000 0.719547 0.040769</td></tr><tr><td colspan="6">0.270116 0.324740 -1.189712 0.324740 0.270116</td></tr><tr><td colspan="6">-0.454129 0.719547 0.000000 -0.719547 0.454129</td></tr><tr><td rowspan="5">5</td><td colspan="6">0.224324 -0.557086 0.665525 -0.557086 0.224324</td></tr><tr><td colspan="6">0.000548 0.087040 0.412412 0.412412 0.087040 0.000548</td></tr><tr><td colspan="6">-0.013185 -0.484330 -0.611879 0.611879 0.484330 0.013185</td></tr><tr><td colspan="6">0.104674 0.774368 -0.879042 -0.879042 0.774368 0.104674</td></tr><tr><td colspan="6">-0.313820 0.006407 1.345160 -1.345160 -0.006407 0.313820</td></tr><tr><td rowspan="6">6</td><td colspan="6">0.390724 -0.861075 0.470351 0.470351 -0.861075 0.390724</td></tr><tr><td colspan="6">-0.168943 0.478256 -0.737001 0.737001 -0.478256 0.168943</td></tr><tr><td colspan="6">0.000246 0.039541 0.234007 0.452411 0.234007 0.000246</td></tr><tr><td colspan="6">-0.004996 -0.286016 -0.769529 0.000000 0.769529 0.286016 0.004996</td></tr><tr><td colspan="6">0.040290 0.715108 0.132833 -1.776461 0.132833 0.715108 0.040290</td></tr><tr><td colspan="6">-0.162385 -0.572706 1.643712 0.000000 -1.643712 0.572706 0.162385</td></tr><tr><td rowspan="10">B<sup>-1</sup></td><td rowspan="3">1</td><td colspan="6">1.000000 -1.000000</td></tr><tr><td colspan="6">1.000000 1.000000</td></tr><tr><td colspan="6">1.000000 1.000000</td></tr><tr><td rowspan="3">2</td><td colspan="6">1.000000 -1.043194 1.000000</td></tr><tr><td colspan="6">1.000000 0.000000 -0.556693</td></tr><tr><td colspan="6">1.000000 1.043194 1.000000</td></tr><tr><td rowspan="4">3</td><td colspan="6">1.000000 -1.016520 1.011659 -1.006966</td></tr><tr><td colspan="6">1.000000 -0.502965 -0.060748 0.366237</td></tr><tr><td colspan="6">1.000000 0.502965 -0.060748 -0.366237</td></tr><tr><td colspan="6">1.000000 1.016520 1.011659 1.006966</td></tr><tr><td rowspan="5">4</td><td colspan="6">1.000000 -1.010308 1.009359 -1.010308 1.005456</td></tr><tr><td colspan="6">1.000000 -0.637638 0.264341 0.057244 -0.326362</td></tr><tr><td colspan="6">1.000000 -0.000000 -0.237895 -0.000000 0.278399</td></tr><tr><td colspan="6">1.000000 0.637638 0.264341 -0.057244 -0.326362</td></tr><tr><td colspan="6">1.000000 1.010308 1.009359 1.010308 1.005456</td></tr><tr><td rowspan="6">5</td><td colspan="6">1.000000 -1.006914 1.008556 -1.009297 1.008081 -1.006181</td></tr><tr><td colspan="6">1.000000 -0.712459 0.409683 -0.145518 -0.111157 0.325908</td></tr><tr><td colspan="6">1.000000 -0.231515 -0.087805 0.136932 0.022119 -0.236290</td></tr><tr><td colspan="6">1.000000 0.231515 -0.087805 -0.136932 0.022119 0.236290</td></tr><tr><td colspan="6">1.000000 0.712459 0.409683 0.145518 -0.111157 -0.325908</td></tr><tr><td colspan="6">1.000000 1.006914 1.008556 1.009297 1.008081 1.006181</td></tr><tr><td rowspan="10">6</td><td colspan="6">1.000000 -1.005823 1.007968 -1.009204 1.008783 -1.008078 1.006174</td></tr><tr><td colspan="6">1.000000 -0.755246 0.509137 -0.274875 0.056328 0.147388 -0.333314</td></tr><tr><td colspan="6">1.000000 -0.362511 0.014527 0.108716 -0.086189 -0.048237 0.236376</td></tr><tr><td colspan="6">1.000000 -0.000000 -0.105121 0.000000 0.078218 -0.000000 -0.187359</td></tr><tr><td colspan="6">1.000000 0.362511 0.014527 -0.108716 -0.086189 0.048237 0.236376</td></tr><tr><td colspan="6">1.000000 0.755246 0.509137 0.274875 0.056328 -0.147388 -0.333314</td></tr><tr><td colspan="6">1.000000 1.005823 1.007968 1.009204 1.008783 1.008078 1.006174</td></tr></tbody></table>
<table><caption>Table 5: Values of the matrices **B** and **B**⁻¹ for the MinVolume basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="2">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td>1</td><td colspan="2">0.500000 0.500000<br>-0.500000 0.500000</td></tr><tr><td>2</td><td colspan="2">0.125000 0.750000 0.125000<br>-0.433013 0.000000 0.433013<br>0.375000 -0.750000 0.375000</td></tr><tr><td>3</td><td colspan="2">0.000410 0.499590 0.499590 0.000410<br>-0.026982 -0.792093 0.792093 0.026982<br>0.456775 -0.456775 -0.456775 0.456775<br>-0.430204 0.834907 -0.834907 0.430204</td></tr><tr><td>4</td><td colspan="2">0.030227 0.148259 0.643029 0.148259 0.030227<br>0.138091 -0.810775 0.000000 0.810775 -0.138091<br>-0.094352 0.960202 -1.731700 0.960202 -0.094352<br>-0.575789 0.810775 0.000000 -0.810775 0.575789<br>0.525520 -1.108461 1.165882 -1.108461 0.525520</td></tr><tr><td>5</td><td colspan="2">0.045894 0.002895 0.451210 0.451210 0.002895 0.045894<br>-0.036501 -0.120977 -1.064403 1.064403 0.120977 0.036501<br>-0.377284 1.333003 -0.955718 -0.955718 1.333003 -0.377284<br>0.330192 -1.365550 2.776543 -2.776543 1.365550 -0.330192<br>0.776885 -1.319318 0.542433 0.542433 -1.319318 0.776885<br>-0.739186 1.503106 -1.750066 1.750066 -1.503106 0.739186</td></tr><tr><td>6</td><td colspan="2">0.018328 0.031209 0.152009 0.596907 0.152009 0.031209 0.018328<br>-0.136709 0.243263 -1.080915 0.000000 1.080915 -0.243263 0.136709<br>0.105859 -0.084538 1.580291 -3.203224 1.580291 -0.084538 0.105859<br>0.834817 -2.298575 2.468184 0.000000 -2.468184 2.298575 -0.834817<br>-0.736850 2.281168 -4.269993 5.451350 -4.269993 2.281168 -0.736850<br>-1.134019 2.055312 -1.408577 0.000000 1.408577 -2.055312 1.134019<br>1.060626 -2.227839 2.589730 -2.845034 2.589730 -2.227839 1.060626</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td>1</td><td colspan="2">1.000000 -1.000000<br>1.000000 1.000000</td></tr><tr><td>2</td><td colspan="2">1.000000 -1.154701 1.000000<br>1.000000 0.000000 -0.333333<br>1.000000 1.154701 1.000000</td></tr><tr><td>3</td><td colspan="2">1.000000 -1.149096 1.093732 -1.090170<br>1.000000 -0.592096 -0.000898 0.037136<br>1.000000 0.592096 -0.000898 -0.037136<br>1.000000 1.149096 1.093732 1.090170</td></tr><tr><td>4</td><td colspan="2">1.000000 -1.142338 1.106977 -1.142338 1.092669<br>1.000000 -0.811256 0.378336 -0.194562 0.010409<br>1.000000 -0.000000 -0.278531 -0.000000 -0.107525<br>1.000000 0.811256 0.378336 0.194562 0.010409<br>1.000000 1.142338 1.106977 1.142338 1.092669</td></tr><tr><td>5</td><td colspan="2">1.000000 -1.129437 1.109754 -1.147162 1.123459 -1.133082<br>1.000000 -0.933700 0.605476 -0.457764 0.234967 -0.158376<br>1.000000 -0.324894 -0.116762 0.091367 -0.115779 0.056856<br>1.000000 0.324894 -0.116762 -0.091367 -0.115779 -0.056856<br>1.000000 0.933700 0.605476 0.457764 0.234967 0.158376<br>1.000000 1.129437 1.109754 1.147162 1.123459 1.133082</td></tr><tr><td>6</td><td colspan="2">1.000000 -1.120437 1.105897 -1.147175 1.129137 -1.150340 1.128210<br>1.000000 -0.990939 0.751557 -0.630814 0.445470 -0.344918 0.217191<br>1.000000 -0.543877 0.088407 0.003123 -0.111657 0.067865 -0.103675<br>1.000000 0.000000 -0.191530 0.000000 -0.059053 0.000000 -0.039190<br>1.000000 0.543877 0.088407 -0.003123 -0.111657 -0.067865 -0.103675<br>1.000000 0.990939 0.751557 0.630814 0.445470 0.344918 0.217191<br>1.000000 1.120437 1.105897 1.147175 1.129137 1.150340 1.128210</td></tr></tbody></table>
<table><caption>Table 6: Values of the matrices **B** and **B**<sup>-1</sup> for the ApproxMinVo basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="2">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td>1</td><td colspan="2">0.500000 0.500000<br>-0.500000 0.500000</td></tr><tr><td>2</td><td colspan="2">0.124377 0.751246 0.124377<br>-0.432291 0.000000 0.432291<br>0.375623 -0.751246 0.375623</td></tr><tr><td>3</td><td colspan="2">0.000649 0.499351 0.499351 0.000649<br>-0.033676 -0.803260 0.803260 0.033676<br>0.453110 -0.453110 -0.453110 0.453110<br>-0.420083 0.849501 -0.849501 0.420083</td></tr><tr><td>4</td><td colspan="2">0.035652 0.146138 0.636420 0.146138 0.035652<br>0.149222 -0.808622 0.000000 0.808622 -0.149222<br>-0.125047 0.972447 -1.694798 0.972447 -0.125047<br>-0.588448 0.808622 0.000000 -0.808622 0.588448<br>0.554424 -1.118584 1.128320 -1.118584 0.554424</td></tr><tr><td>5</td><td colspan="2">0.046170 0.000790 0.453040 0.453040 0.000790 0.046170<br>-0.030551 -0.064281 -1.047189 1.047189 0.064281 0.030551<br>-0.376843 1.346724 -0.969881 -0.969881 1.346724 -0.376843<br>0.299901 -1.536987 2.717358 -2.717358 1.536987 -0.299901<br>0.773030 -1.326775 0.553745 0.553745 -1.326775 0.773030<br>-0.711707 1.622007 -1.707074 1.707074 -1.622007 0.711707</td></tr><tr><td>6</td><td colspan="2">0.018148 0.034239 0.143952 0.607323 0.143952 0.034239 0.018148<br>-0.137219 0.252439 -1.058872 0.000000 1.058872 -0.252439 0.137219<br>0.112414 -0.122731 1.632940 -3.245245 1.632940 -0.122731 0.112414<br>0.836717 -2.293984 2.375355 0.000000 -2.375355 2.293984 -0.836717<br>-0.765241 2.327842 -4.313797 5.502391 -4.313797 2.327842 -0.765241<br>-1.138248 2.041545 -1.331193 0.000000 1.331193 -2.041545 1.138248<br>1.088642 -2.239350 2.582942 -2.864469 2.582942 -2.239350 1.088642</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td>1</td><td colspan="2">1.000000 -1.000000<br>1.000000 1.000000</td></tr><tr><td>2</td><td colspan="2">1.000000 -1.156628 1.000000<br>1.000000 0.000000 -0.331122<br>1.000000 1.156628 1.000000</td></tr><tr><td>3</td><td colspan="2">1.000000 -1.160380 1.102051 -1.097218<br>1.000000 -0.573815 -0.001433 0.046001<br>1.000000 0.573815 -0.001433 -0.046001<br>1.000000 1.160380 1.102051 1.097218</td></tr><tr><td>4</td><td colspan="2">1.000000 -1.138365 1.097973 -1.138365 1.085173<br>1.000000 -0.828408 0.391487 -0.210072 0.023992<br>1.000000 -0.000000 -0.302808 0.000000 -0.132602<br>1.000000 0.828408 0.391487 0.210072 0.023992<br>1.000000 1.138365 1.097973 1.138365 1.085173</td></tr><tr><td>5</td><td colspan="2">1.000000 -1.140533 1.111718 -1.156170 1.129029 -1.140767<br>1.000000 -0.909198 0.600007 -0.443176 0.232770 -0.147718<br>1.000000 -0.388384 -0.114344 0.060935 -0.115468 0.042349<br>1.000000 0.388384 -0.114344 -0.060935 -0.115468 -0.042349<br>1.000000 0.909198 0.600007 0.443176 0.232770 0.147718<br>1.000000 1.140533 1.111718 1.156170 1.129029 1.140767</td></tr><tr><td>6</td><td colspan="2">1.000000 -1.120708 1.091898 -1.139451 1.112410 -1.141768 1.111816<br>1.000000 -0.992297 0.764313 -0.638222 0.458383 -0.349528 0.226618<br>1.000000 -0.563536 0.093807 -0.004494 -0.108458 0.064632 -0.102596<br>1.000000 0.000000 -0.195905 0.000000 -0.066752 0.000000 -0.043363<br>1.000000 0.563536 0.093807 0.004494 -0.108458 -0.064632 -0.102596<br>1.000000 0.992297 0.764313 0.638222 0.458383 0.349528 0.226618<br>1.000000 1.120708 1.091898 1.139451 1.112410 1.141768 1.111816</td></tr></tbody></table>
<table><caption>Table 7: Values of the matrices $B$ and $B^{-1}$ for the Recursive basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="2">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td>1</td><td colspan="2">0.500000 0.500000<br>-0.500000 0.500000</td></tr><tr><td>2</td><td colspan="2">0.500000 0.250000 0.250000<br>-0.500000 0.000000 0.500000<br>0.000000 -0.250000 0.250000</td></tr><tr><td>3</td><td colspan="2">0.500000 0.250000 0.125000 0.125000<br>-0.500000 0.000000 0.125000 0.375000<br>0.000000 -0.250000 -0.125000 0.375000<br>0.000000 0.000000 -0.125000 0.125000</td></tr><tr><td>4</td><td colspan="2">0.500000 0.250000 0.125000 0.062500 0.062500<br>-0.500000 0.000000 0.125000 0.125000 0.250000<br>0.000000 -0.250000 -0.125000 0.000000 0.375000<br>0.000000 0.000000 -0.125000 -0.125000 0.250000<br>0.000000 0.000000 0.000000 -0.062500 -0.062500 0.062500</td></tr><tr><td>5</td><td colspan="2">0.500000 0.250000 0.125000 0.062500 0.031250 0.031250<br>-0.500000 0.000000 0.125000 0.125000 0.093750 0.156250<br>0.000000 -0.250000 -0.125000 0.000000 0.062500 0.312500<br>0.000000 0.000000 -0.125000 -0.125000 -0.062500 0.312500<br>0.000000 0.000000 0.000000 -0.062500 -0.093750 0.156250<br>0.000000 0.000000 0.000000 0.000000 -0.031250 -0.031250 0.031250</td></tr><tr><td>6</td><td colspan="2">0.500000 0.250000 0.125000 0.062500 0.031250 0.015625 0.015625<br>-0.500000 0.000000 0.125000 0.125000 0.093750 0.062500 0.093750<br>0.000000 -0.250000 -0.125000 0.000000 0.062500 0.078125 0.234375<br>0.000000 0.000000 -0.125000 -0.125000 -0.062500 0.000000 0.312500<br>0.000000 0.000000 0.000000 -0.062500 -0.093750 -0.078125 0.234375<br>0.000000 0.000000 0.000000 0.000000 -0.031250 -0.062500 0.093750<br>0.000000 0.000000 0.000000 0.000000 0.000000 -0.015625 0.015625</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td>1</td><td colspan="2">1.000000 -1.000000<br>1.000000 1.000000</td></tr><tr><td>2</td><td colspan="2">1.000000 -1.000000 1.000000<br>1.000000 1.000000 -3.000000<br>1.000000 1.000000 1.000000</td></tr><tr><td>3</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000<br>1.000000 1.000000 -3.000000 5.000000<br>1.000000 1.000000 1.000000 -7.000000<br>1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>4</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000<br>1.000000 1.000000 -3.000000 5.000000 -7.000000<br>1.000000 1.000000 1.000000 -7.000000 17.000000<br>1.000000 1.000000 1.000000 1.000000 -15.000000 49.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 -31.000000</td></tr><tr><td>5</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000<br>1.000000 1.000000 -3.000000 5.000000 -7.000000 9.000000<br>1.000000 1.000000 1.000000 -7.000000 17.000000 -31.000000<br>1.000000 1.000000 1.000000 1.000000 -15.000000 49.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 -31.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>6</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000 1.000000<br>1.000000 1.000000 -3.000000 5.000000 -7.000000 9.000000 -11.000000<br>1.000000 1.000000 1.000000 -7.000000 17.000000 -31.000000 49.000000<br>1.000000 1.000000 1.000000 1.000000 -15.000000 49.000000 -111.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 -31.000000 129.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 -63.000000<br>1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000</td></tr></tbody></table>
<table><caption>Table 8: Values of the matrices $B$ and $B^{-1}$ for the Recursive2 basis functions from order 1 to 6.</caption><thead><tr><th>matrice</th><th>order</th><th colspan="2">value</th></tr></thead><tbody><tr><td rowspan="12">B</td><td>1</td><td colspan="2">0.500000 0.500000</td></tr><tr><td></td><td colspan="2">-0.500000 0.500000</td></tr><tr><td>2</td><td colspan="2">0.250000 0.250000 0.500000</td></tr><tr><td></td><td colspan="2">-0.500000 0.000000 0.500000</td></tr><tr><td></td><td colspan="2">0.250000 -0.250000 0.000000</td></tr><tr><td>3</td><td colspan="2">0.125000 0.125000 0.250000 0.500000</td></tr><tr><td></td><td colspan="2">-0.375000 -0.125000 0.000000 0.500000</td></tr><tr><td></td><td colspan="2">0.375000 -0.125000 -0.250000 0.000000</td></tr><tr><td></td><td colspan="2">-0.125000 0.125000 0.000000 0.000000</td></tr><tr><td>4</td><td colspan="2">0.062500 0.062500 0.125000 0.250000 0.500000</td></tr><tr><td></td><td colspan="2">-0.250000 -0.125000 -0.125000 0.000000 0.500000</td></tr><tr><td></td><td colspan="2">0.375000 0.000000 -0.125000 -0.250000 0.000000</td></tr><tr><td></td><td colspan="2">-0.250000 0.125000 0.125000 0.000000 0.000000</td></tr><tr><td></td><td colspan="2">0.062500 -0.062500 0.000000 0.000000 0.000000</td></tr><tr><td>5</td><td colspan="2">0.031250 0.031250 0.062500 0.125000 0.250000 0.500000</td></tr><tr><td></td><td colspan="2">-0.156250 -0.093750 -0.125000 -0.125000 0.000000 0.500000</td></tr><tr><td></td><td colspan="2">0.312500 0.062500 0.000000 -0.125000 -0.250000 0.000000</td></tr><tr><td></td><td colspan="2">-0.312500 0.062500 0.125000 0.125000 0.000000 0.000000</td></tr><tr><td></td><td colspan="2">0.156250 -0.093750 -0.062500 0.000000 0.000000 0.000000</td></tr><tr><td></td><td colspan="2">-0.031250 0.031250 0.000000 0.000000 0.000000 0.000000</td></tr><tr><td>6</td><td colspan="2">0.015625 0.015625 0.031250 0.062500 0.125000 0.250000 0.500000</td></tr><tr><td></td><td colspan="2">-0.093750 -0.062500 -0.093750 -0.125000 -0.125000 0.000000 0.500000</td></tr><tr><td></td><td colspan="2">0.234375 0.078125 0.062500 0.000000 -0.125000 -0.250000 0.000000</td></tr><tr><td></td><td colspan="2">-0.312500 0.000000 0.062500 0.125000 0.125000 0.000000 0.000000</td></tr><tr><td></td><td colspan="2">0.234375 -0.078125 -0.093750 -0.062500 0.000000 0.000000 0.000000</td></tr><tr><td></td><td colspan="2">-0.093750 0.062500 0.031250 0.000000 0.000000 0.000000 0.000000</td></tr><tr><td></td><td colspan="2">0.015625 -0.015625 0.000000 0.000000 0.000000 0.000000 0.000000</td></tr><tr><td rowspan="12">B<sup>-1</sup></td><td>1</td><td colspan="2">1.000000 -1.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000</td></tr><tr><td>2</td><td colspan="2">1.000000 -1.000000 1.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -3.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 1.000000</td></tr><tr><td>3</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -3.000000 -5.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>4</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 -15.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 7.000000 17.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -3.000000 -5.000000 -7.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>5</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 31.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 -15.000000 -49.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 7.000000 17.000000 31.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -3.000000 -5.000000 -7.000000 -9.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 1.000000 1.000000 1.000000 1.000000</td></tr><tr><td>6</td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000 1.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 -1.000000 -63.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 1.000000 31.000000 129.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 -1.000000 -15.000000 -49.000000 -111.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 1.000000 7.000000 17.000000 31.000000 49.000000</td></tr><tr><td></td><td colspan="2">1.000000 -1.000000 -3.000000 -5.000000 -7.000000 -9.000000 -11.000000</td></tr><tr><td></td><td colspan="2">1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000</td></tr></tbody></table>
<table><caption>Table 9: Representation of the basis functions for the order 2 to 6</caption><thead><tr><th>order :</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><td>Bernstein</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MinVariance</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BSplines</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MinNorm</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MinVolume</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ApproxMinVo</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Recursive1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Recursive2</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table 10: Values of the different criteria from the basis function from order 1 to 6.</caption><thead><tr><th>order</th><th>type</th><th>MinVolume<br/>-det(|B|)<sup>2</sup></th><th>MinNorm<br/>||B<sup>-1</sup>||<sup>2</sup></th><th>MinVariance<br/>||B<sup>-1</sup> - Average (B<sup>-1</sup>)||<sup>2</sup></th></tr></thead><tbody><tr><td rowspan="8">1</td><td>Bernstein</td><td>-0.25</td><td>4</td><td>2</td></tr><tr><td>MinVariance</td><td>-0.25</td><td>4</td><td><b>2</b></td></tr><tr><td>BSplines</td><td>-0.25</td><td>4</td><td>2</td></tr><tr><td>MinNorm</td><td>-0.25</td><td><b>4</b></td><td>2</td></tr><tr><td>MinVolume</td><td><b>-0.25</b></td><td>4</td><td>2</td></tr><tr><td>ApproxMinVo</td><td>-0.25</td><td>4</td><td>2</td></tr><tr><td>Recursive</td><td>-0.25</td><td>4</td><td>2</td></tr><tr><td>Recursive2</td><td>-0.25</td><td>4</td><td>2</td></tr><tr><td rowspan="8">2</td><td>Bernstein</td><td>-0.0625</td><td>8</td><td>4.66667</td></tr><tr><td>MinVariance</td><td>-0.101637</td><td>7.53553</td><td><b>3.73773</b></td></tr><tr><td>BSplines</td><td>-0.00390625</td><td>30</td><td>18.6667</td></tr><tr><td>MinNorm</td><td>-0.094799</td><td><b>7.48641</b></td><td>3.79204</td></tr><tr><td>MinVolume</td><td><b>-0.105469</b></td><td>7.77778</td><td>3.85185</td></tr><tr><td>ApproxMinVo</td><td>-0.105467</td><td>7.78522</td><td>3.85684</td></tr><tr><td>Recursive</td><td>-0.015625</td><td>17</td><td>13.3333</td></tr><tr><td>Recursive2</td><td>-0.015625</td><td>17</td><td>13.3333</td></tr><tr><td rowspan="8">3</td><td>Bernstein</td><td>-0.0197754</td><td>12.4444</td><td>8</td></tr><tr><td>MinVariance</td><td>-0.0776098</td><td>10.9429</td><td><b>5.99574</b></td></tr><tr><td>BSplines</td><td>-1.69542e-06</td><td>609.778</td><td>552</td></tr><tr><td>MinNorm</td><td>-0.0703759</td><td><b>10.9231</b></td><td>6.01885</td></tr><tr><td>MinVolume</td><td><b>-0.110146</b></td><td>12.1142</td><td>6.91991</td></tr><tr><td>ApproxMinVo</td><td>-0.110036</td><td>12.1925</td><td>6.98117</td></tr><tr><td>Recursive</td><td>-0.000244141</td><td>96</td><td>90</td></tr><tr><td>Recursive2</td><td>-0.000244141</td><td>96</td><td>90</td></tr><tr><td rowspan="8">4</td><td>Bernstein</td><td>-0.00878906</td><td>17.1111</td><td>11.3556</td></tr><tr><td>MinVariance</td><td>-0.138639</td><td>14.5243</td><td><b>7.75466</b></td></tr><tr><td>BSplines</td><td>-1.14978e-11</td><td>26930.6</td><td>19580.8</td></tr><tr><td>MinNorm</td><td>-0.101192</td><td><b>14.449</b></td><td>7.84653</td></tr><tr><td>MinVolume</td><td><b>-0.322429</b></td><td>16.826</td><td>9.49568</td></tr><tr><td>ApproxMinVo</td><td>-0.321755</td><td>16.8275</td><td>9.52516</td></tr><tr><td>Recursive</td><td>-9.53674e-07</td><td>665</td><td>656</td></tr><tr><td>Recursive2</td><td>-9.53674e-07</td><td>665</td><td>656</td></tr><tr><td rowspan="8">5</td><td>Bernstein</td><td>-0.00582077</td><td>21.92</td><td>15.0133</td></tr><tr><td>MinVariance</td><td>-0.477859</td><td>18.1917</td><td><b>10.1726</b></td></tr><tr><td>BSplines</td><td>-7.79745e-19</td><td>2.06027e+06</td><td>1.98912e+06</td></tr><tr><td>MinNorm</td><td>-0.255511</td><td><b>18.0599</b></td><td>10.3167</td></tr><tr><td>MinVolume</td><td><b>-2.88564</b></td><td>22.0832</td><td>13.3504</td></tr><tr><td>ApproxMinVo</td><td>-2.86456</td><td>22.1827</td><td>13.446</td></tr><tr><td>Recursive</td><td>-9.31323e-10</td><td>5076</td><td>5063.33</td></tr><tr><td>Recursive2</td><td>-9.31323e-10</td><td>5076</td><td>5063.33</td></tr><tr><td rowspan="8">6</td><td>Bernstein</td><td>-0.00596719</td><td>26.8444</td><td>18.6438</td></tr><tr><td>MinVariance</td><td>-3.65267</td><td>22.0119</td><td><b>12.2725</b></td></tr><tr><td>BSplines</td><td>-3.67221e-28</td><td>2.40738e+08</td><td>1.87088e+08</td></tr><tr><td>MinNorm</td><td>-1.19714</td><td><b>21.7376</b></td><td>12.5152</td></tr><tr><td>MinVolume</td><td><b>-82.8592</b></td><td>27.6542</td><td>16.6707</td></tr><tr><td>ApproxMinVo</td><td>-82.3749</td><td>27.5176</td><td>16.5483</td></tr><tr><td>Recursive</td><td>-2.27374e-13</td><td>40537</td><td>40520</td></tr><tr><td>Recursive2</td><td>-2.27374e-13</td><td>40537</td><td>40520</td></tr></tbody></table>
<table><caption>Table 11: Problem list and their specificities.</caption><thead><tr><th>problem</th><th>center of the target</th><th>shape of the target</th><th>criteria</th></tr></thead><tbody><tr><td>1</td><td>{0.2; 1.7}</td><td>square</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>2</td><td>{0.4; 1.4}</td><td>square</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>3</td><td>{0.6; 1.1}</td><td>square</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>4</td><td>{0.2; 1.7}</td><td>circle</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>5</td><td>{0.4; 1.4}</td><td>circle</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>6</td><td>{0.6; 1.1}</td><td>circle</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>7</td><td>{0.2; 1.7}</td><td>ring</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>8</td><td>{0.4; 1.4}</td><td>ring</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>9</td><td>{0.6; 1.1}</td><td>ring</td><td>∑<sub>i</sub> q<sup>2</sup><sub>i</sub></td></tr><tr><td>11</td><td>{0.2; 1.7}</td><td>square</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>12</td><td>{0.4; 1.4}</td><td>square</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>13</td><td>{0.6; 1.1}</td><td>square</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>14</td><td>{0.2; 1.7}</td><td>circle</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>15</td><td>{0.4; 1.4}</td><td>circle</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>16</td><td>{0.6; 1.1}</td><td>circle</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>17</td><td>{0.2; 1.7}</td><td>ring</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>18</td><td>{0.4; 1.4}</td><td>ring</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr><tr><td>19</td><td>{0.6; 1.1}</td><td>ring</td><td>∑<sub>i</sub> Γ<sup>2</sup><sub>i</sub></td></tr></tbody></table>
<table><caption>Table 12: Bissection of problem 2D with problem: 1, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time.per.iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>28.85e-6</td><td>22.30e-3</td><td>22.33e-3</td><td>0.022</td><td>3367</td><td>6.623e-6</td><td>955.68e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>706.9e-3</td><td>5.474e-3</td><td>712.4e-3</td><td>0.712</td><td>991</td><td>5.524e-6</td><td>951.92e-3</td><td>24.55</td><td>3190.4</td><td>29.43</td></tr><tr><td>MinVariance</td><td>726.6e-3</td><td>5.426e-3</td><td>732.1e-3</td><td>0.732</td><td>1029</td><td>5.273e-6</td><td>952.00e-3</td><td>24.33</td><td>3278.6</td><td>30.56</td></tr><tr><td>BSplines</td><td>729.4e-3</td><td>6.341e-3</td><td>735.8e-3</td><td>0.736</td><td>1535</td><td>4.131e-6</td><td>952.86e-3</td><td>28.43</td><td>3205.1</td><td>45.59</td></tr><tr><td>MinNorm</td><td>737.9e-3</td><td>5.136e-3</td><td>743.0e-3</td><td>0.743</td><td>1017</td><td>5.050e-6</td><td>951.96e-3</td><td>23.03</td><td>3327.7</td><td>30.20</td></tr><tr><td>MinVolume</td><td>723.2e-3</td><td>5.240e-3</td><td>728.4e-3</td><td>0.728</td><td>1047</td><td>5.005e-6</td><td>952.07e-3</td><td>23.50</td><td>3262.3</td><td>31.10</td></tr><tr><td>ApproxMinVolume</td><td>728.5e-3</td><td>5.654e-3</td><td>734.1e-3</td><td>0.734</td><td>1049</td><td>5.390e-6</td><td>952.07e-3</td><td>25.35</td><td>3287.9</td><td>31.16</td></tr><tr><td>Recursive</td><td>728.3e-3</td><td>5.229e-3</td><td>733.6e-3</td><td>0.734</td><td>1005</td><td>5.203e-6</td><td>951.92e-3</td><td>23.45</td><td>3285.3</td><td>29.85</td></tr><tr><td>Recursive2</td><td>720.3e-3</td><td>4.852e-3</td><td>725.2e-3</td><td>0.725</td><td>1007</td><td>4.818e-6</td><td>951.92e-3</td><td>21.76</td><td>3247.9</td><td>29.91</td></tr><tr><td>Interval</td><td>28.85e-6</td><td>4.499</td><td>4.499</td><td>4.499</td><td>703327</td><td>6.397e-6</td><td>656.18e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.218</td><td>568.2e-3</td><td>1.786</td><td>1.786</td><td>116043</td><td>4.897e-6</td><td>653.31e-3</td><td>12.63</td><td>39.71</td><td>16.50</td></tr><tr><td>MinVariance</td><td>1.307</td><td>604.3e-3</td><td>1.912</td><td>1.912</td><td>124251</td><td>4.863e-6</td><td>653.39e-3</td><td>13.43</td><td>42.49</td><td>17.67</td></tr><tr><td>BSplines</td><td>1.224</td><td>1.033</td><td>2.257</td><td>2.257</td><td>238083</td><td>4.337e-6</td><td>654.29e-3</td><td>22.95</td><td>50.17</td><td>33.85</td></tr><tr><td>MinNorm</td><td>1.370</td><td>609.4e-3</td><td>1.980</td><td>1.980</td><td>120347</td><td>5.064e-6</td><td>653.35e-3</td><td>13.55</td><td>44.00</td><td>17.11</td></tr><tr><td>MinVolume</td><td>1.272</td><td>598.1e-3</td><td>1.870</td><td>1.870</td><td>132195</td><td>4.524e-6</td><td>653.46e-3</td><td>13.29</td><td>41.56</td><td>18.80</td></tr><tr><td>ApproxMinVolume</td><td>1.286</td><td>644.7e-3</td><td>1.930</td><td>1.930</td><td>132393</td><td>4.869e-6</td><td>653.46e-3</td><td>14.33</td><td>42.91</td><td>18.82</td></tr><tr><td>Recursive</td><td>1.217</td><td>559.5e-3</td><td>1.776</td><td>1.776</td><td>116609</td><td>4.798e-6</td><td>653.31e-3</td><td>12.44</td><td>39.49</td><td>16.58</td></tr><tr><td>Recursive2</td><td>1.258</td><td>579.5e-3</td><td>1.838</td><td>1.838</td><td>116689</td><td>4.966e-6</td><td>653.31e-3</td><td>12.88</td><td>40.85</td><td>16.59</td></tr><tr><td>Interval</td><td>24.08e-6</td><td>845.2</td><td>845.2</td><td>14.05</td><td>98257105</td><td>8.602e-6</td><td>546.04e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.932</td><td>114.7</td><td>118.6</td><td>01.58</td><td>9779187</td><td>11.73e-6</td><td>543.02e-3</td><td>13.57</td><td>14.04</td><td>9.953</td></tr><tr><td>MinVariance</td><td>4.808</td><td>130.2</td><td>135.0</td><td>02.15</td><td>10554147</td><td>12.34e-6</td><td>543.10e-3</td><td>15.41</td><td>15.98</td><td>10.74</td></tr><tr><td>BSplines</td><td>3.878</td><td>235.0</td><td>238.9</td><td>03.58</td><td>22194981</td><td>10.59e-6</td><td>544.03e-3</td><td>27.80</td><td>28.26</td><td>22.59</td></tr><tr><td>MinNorm</td><td>4.704</td><td>132.8</td><td>137.5</td><td>02.17</td><td>10195429</td><td>13.03e-6</td><td>543.06e-3</td><td>15.71</td><td>16.27</td><td>10.38</td></tr><tr><td>MinVolume</td><td>4.165</td><td>133.8</td><td>138.0</td><td>02.17</td><td>11323765</td><td>11.82e-6</td><td>543.17e-3</td><td>15.83</td><td>16.33</td><td>11.52</td></tr><tr><td>ApproxMinVolume</td><td>4.317</td><td>136.3</td><td>140.6</td><td>02.20</td><td>11343677</td><td>12.02e-6</td><td>543.18e-3</td><td>16.13</td><td>16.64</td><td>11.54</td></tr><tr><td>Recursive</td><td>3.911</td><td>130.0</td><td>134.0</td><td>02.13</td><td>9800513</td><td>13.27e-6</td><td>543.02e-3</td><td>15.39</td><td>15.85</td><td>9.974</td></tr><tr><td>Recursive2</td><td>3.887</td><td>114.6</td><td>118.5</td><td>01.58</td><td>9803491</td><td>11.69e-6</td><td>543.02e-3</td><td>13.56</td><td>14.02</td><td>9.977</td></tr><tr><td>Interval</td><td>31.00e-6</td><td>173650</td><td>173650</td><td>2-00:14:10</td><td>18810829203</td><td>9.231e-6</td><td>463.96e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>57.31</td><td>51490</td><td>51547</td><td>14:19:07</td><td>1682985771</td><td>30.59e-6</td><td>461.15e-3</td><td>29.65</td><td>29.68</td><td>8.947</td></tr><tr><td>MinVariance</td><td>60.87</td><td>57862</td><td>57923</td><td>16:05:23</td><td>1817885635</td><td>31.83e-6</td><td>461.22e-3</td><td>33.32</td><td>33.36</td><td>9.664</td></tr><tr><td>BSplines</td><td>57.88</td><td>110975</td><td>110975</td><td>10:64:39:53</td><td>4035420446</td><td>27.50e-6</td><td>462.14e-3</td><td>63.91</td><td>63.91</td><td>21.45</td></tr><tr><td>MinNorm</td><td>59.77</td><td>55678</td><td>55737</td><td>15:28:57</td><td>1755309023</td><td>31.72e-6</td><td>461.19e-3</td><td>32.06</td><td>32.10</td><td>9.331</td></tr><tr><td>MinVolume</td><td>58.58</td><td>60358</td><td>60417</td><td>16:46:57</td><td>1953182331</td><td>30.90e-6</td><td>461.30e-3</td><td>34.76</td><td>34.79</td><td>10.38</td></tr><tr><td>ApproxMinVolume</td><td>59.84</td><td>60693</td><td>60753</td><td>16:52:33</td><td>1956747259</td><td>31.02e-6</td><td>461.30e-3</td><td>34.95</td><td>34.99</td><td>10.40</td></tr><tr><td>Recursive</td><td>58.09</td><td>51883</td><td>51941</td><td>14:25:41</td><td>1685327733</td><td>30.79e-6</td><td>461.15e-3</td><td>29.88</td><td>29.91</td><td>8.959</td></tr><tr><td>Recursive2</td><td>57.64</td><td>52007</td><td>52065</td><td>14:27:45</td><td>1686154457</td><td>30.84e-6</td><td>461.15e-3</td><td>29.95</td><td>29.98</td><td>8.964</td></tr><tr><td>Interval</td><td>30.04e-6</td><td>9.402e-3</td><td>9.432e-3</td><td>0.009</td><td>1001</td><td>9.392e-6</td><td>1.1075</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>998.4e-3</td><td>7.095e-3</td><td>1.005</td><td>1.005</td><td>549</td><td>12.92e-6</td><td>1.0801</td><td>75.46</td><td>10660.5</td><td>54.85</td></tr><tr><td>MinVariance</td><td>998.2e-3</td><td>9.334e-3</td><td>1.008</td><td>1.008</td><td>557</td><td>16.76e-6</td><td>1.0808</td><td>99.28</td><td>10682.1</td><td>55.64</td></tr><tr><td>BSplines</td><td>998.3e-3</td><td>10.52e-3</td><td>1.009</td><td>1.009</td><td>709</td><td>14.84e-6</td><td>1.0884</td><td>111.9</td><td>10695.6</td><td>70.83</td></tr><tr><td>MinNorm</td><td>997.4e-3</td><td>4.675e-3</td><td>1.002</td><td>1.002</td><td>551</td><td>8.484e-6</td><td>1.0804</td><td>49.72</td><td>10624.2</td><td>55.04</td></tr><tr><td>MinVolume</td><td>997.8e-3</td><td>5.835e-3</td><td>1.004</td><td>1.004</td><td>569</td><td>10.25e-6</td><td>1.0814</td><td>62.06</td><td>10641.5</td><td>56.84</td></tr><tr><td>ApproxMinVolume</td><td>998.1e-3</td><td>8.696e-3</td><td>1.007</td><td>1.007</td><td>569</td><td>15.28e-6</td><td>1.0814</td><td>92.49</td><td>10674.4</td><td>56.84</td></tr><tr><td>Recursive</td><td>981.0e-3</td><td>5.225e-3</td><td>986.3e-3</td><td>0.986</td><td>553</td><td>9.448e-6</td><td>1.0801</td><td>55.57</td><td>10456.8</td><td>55.24</td></tr><tr><td>Recursive2</td><td>997.7e-3</td><td>4.313e-3</td><td>1.002</td><td>1.002</td><td>553</td><td>7.799e-6</td><td>1.0801</td><td>45.87</td><td>10623.7</td><td>55.24</td></tr><tr><td>Interval</td><td>30.04e-6</td><td>730.5e-3</td><td>730.5e-3</td><td>0.731</td><td>103227</td><td>7.077e-6</td><td>708.82e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.203</td><td>130.8e-3</td><td>1.334</td><td>1.334</td><td>24313</td><td>5.381e-6</td><td>684.89e-3</td><td>17.91</td><td>182.6</td><td>23.55</td></tr><tr><td>MinVariance</td><td>1.352</td><td>136.6e-3</td><td>1.488</td><td>1.488</td><td>25717</td><td>5.310e-6</td><td>685.53e-3</td><td>18.70</td><td>203.7</td><td>24.91</td></tr><tr><td>BSplines</td><td>1.205</td><td>204.7e-3</td><td>1.410</td><td>1.410</td><td>43997</td><td>4.652e-6</td><td>692.89e-3</td><td>28.02</td><td>193.0</td><td>42.62</td></tr><tr><td>MinNorm</td><td>1.330</td><td>142.2e-3</td><td>1.472</td><td>1.472</td><td>25099</td><td>5.666e-6</td><td>685.24e-3</td><td>19.47</td><td>201.5</td><td>24.31</td></tr><tr><td>MinVolume</td><td>1.264</td><td>151.3e-3</td><td>1.415</td><td>1.415</td><td>27047</td><td>5.595e-6</td><td>686.12e-3</td><td>20.72</td><td>193.7</td><td>26.20</td></tr><tr><td>ApproxMinVolume</td><td>1.258</td><td>152.3e-3</td><td>1.411</td><td>1.411</td><td>27093</td><td>5.623e-6</td><td>686.14e-3</td><td>20.85</td><td>193.1</td><td>26.25</td></tr><tr><td>Recursive</td><td>1.278</td><td>137.9e-3</td><td>1.416</td><td>1.416</td><td>24743</td><td>5.572e-6</td><td>684.89e-3</td><td>18.87</td><td>193.8</td><td>23.97</td></tr><tr><td>Recursive2</td><td>1.264</td><td>142.6e-3</td><td>1.406</td><td>1.406</td><td>24791</td><td>5.751e-6</td><td>684.89e-3</td><td>19.52</td><td>192.5</td><td>24.02</td></tr><tr><td>Interval</td><td>22.17e-6</td><td>58.27</td><td>58.27</td><td>58.269</td><td>7499637</td><td>7.770e-6</td><td>600.02e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.931</td><td>12.21</td><td>16.14</td><td>16.144</td><td>970109</td><td>12.59e-6</td><td>574.62e-3</td><td>20.96</td><td>27.71</td><td>12.94</td></tr><tr><td>MinVariance</td><td>4.914</td><td>14.06</td><td>18.98</td><td>18.976</td><td>1041705</td><td>13.50e-6</td><td>575.28e-3</td><td>24.13</td><td>32.57</td><td>13.89</td></tr><tr><td>BSplines</td><td>3.873</td><td>23.72</td><td>27.59</td><td>27.589</td><td>2080791</td><td>11.35e-6</td><td>582.96e-3</td><td>40.70</td><td>47.35</td><td>27.87</td></tr><tr><td>MinNorm</td><td>4.756</td><td>13.38</td><td>18.13</td><td>18.135</td><td>1008885</td><td>13.26e-6</td><td>574.98e-3</td><td>22.96</td><td>31.12</td><td>13.45</td></tr><tr><td>MinVolume</td><td>4.194</td><td>13.95</td><td>18.15</td><td>18.148</td><td>1113227</td><td>12.53e-6</td><td>575.90e-3</td><td>23.95</td><td>31.14</td><td>14.84</td></tr><tr><td>ApproxMinVolume</td><td>4.438</td><td>14.08</td><td>18.51</td><td>18.514</td><td>1115167</td><td>12.62e-6</td><td>575.91e-3</td><td>24.16</td><td>31.77</td><td>14.87</td></tr><tr><td>Recursive</td><td>4.064</td><td>12.07</td><td>16.14</td><td>16.136</td><td>985009</td><td>12.26e-6</td><td>574.62e-3</td><td>20.72</td><td>27.69</td><td>13.13</td></tr><tr><td>Recursive2</td><td>3.963</td><td>12.33</td><td>16.29</td><td>16.290</td><td>985651</td><td>12.51e-6</td><td>574.62e-3</td><td>21.16</td><td>27.96</td><td>13.14</td></tr><tr><td>Interval</td><td>26.94e-6</td><td>5051</td><td>5051</td><td>01:24:11</td><td>580056931</td><td>8.709e-6</td><td>518.93e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>57.09</td><td>2720</td><td>2777</td><td>46:17</td><td>86138425</td><td>31.58e-6</td><td>494.87e-3</td><td>53.85</td><td>54.98</td><td>14.85</td></tr><tr><td>MinVariance</td><td>60.34</td><td>3072</td><td>3132</td><td>52:12</td><td>91934425</td><td>33.42e-6</td><td>495.51e-3</td><td>60.82</td><td>62.02</td><td>15.85</td></tr><tr><td>BSplines</td><td>57.63</td><td>5523</td><td>5580</td><td>01:33:00</td><td>180189305</td><td>30.65e-6</td><td>503.11e-3</td><td>109.3</td><td>110.5</td><td>31.06</td></tr><tr><td>MinNorm</td><td>60.35</td><td>2963</td><td>3023</td><td>50:23</td><td>89237027</td><td>33.20e-6</td><td>495.22e-3</td><td>58.65</td><td>59.85</td><td>15.38</td></tr><tr><td>MinVolume</td><td>58.38</td><td>3314</td><td>3372</td><td>56:12</td><td>97738955</td><td>33.91e-6</td><td>496.13e-3</td><td>65.61</td><td>66.77</td><td>16.85</td></tr><tr><td>ApproxMinVolume</td><td>59.59</td><td>3178</td><td>3238</td><td>53:58</td><td>97890755</td><td>32.47e-6</td><td>496.14e-3</td><td>62.92</td><td>64.10</td><td>16.88</td></tr><tr><td>Recursive</td><td>57.54</td><td>2823</td><td>2881</td><td>48:01</td><td>86945127</td><td>32.48e-6</td><td>494.87e-3</td><td>55.90</td><td>57.04</td><td>14.99</td></tr><tr><td>Recursive2</td><td>57.78</td><td>2830</td><td>2888</td><td>48:08</td><td>87044285</td><td>32.51e-6</td><td>494.87e-3</td><td>56.02</td><td>57.17</td><td>15.01</td></tr></tbody></table>
<table><caption>Table 13: Bisection of problem 2D with problem: 2, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time-per.iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>45.06e-6</td><td>7.891e-3</td><td>7.936e-3</td><td>0.008</td><td>857</td><td>9.208e-6</td><td>2.7090</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>723.3e-3</td><td>4.526e-3</td><td>727.8e-3</td><td>0.728</td><td>451</td><td>10.04e-6</td><td>2.7057</td><td>57.36</td><td>9170.7</td><td>52.63</td></tr><tr><td>MinVariance</td><td>747.6e-3</td><td>4.444e-3</td><td>752.0e-3</td><td>0.752</td><td>471</td><td>9.435e-6</td><td>2.7058</td><td>56.32</td><td>9476.2</td><td>54.96</td></tr><tr><td>BSplines</td><td>719.7e-3</td><td>4.730e-3</td><td>724.5e-3</td><td>0.724</td><td>703</td><td>6.728e-6</td><td>2.7073</td><td>59.94</td><td>9129.1</td><td>82.03</td></tr><tr><td>MinNorm</td><td>755.4e-3</td><td>4.881e-3</td><td>760.3e-3</td><td>0.760</td><td>459</td><td>10.63e-6</td><td>2.7057</td><td>61.85</td><td>9580.6</td><td>53.56</td></tr><tr><td>MinVolume</td><td>753.2e-3</td><td>4.988e-3</td><td>758.2e-3</td><td>0.758</td><td>489</td><td>10.20e-6</td><td>2.7059</td><td>63.21</td><td>9553.6</td><td>57.06</td></tr><tr><td>ApproxMinVolume</td><td>741.2e-3</td><td>4.047e-3</td><td>745.3e-3</td><td>0.745</td><td>489</td><td>8.276e-6</td><td>2.7059</td><td>51.29</td><td>9391.1</td><td>57.06</td></tr><tr><td>Recursive</td><td>735.8e-3</td><td>4.295e-3</td><td>740.1e-3</td><td>0.740</td><td>451</td><td>9.523e-6</td><td>2.7057</td><td>54.43</td><td>9326.3</td><td>52.63</td></tr><tr><td>Recursive2</td><td>735.4e-3</td><td>4.268e-3</td><td>739.7e-3</td><td>0.740</td><td>451</td><td>9.464e-6</td><td>2.7057</td><td>54.09</td><td>9320.5</td><td>52.63</td></tr><tr><td>Interval</td><td>29.09e-6</td><td>2.089</td><td>2.089</td><td>2.089</td><td>302195</td><td>6.912e-6</td><td>2.4369</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.235</td><td>244.8e-3</td><td>1.480</td><td>1.480</td><td>44913</td><td>5.450e-6</td><td>2.4310</td><td>11.72</td><td>70.84</td><td>14.86</td></tr><tr><td>MinVariance</td><td>1.343</td><td>251.6e-3</td><td>1.595</td><td>1.595</td><td>48259</td><td>5.213e-6</td><td>2.4312</td><td>12.04</td><td>76.36</td><td>15.97</td></tr><tr><td>BSplines</td><td>1.235</td><td>468.0e-3</td><td>1.703</td><td>1.703</td><td>99953</td><td>4.682e-6</td><td>2.4329</td><td>22.40</td><td>81.51</td><td>33.08</td></tr><tr><td>MinNorm</td><td>1.371</td><td>257.9e-3</td><td>1.629</td><td>1.629</td><td>46669</td><td>5.527e-6</td><td>2.4311</td><td>12.35</td><td>77.98</td><td>15.44</td></tr><tr><td>MinVolume</td><td>1.324</td><td>277.6e-3</td><td>1.601</td><td>1.601</td><td>51647</td><td>5.376e-6</td><td>2.4313</td><td>13.29</td><td>76.66</td><td>17.09</td></tr><tr><td>ApproxMinVolume</td><td>1.267</td><td>265.7e-3</td><td>1.533</td><td>1.533</td><td>51749</td><td>5.134e-6</td><td>2.4313</td><td>12.72</td><td>73.40</td><td>17.12</td></tr><tr><td>Recursive</td><td>1.237</td><td>244.3e-3</td><td>1.481</td><td>1.481</td><td>45017</td><td>5.428e-6</td><td>2.4310</td><td>11.70</td><td>70.90</td><td>14.90</td></tr><tr><td>Recursive2</td><td>1.233</td><td>248.8e-3</td><td>1.482</td><td>1.482</td><td>45173</td><td>5.507e-6</td><td>2.4310</td><td>11.91</td><td>70.93</td><td>14.95</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>29.09e-6</td><td>672.1</td><td>672.1</td><td>11:12</td><td>83653481</td><td>8.034e-6</td><td>1.2775</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.891</td><td>110.9</td><td>114.8</td><td>01:54</td><td>9140617</td><td>12.13e-6</td><td>1.2731</td><td>16.50</td><td>17.08</td><td>10.93</td></tr><tr><td>MinVariance</td><td>4.791</td><td>126.4</td><td>131.2</td><td>02:11</td><td>9887775</td><td>12.78e-6</td><td>1.2732</td><td>18.81</td><td>19.52</td><td>11.82</td></tr><tr><td>BSplines</td><td>3.902</td><td>234.6</td><td>238.5</td><td>03:58</td><td>21844859</td><td>10.74e-6</td><td>1.2746</td><td>34.90</td><td>35.48</td><td>26.11</td></tr><tr><td>MinNorm</td><td>4.663</td><td>123.4</td><td>128.0</td><td>02:08</td><td>9542241</td><td>12.93e-6</td><td>1.2731</td><td>18.35</td><td>19.05</td><td>11.41</td></tr><tr><td>MinVolume</td><td>4.186</td><td>129.2</td><td>133.3</td><td>02:13</td><td>10629611</td><td>12.15e-6</td><td>1.2733</td><td>19.22</td><td>19.84</td><td>12.71</td></tr><tr><td>ApproxMinVolume</td><td>4.334</td><td>131.2</td><td>135.5</td><td>02:15</td><td>10649137</td><td>12.32e-6</td><td>1.2733</td><td>19.52</td><td>20.16</td><td>12.73</td></tr><tr><td>Recursive</td><td>3.961</td><td>110.1</td><td>114.0</td><td>01:54</td><td>9159043</td><td>12.02e-6</td><td>1.2731</td><td>16.38</td><td>16.97</td><td>10.95</td></tr><tr><td>Recursive2</td><td>3.933</td><td>112.9</td><td>116.8</td><td>01:56</td><td>9155733</td><td>12.33e-6</td><td>1.2731</td><td>16.79</td><td>17.38</td><td>10.94</td></tr><tr><td>Interval</td><td>27.00e-6</td><td>544923</td><td>544923</td><td>6:07:22:03</td><td>54998575207</td><td>9.908e-6</td><td>1.0796</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>57.47</td><td>126707</td><td>126707</td><td>11:11:11:47</td><td>4379742224</td><td>28.93e-6</td><td>1.0754</td><td>23.25</td><td>23.25</td><td>7.963</td></tr><tr><td>MinVariance</td><td>60.76</td><td>143554</td><td>143554</td><td>11:15:52:34</td><td>4778229866</td><td>30.04e-6</td><td>1.0755</td><td>26.34</td><td>26.34</td><td>6.868</td></tr><tr><td>BSplines</td><td>57.78</td><td>307706</td><td>307707</td><td>3:13:28:27</td><td>11931768794</td><td>25.79e-6</td><td>1.0770</td><td>56.47</td><td>56.47</td><td>21.69</td></tr><tr><td>MinNorm</td><td>60.24</td><td>137930</td><td>137930</td><td>11:14:18:50</td><td>4592853204</td><td>30.03e-6</td><td>1.0755</td><td>25.31</td><td>25.31</td><td>8.351</td></tr><tr><td>MinVolume</td><td>58.21</td><td>152631</td><td>152631</td><td>11:18:23:51</td><td>5182587620</td><td>29.45e-6</td><td>1.0757</td><td>28.01</td><td>28.01</td><td>9.423</td></tr><tr><td>ApproxMinVolume</td><td>60.04</td><td>152228</td><td>152229</td><td>11:18:17:09</td><td>5193302588</td><td>29.31e-6</td><td>1.0757</td><td>27.94</td><td>27.94</td><td>9.443</td></tr><tr><td>Recursive</td><td>57.99</td><td>129026</td><td>129027</td><td>11:11:50:27</td><td>4383611222</td><td>29.43e-6</td><td>1.0754</td><td>23.68</td><td>23.68</td><td>7.970</td></tr><tr><td>Recursive2</td><td>57.81</td><td>129202</td><td>129292</td><td>11:11:54:52</td><td>4386111558</td><td>29.48e-6</td><td>1.0754</td><td>23.73</td><td>23.73</td><td>7.975</td></tr><tr><td>Interval</td><td>36.95e-6</td><td>19.04e-3</td><td>19.08e-3</td><td>0.019</td><td>603</td><td>31.58e-6</td><td>2.7694</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.008</td><td>4.885e-3</td><td>1.013</td><td>1.013</td><td>385</td><td>12.69e-6</td><td>2.7381</td><td>25.65</td><td>5306.5</td><td>63.85</td></tr><tr><td>MinVariance</td><td>1.008</td><td>18.90e-3</td><td>1.027</td><td>1.027</td><td>389</td><td>48.60e-6</td><td>2.7392</td><td>99.26</td><td>5381.2</td><td>64.51</td></tr><tr><td>BSplines</td><td>1.007</td><td>4.766e-3</td><td>1.012</td><td>1.012</td><td>509</td><td>9.363e-6</td><td>2.7517</td><td>25.03</td><td>5304.6</td><td>84.41</td></tr><tr><td>MinNorm</td><td>1.007</td><td>16.94e-3</td><td>1.024</td><td>1.024</td><td>389</td><td>43.56e-6</td><td>2.7387</td><td>88.96</td><td>5365.2</td><td>64.51</td></tr><tr><td>MinVolume</td><td>1.007</td><td>17.52e-3</td><td>1.025</td><td>1.025</td><td>397</td><td>44.14e-6</td><td>2.7402</td><td>92.00</td><td>5370.7</td><td>65.84</td></tr><tr><td>ApproxMinVolume</td><td>1.007</td><td>15.83e-3</td><td>1.023</td><td>1.023</td><td>397</td><td>39.88e-6</td><td>2.7402</td><td>83.14</td><td>5361.0</td><td>65.84</td></tr><tr><td>Recursive</td><td>1.008</td><td>18.40e-3</td><td>1.026</td><td>1.026</td><td>385</td><td>47.79e-6</td><td>2.7381</td><td>96.61</td><td>5377.3</td><td>63.85</td></tr><tr><td>Recursive2</td><td>667.8e-3</td><td>4.380e-3</td><td>672.2e-3</td><td>0.672</td><td>385</td><td>11.38e-6</td><td>2.7381</td><td>23.00</td><td>3522.9</td><td>63.85</td></tr><tr><td rowspan="10">4</td><td rowspan="10">2</td><td>Interval</td><td>30.04e-6</td><td>549.0e-3</td><td>549.1e-3</td><td>0.549</td><td>82629</td><td>6.644e-6</td><td>2.5140</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.173</td><td>119.2e-3</td><td>1.292</td><td>1.292</td><td>21145</td><td>5.637e-6</td><td>2.4634</td><td>21.71</td><td>235.3</td><td>25.59</td></tr><tr><td>MinVariance</td><td>1.306</td><td>123.8e-3</td><td>1.429</td><td>1.429</td><td>22287</td><td>5.554e-6</td><td>2.4646</td><td>22.54</td><td>260.3</td><td>26.97</td></tr><tr><td>BSplines</td><td>1.201</td><td>188.0e-3</td><td>1.389</td><td>1.389</td><td>37621</td><td>4.998e-6</td><td>2.4783</td><td>34.25</td><td>252.9</td><td>45.53</td></tr><tr><td>MinNorm</td><td>1.357</td><td>128.2e-3</td><td>1.485</td><td>1.485</td><td>21751</td><td>5.893e-6</td><td>2.4641</td><td>23.35</td><td>270.5</td><td>26.32</td></tr><tr><td>MinVolume</td><td>1.316</td><td>136.4e-3</td><td>1.453</td><td>1.453</td><td>23517</td><td>5.801e-6</td><td>2.4657</td><td>24.85</td><td>264.5</td><td>28.46</td></tr><tr><td>ApproxMinVolume</td><td>1.311</td><td>143.5e-3</td><td>1.454</td><td>1.454</td><td>23551</td><td>6.093e-6</td><td>2.4658</td><td>26.14</td><td>264.9</td><td>28.50</td></tr><tr><td>Recursive</td><td>1.252</td><td>121.1e-3</td><td>1.374</td><td>1.374</td><td>21245</td><td>5.702e-6</td><td>2.4634</td><td>22.06</td><td>250.2</td><td>25.71</td></tr><tr><td>Recursive2</td><td>1.236</td><td>125.2e-3</td><td>1.361</td><td>1.361</td><td>21381</td><td>5.857e-6</td><td>2.4634</td><td>22.81</td><td>247.9</td><td>25.88</td></tr><tr><td>Interval</td><td>30.04e-6</td><td>54.89</td><td>54.89</td><td>54.890</td><td>6777589</td><td>8.099e-6</td><td>1.3554</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.992</td><td>14.87</td><td>18.86</td><td>18.857</td><td>1160499</td><td>12.81e-6</td><td>1.3191</td><td>27.08</td><td>34.35</td><td>17.12</td></tr><tr><td>MinVariance</td><td>4.906</td><td>16.79</td><td>21.70</td><td>21.697</td><td>1243777</td><td>13.50e-6</td><td>1.3201</td><td>30.59</td><td>39.53</td><td>18.35</td></tr><tr><td>BSplines</td><td>4.003</td><td>29.11</td><td>33.11</td><td>33.114</td><td>2450441</td><td>11.88e-6</td><td>1.3315</td><td>53.04</td><td>60.33</td><td>36.16</td></tr><tr><td>MinNorm</td><td>4.822</td><td>16.20</td><td>21.03</td><td>21.026</td><td>1205215</td><td>13.44e-6</td><td>1.3196</td><td>29.52</td><td>38.31</td><td>17.78</td></tr><tr><td>MinVolume</td><td>4.252</td><td>16.91</td><td>21.17</td><td>21.166</td><td>1324903</td><td>12.77e-6</td><td>1.3210</td><td>30.81</td><td>38.56</td><td>19.55</td></tr><tr><td>ApproxMinVolume</td><td>4.483</td><td>17.09</td><td>21.57</td><td>21.573</td><td>1326975</td><td>12.88e-6</td><td>1.3211</td><td>31.14</td><td>39.30</td><td>19.58</td></tr><tr><td>Recursive</td><td>4.067</td><td>14.72</td><td>18.78</td><td>18.783</td><td>1173425</td><td>12.54e-6</td><td>1.3191</td><td>26.81</td><td>34.22</td><td>17.31</td></tr><tr><td>Recursive2</td><td>3.987</td><td>14.75</td><td>18.74</td><td>18.739</td><td>1169849</td><td>12.61e-6</td><td>1.3191</td><td>26.88</td><td>34.14</td><td>17.26</td></tr><tr><td rowspan="10">5</td><td rowspan="10">2</td><td>Interval</td><td>24.80e-6</td><td>9826</td><td>9826</td><td>02:43:46</td><td>1087998677</td><td>9.031e-6</td><td>1.1612</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>57.09</td><td>6345</td><td>6402</td><td>01:46:42</td><td>194269469</td><td>32.66e-6</td><td>1.1267</td><td>64.58</td><td>65.16</td><td>17.86</td></tr><tr><td>MinVariance</td><td>60.35</td><td>6700</td><td>6760</td><td>01:52:40</td><td>207619475</td><td>32.27e-6</td><td>1.1277</td><td>68.19</td><td>68.80</td><td>19.08</td></tr><tr><td>BSplines</td><td>57.29</td><td>11515</td><td>11572</td><td>03:12:52</td><td>405146443</td><td>28.42e-6</td><td>1.1391</td><td>117.2</td><td>117.8</td><td>37.24</td></tr><tr><td>MinNorm</td><td>60.57</td><td>6694</td><td>6754</td><td>01:52:34</td><td>201431467</td><td>33.23e-6</td><td>1.1272</td><td>68.13</td><td>68.74</td><td>18.51</td></tr><tr><td>MinVolume</td><td>58.83</td><td>6861</td><td>6920</td><td>01:55:20</td><td>220846803</td><td>31.07e-6</td><td>1.1286</td><td>69.83</td><td>70.43</td><td>20.30</td></tr><tr><td>ApproxMinVolume</td><td>59.63</td><td>6863</td><td>6922</td><td>01:55:22</td><td>221191743</td><td>31.03e-6</td><td>1.1286</td><td>69.85</td><td>70.45</td><td>20.33</td></tr><tr><td>Recursive</td><td>58.51</td><td>6388</td><td>6446</td><td>01:47:26</td><td>195322565</td><td>32.71e-6</td><td>1.1267</td><td>65.01</td><td>65.61</td><td>17.95</td></tr><tr><td>Recursive2</td><td>58.03</td><td>6076</td><td>6134</td><td>01:42:14</td><td>195595667</td><td>31.07e-6</td><td>1.1267</td><td>61.84</td><td>62.43</td><td>17.98</td></tr></tbody></table>
<table><caption>Table 14: Bissection of problem 2D with problem: 3, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep_time</th><th>comput_time</th><th>total_time</th><th>(D-H:M:S:ms)</th><th>nb_iter</th><th>time_per_iter</th><th>criteria</th><th>comput_time (%)</th><th>total_time (%)</th><th>nb_iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>40.05e-6</td><td>2.936e-3</td><td>2.976e-3</td><td>0.003</td><td>59</td><td>49.76e-6</td><td>infeasible</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>717.1e-3</td><td>3.223e-3</td><td>720.3e-3</td><td>0.720</td><td>41</td><td>78.61e-6</td><td>infeasible</td><td>109.8</td><td>24204.2</td><td>69.49</td></tr><tr><td>MinVariance</td><td>732.1e-3</td><td>3.017e-3</td><td>735.2e-3</td><td>0.735</td><td>41</td><td>73.58e-6</td><td>infeasible</td><td>102.8</td><td>24703.8</td><td>69.49</td></tr><tr><td>B SPLines</td><td>723.2e-3</td><td>3.740e-3</td><td>727.0e-3</td><td>0.727</td><td>41</td><td>91.22e-6</td><td>infeasible</td><td>127.4</td><td>24427.8</td><td>69.49</td></tr><tr><td>MinNorm</td><td>744.1e-3</td><td>3.214e-3</td><td>747.3e-3</td><td>0.747</td><td>41</td><td>78.39e-6</td><td>infeasible</td><td>109.5</td><td>25111.6</td><td>69.49</td></tr><tr><td>MinVolume</td><td>742.6e-3</td><td>3.122e-3</td><td>745.7e-3</td><td>0.746</td><td>41</td><td>76.15e-6</td><td>infeasible</td><td>106.3</td><td>25058.3</td><td>69.49</td></tr><tr><td>ApproxMinVolume</td><td>732.7e-3</td><td>3.145e-3</td><td>735.8e-3</td><td>0.736</td><td>41</td><td>76.71e-6</td><td>infeasible</td><td>107.1</td><td>24726.5</td><td>69.49</td></tr><tr><td>Recursive</td><td>725.3e-3</td><td>3.010e-3</td><td>728.3e-3</td><td>0.728</td><td>41</td><td>73.42e-6</td><td>infeasible</td><td>102.5</td><td>24472.8</td><td>69.49</td></tr><tr><td>Recursive2</td><td>727.3e-3</td><td>3.382e-3</td><td>730.7e-3</td><td>0.731</td><td>41</td><td>82.49e-6</td><td>infeasible</td><td>115.2</td><td>24552.2</td><td>69.49</td></tr><tr><td>Interval</td><td>25.03e-6</td><td>512.5e-3</td><td>512.5e-3</td><td>0.513</td><td>74295</td><td>6.898e-6</td><td>3.9157</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.184</td><td>97.64e-3</td><td>1.282</td><td>1.282</td><td>18541</td><td>5.266e-6</td><td>3.9108</td><td>19.05</td><td>250.1</td><td>24.96</td></tr><tr><td>MinVariance</td><td>1.384</td><td>113.3e-3</td><td>1.497</td><td>1.497</td><td>19747</td><td>5.735e-6</td><td>3.9110</td><td>22.10</td><td>292.1</td><td>26.58</td></tr><tr><td>B SPLines</td><td>1.241</td><td>211.7e-3</td><td>1.452</td><td>1.452</td><td>38723</td><td>5.468e-6</td><td>3.9131</td><td>41.31</td><td>283.4</td><td>52.12</td></tr><tr><td>MinNorm</td><td>1.375</td><td>110.8e-3</td><td>1.486</td><td>1.486</td><td>19189</td><td>5.774e-6</td><td>3.9109</td><td>21.62</td><td>289.9</td><td>25.83</td></tr><tr><td>MinVolume</td><td>1.278</td><td>117.3e-3</td><td>1.396</td><td>1.396</td><td>21011</td><td>5.584e-6</td><td>3.9111</td><td>22.89</td><td>272.3</td><td>28.28</td></tr><tr><td>ApproxMinVolume</td><td>1.217</td><td>112.4e-3</td><td>1.329</td><td>1.329</td><td>21033</td><td>5.344e-6</td><td>3.9111</td><td>21.93</td><td>259.4</td><td>28.31</td></tr><tr><td>Recursive</td><td>1.177</td><td>98.20e-3</td><td>1.275</td><td>1.275</td><td>18541</td><td>5.296e-6</td><td>3.9108</td><td>19.16</td><td>248.8</td><td>24.96</td></tr><tr><td>Recursive2</td><td>1.105</td><td>98.07e-3</td><td>1.203</td><td>1.203</td><td>18611</td><td>5.269e-6</td><td>3.9108</td><td>19.13</td><td>234.6</td><td>25.05</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>27.89e-6</td><td>783.3</td><td>783.3</td><td>13.03</td><td>96586019</td><td>8.110e-6</td><td>2.2271</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.921</td><td>119.8</td><td>123.8</td><td>02:03</td><td>9223587</td><td>12.99e-6</td><td>2.2173</td><td>15.30</td><td>15.80</td><td>9.550</td></tr><tr><td>MinVariance</td><td>4.758</td><td>134.8</td><td>139.5</td><td>02:19</td><td>9979647</td><td>13.51e-6</td><td>2.2174</td><td>17.21</td><td>17.82</td><td>10.33</td></tr><tr><td>B SPLines</td><td>3.947</td><td>254.1</td><td>258.1</td><td>04:18</td><td>21887309</td><td>11.61e-6</td><td>2.2193</td><td>32.44</td><td>32.95</td><td>22.66</td></tr><tr><td>MinNorm</td><td>4.647</td><td>130.8</td><td>135.5</td><td>02:15</td><td>9627957</td><td>13.59e-6</td><td>2.2174</td><td>16.70</td><td>17.29</td><td>9.968</td></tr><tr><td>MinVolume</td><td>4.139</td><td>137.4</td><td>141.5</td><td>02:21</td><td>10737257</td><td>12.80e-6</td><td>2.2176</td><td>17.54</td><td>18.07</td><td>11.12</td></tr><tr><td>ApproxMinVolume</td><td>4.360</td><td>140.2</td><td>144.6</td><td>02:24</td><td>10756905</td><td>13.04e-6</td><td>2.2176</td><td>17.90</td><td>18.46</td><td>11.14</td></tr><tr><td>Recursive</td><td>3.948</td><td>118.5</td><td>122.4</td><td>02:02</td><td>9236405</td><td>12.83e-6</td><td>2.2173</td><td>15.12</td><td>15.63</td><td>9.563</td></tr><tr><td>Recursive2</td><td>3.911</td><td>120.0</td><td>123.9</td><td>02:03</td><td>9233981</td><td>13.00e-6</td><td>2.2173</td><td>15.32</td><td>15.82</td><td>9.560</td></tr><tr><td>Interval</td><td>23.13e-6</td><td>47226</td><td>47226</td><td>13:07:06</td><td>5048693363</td><td>9.354e-6</td><td>1.4911</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>57.81</td><td>39298</td><td>39356</td><td>10:55:56</td><td>1108947741</td><td>35.44e-6</td><td>1.4862</td><td>83.21</td><td>83.34</td><td>21.97</td></tr><tr><td>MinVariance</td><td>60.79</td><td>43141</td><td>43202</td><td>12:00:02</td><td>1184923479</td><td>36.41e-6</td><td>1.4864</td><td>91.35</td><td>91.48</td><td>23.47</td></tr><tr><td>B SPLines</td><td>66.31</td><td>74744</td><td>74810</td><td>20:46:50</td><td>2353185399</td><td>31.76e-6</td><td>1.4881</td><td>158.3</td><td>158.4</td><td>46.61</td></tr><tr><td>MinNorm</td><td>69.03</td><td>41340</td><td>41409</td><td>11:30:09</td><td>1149775885</td><td>35.96e-6</td><td>1.4863</td><td>87.54</td><td>87.68</td><td>22.77</td></tr><tr><td>MinVolume</td><td>62.62</td><td>44406</td><td>44469</td><td>12:21:09</td><td>1260200473</td><td>35.24e-6</td><td>1.4865</td><td>94.03</td><td>94.16</td><td>24.96</td></tr><tr><td>ApproxMinVolume</td><td>73.58</td><td>44958</td><td>45031</td><td>12:30:31</td><td>1262169197</td><td>35.62e-6</td><td>1.4865</td><td>95.20</td><td>95.35</td><td>25.00</td></tr><tr><td>Recursive</td><td>71.40</td><td>41076</td><td>41147</td><td>11:25:47</td><td>1110759591</td><td>36.98e-6</td><td>1.4862</td><td>86.98</td><td>87.13</td><td>22.00</td></tr><tr><td>Recursive2</td><td>70.65</td><td>38534</td><td>38605</td><td>10:43:25</td><td>1109214085</td><td>34.74e-6</td><td>1.4862</td><td>81.60</td><td>81.74</td><td>21.97</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>26.94e-6</td><td>3.105e-3</td><td>3.132e-3</td><td>0.003</td><td>59</td><td>52.63e-6</td><td>infeasible</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>715.4e-3</td><td>3.829e-3</td><td>719.2e-3</td><td>0.719</td><td>41</td><td>93.39e-6</td><td>infeasible</td><td>123.3</td><td>22965.1</td><td>69.49</td></tr><tr><td>MinVariance</td><td>944.4e-3</td><td>2.939e-3</td><td>947.3e-3</td><td>0.947</td><td>41</td><td>71.68e-6</td><td>infeasible</td><td>94.66</td><td>30248.2</td><td>69.49</td></tr><tr><td>B SPLines</td><td>944.2e-3</td><td>4.258e-3</td><td>948.5e-3</td><td>0.948</td><td>41</td><td>103.9e-6</td><td>infeasible</td><td>137.1</td><td>30284.3</td><td>69.49</td></tr><tr><td>MinNorm</td><td>941.2e-3</td><td>2.811e-3</td><td>944.0e-3</td><td>0.944</td><td>41</td><td>68.56e-6</td><td>infeasible</td><td>90.53</td><td>30142.1</td><td>69.49</td></tr><tr><td>MinVolume</td><td>944.2e-3</td><td>3.552e-3</td><td>947.8e-3</td><td>0.948</td><td>41</td><td>86.63e-6</td><td>infeasible</td><td>114.4</td><td>30261.5</td><td>69.49</td></tr><tr><td>ApproxMinVolume</td><td>941.8e-3</td><td>2.734e-3</td><td>944.6e-3</td><td>0.945</td><td>41</td><td>66.69e-6</td><td>infeasible</td><td>88.06</td><td>30160.7</td><td>69.49</td></tr><tr><td>Recursive</td><td>943.3e-3</td><td>2.642e-3</td><td>946.0e-3</td><td>0.946</td><td>41</td><td>64.44e-6</td><td>infeasible</td><td>85.09</td><td>30204.2</td><td>69.49</td></tr><tr><td>Recursive2</td><td>943.0e-3</td><td>2.698e-3</td><td>945.7e-3</td><td>0.946</td><td>41</td><td>65.80e-6</td><td>infeasible</td><td>86.89</td><td>30197.1</td><td>69.49</td></tr><tr><td>Interval</td><td>27.18e-6</td><td>151.5e-3</td><td>151.5e-3</td><td>0.152</td><td>21335</td><td>7.100e-6</td><td>3.9989</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.241</td><td>38.74e-3</td><td>1.280</td><td>1.280</td><td>6171</td><td>6.278e-6</td><td>3.9588</td><td>25.58</td><td>844.7</td><td>28.92</td></tr><tr><td>MinVariance</td><td>1.365</td><td>40.34e-3</td><td>1.405</td><td>1.405</td><td>6463</td><td>6.242e-6</td><td>3.9602</td><td>26.63</td><td>927.5</td><td>30.29</td></tr><tr><td>B SPLines</td><td>1.233</td><td>59.76e-3</td><td>1.292</td><td>1.292</td><td>11131</td><td>5.369e-6</td><td>3.9774</td><td>39.45</td><td>853.0</td><td>52.17</td></tr><tr><td>MinNorm</td><td>1.374</td><td>39.50e-3</td><td>1.413</td><td>1.413</td><td>6349</td><td>6.221e-6</td><td>3.9596</td><td>26.08</td><td>932.8</td><td>29.76</td></tr><tr><td>MinVolume</td><td>1.310</td><td>42.41e-3</td><td>1.352</td><td>1.352</td><td>6833</td><td>6.207e-6</td><td>3.9616</td><td>28.00</td><td>892.5</td><td>32.03</td></tr><tr><td>ApproxMinVolume</td><td>1.299</td><td>41.50e-3</td><td>1.340</td><td>1.340</td><td>6843</td><td>6.065e-6</td><td>3.9617</td><td>27.40</td><td>884.7</td><td>32.07</td></tr><tr><td>Recursive</td><td>1.247</td><td>39.09e-3</td><td>1.286</td><td>1.286</td><td>6173</td><td>6.333e-6</td><td>3.9588</td><td>25.81</td><td>848.8</td><td>28.93</td></tr><tr><td>Recursive2</td><td>1.239</td><td>38.85e-3</td><td>1.278</td><td>1.278</td><td>6201</td><td>6.264e-6</td><td>3.9588</td><td>25.64</td><td>843.2</td><td>29.06</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>33.86e-6</td><td>141.6</td><td>141.6</td><td>02:21</td><td>17502691</td><td>8.091e-6</td><td>2.5375</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.984</td><td>17.94</td><td>21.92</td><td>21.924</td><td>1335393</td><td>13.43e-6</td><td>2.3014</td><td>12.67</td><td>15.48</td><td>7.630</td></tr><tr><td>MinVariance</td><td>4.861</td><td>20.23</td><td>25.09</td><td>25.092</td><td>1423285</td><td>14.21e-6</td><td>2.3026</td><td>14.29</td><td>17.72</td><td>8.132</td></tr><tr><td>B SPLines</td><td>3.959</td><td>33.46</td><td>37.42</td><td>37.422</td><td>2748539</td><td>12.17e-6</td><td>2.3172</td><td>23.63</td><td>26.43</td><td>15.70</td></tr><tr><td>MinNorm</td><td>4.719</td><td>19.88</td><td>24.60</td><td>24.600</td><td>1382493</td><td>14.38e-6</td><td>2.3021</td><td>14.04</td><td>17.37</td><td>7.899</td></tr><tr><td>MinVolume</td><td>4.217</td><td>20.62</td><td>24.84</td><td>24.839</td><td>1509929</td><td>13.66e-6</td><td>2.3038</td><td>14.56</td><td>17.54</td><td>8.627</td></tr><tr><td>ApproxMinVolume</td><td>4.421</td><td>20.78</td><td>25.20</td><td>25.202</td><td>1512139</td><td>13.74e-6</td><td>2.3039</td><td>14.68</td><td>17.80</td><td>8.639</td></tr><tr><td>Recursive</td><td>4.049</td><td>18.28</td><td>22.33</td><td>22.327</td><td>1344243</td><td>13.60e-6</td><td>2.3014</td><td>12.91</td><td>15.77</td><td>7.680</td></tr><tr><td>Recursive2</td><td>3.970</td><td>18.20</td><td>22.16</td><td>22.165</td><td>1342371</td><td>13.55e-6</td><td>2.3014</td><td>12.85</td><td>15.65</td><td>7.670</td></tr><tr><td>Interval</td><td>30.04e-6</td><td>2097</td><td>2097</td><td>34:57</td><td>227373771</td><td>9.225e-6</td><td>1.5822</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>57.59</td><td>1931</td><td>1988</td><td>33:08</td><td>52284461</td><td>36.94e-6</td><td>1.5430</td><td>92.08</td><td>94.83</td><td>22.99</td></tr><tr><td>MinVariance</td><td>60.43</td><td>2020</td><td>2081</td><td>34:41</td><td>55748841</td><td>36.24e-6</td><td>1.5441</td><td>96.33</td><td>99.22</td><td>24.52</td></tr><tr><td>B SPLines</td><td>58.22</td><td>3463</td><td>3521</td><td>58:41</td><td>109297333</td><td>31.69e-6</td><td>1.5578</td><td>165.1</td><td>167.9</td><td>48.07</td></tr><tr><td>MinNorm</td><td>60.35</td><td>2023</td><td>2084</td><td>34:44</td><td>54136971</td><td>37.38e-6</td><td>1.5436</td><td>96.49</td><td>99.37</td><td>23.81</td></tr><tr><td>MinVolume</td><td>57.57</td><td>2062</td><td>2119</td><td>35:19</td><td>59198995</td><td>34.84e-6</td><td>1.5452</td><td>98.32</td><td>101.1</td><td>26.04</td></tr><tr><td>ApproxMinVolume</td><td>59.98</td><td>2184</td><td>2244</td><td>37:24</td><td>59289665</td><td>36.84e-6</td><td>1.5453</td><td>104.1</td><td>107.0</td><td>26.08</td></tr><tr><td>Recursive</td><td>57.39</td><td>1923</td><td>1980</td><td>33:00</td><td>52908543</td><td>36.35e-6</td><td>1.5430</td><td>91.69</td><td>94.43</td><td>23.27</td></tr><tr><td>Recursive2</td><td>57.68</td><td>1925</td><td>1983</td><td>33:03</td><td>52413113</td><td>36.74e-6</td><td>1.5430</td><td>91.80</td><td>94.55</td><td>23.05</td></tr></tbody></table>
<table><caption>Table 15: Bissection of problem 2D with problem: 4, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time.per.iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>30.99e-6</td><td>16.64e-3</td><td>16.67e-3</td><td>0.017</td><td>2383</td><td>6.981e-6</td><td>1.0755</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>764.4e-3</td><td>6.178e-3</td><td>770.6e-3</td><td>0.771</td><td>817</td><td>7.562e-6</td><td>1.0716</td><td>37.14</td><td>4623.5</td><td>34.28</td></tr><tr><td>MinVariance</td><td>842.4e-3</td><td>5.851e-3</td><td>848.2e-3</td><td>0.848</td><td>877</td><td>6.672e-6</td><td>1.0717</td><td>35.17</td><td>5089.2</td><td>36.80</td></tr><tr><td>BSplines</td><td>863.2e-3</td><td>15.61e-3</td><td>878.9e-3</td><td>0.879</td><td>2039</td><td>7.657e-6</td><td>1.0735</td><td>93.85</td><td>5273.0</td><td>85.56</td></tr><tr><td>MinNorm</td><td>954.0e-3</td><td>7.705e-3</td><td>961.8e-3</td><td>0.962</td><td>853</td><td>9.033e-6</td><td>1.0717</td><td>46.31</td><td>5770.4</td><td>35.80</td></tr><tr><td>MinVolume</td><td>920.7e-3</td><td>7.322e-3</td><td>928.0e-3</td><td>0.928</td><td>955</td><td>7.667e-6</td><td>1.0718</td><td>44.01</td><td>5567.9</td><td>40.08</td></tr><tr><td>ApproxMinVolume</td><td>893.4e-3</td><td>6.409e-3</td><td>899.8e-3</td><td>0.900</td><td>957</td><td>6.697e-6</td><td>1.0718</td><td>38.52</td><td>5398.6</td><td>40.16</td></tr><tr><td>Recursive</td><td>873.4e-3</td><td>7.287e-3</td><td>880.7e-3</td><td>0.881</td><td>1033</td><td>7.054e-6</td><td>1.0716</td><td>43.80</td><td>5283.9</td><td>43.35</td></tr><tr><td>Recursive2</td><td>860.3e-3</td><td>7.059e-3</td><td>867.4e-3</td><td>0.867</td><td>1033</td><td>6.833e-6</td><td>1.0716</td><td>42.43</td><td>5204.0</td><td>43.35</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>23.13e-6</td><td>4.811</td><td>4.811</td><td>4.811</td><td>711195</td><td>6.764e-6</td><td>741.67e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.985</td><td>2.027</td><td>5.012</td><td>5.012</td><td>193081</td><td>10.50e-6</td><td>738.33e-3</td><td>42.14</td><td>104.2</td><td>27.15</td></tr><tr><td>MinVariance</td><td>4.365</td><td>2.589</td><td>6.954</td><td>6.954</td><td>223843</td><td>11.56e-6</td><td>738.62e-3</td><td>53.81</td><td>144.5</td><td>31.47</td></tr><tr><td>BSplines</td><td>3.130</td><td>8.459</td><td>11.59</td><td>11.589</td><td>775983</td><td>10.90e-6</td><td>742.24e-3</td><td>175.8</td><td>240.9</td><td>109.1</td></tr><tr><td>MinNorm</td><td>4.151</td><td>2.365</td><td>6.516</td><td>6.516</td><td>208239</td><td>11.36e-6</td><td>738.47e-3</td><td>49.17</td><td>135.5</td><td>29.28</td></tr><tr><td>MinVolume</td><td>3.412</td><td>2.716</td><td>6.128</td><td>6.128</td><td>255471</td><td>10.63e-6</td><td>738.94e-3</td><td>56.46</td><td>127.4</td><td>35.92</td></tr><tr><td>ApproxMinVolume</td><td>3.387</td><td>2.728</td><td>6.116</td><td>6.116</td><td>256251</td><td>10.65e-6</td><td>738.94e-3</td><td>56.72</td><td>127.1</td><td>36.03</td></tr><tr><td>Recursive</td><td>3.437</td><td>2.278</td><td>5.716</td><td>5.716</td><td>209177</td><td>10.89e-6</td><td>738.33e-3</td><td>47.36</td><td>118.8</td><td>29.41</td></tr><tr><td>Recursive2</td><td>3.664</td><td>2.224</td><td>5.888</td><td>5.888</td><td>203103</td><td>10.95e-6</td><td>738.33e-3</td><td>46.23</td><td>122.4</td><td>28.56</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>29.09e-6</td><td>1060</td><td>1060</td><td>17.40</td><td>127570319</td><td>8.315e-6</td><td>621.32e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.980</td><td>372.9</td><td>376.9</td><td>06:16</td><td>21728921</td><td>17.16e-6</td><td>617.74e-3</td><td>35.15</td><td>35.53</td><td>17.03</td></tr><tr><td>MinVariance</td><td>5.156</td><td>449.7</td><td>454.8</td><td>07:34</td><td>26186913</td><td>17.17e-6</td><td>618.08e-3</td><td>42.39</td><td>42.88</td><td>20.53</td></tr><tr><td>BSplines</td><td>3.946</td><td>2073</td><td>2077</td><td>34:37</td><td>123119939</td><td>16.84e-6</td><td>621.91e-3</td><td>195.5</td><td>195.9</td><td>96.51</td></tr><tr><td>MinNorm</td><td>5.077</td><td>405.0</td><td>410.1</td><td>06:50</td><td>24001845</td><td>16.87e-6</td><td>617.91e-3</td><td>38.18</td><td>38.66</td><td>18.81</td></tr><tr><td>MinVolume</td><td>4.473</td><td>537.7</td><td>542.2</td><td>09:02</td><td>30954991</td><td>17.37e-6</td><td>618.38e-3</td><td>50.70</td><td>51.12</td><td>24.27</td></tr><tr><td>ApproxMinVolume</td><td>4.623</td><td>523.6</td><td>528.2</td><td>08:48</td><td>31078695</td><td>16.85e-6</td><td>618.38e-3</td><td>49.36</td><td>49.80</td><td>24.36</td></tr><tr><td>Recursive</td><td>4.062</td><td>396.9</td><td>400.9</td><td>06:40</td><td>23405821</td><td>16.96e-6</td><td>617.74e-3</td><td>37.42</td><td>37.80</td><td>18.35</td></tr><tr><td>Recursive2</td><td>4.028</td><td>449.1</td><td>453.1</td><td>07:33</td><td>25972355</td><td>17.29e-6</td><td>617.74e-3</td><td>42.34</td><td>42.72</td><td>20.36</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>27.00e-6</td><td>279920</td><td>279920</td><td>3-05:45:20</td><td>28828576484</td><td>9.710e-6</td><td>529.85e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>59.44</td><td>184551</td><td>184551</td><td>2-03:15:51</td><td>4449324341</td><td>41.48e-6</td><td>526.69e-3</td><td>65.93</td><td>65.93</td><td>15.43</td></tr><tr><td>MinVariance</td><td>65.23</td><td>234787</td><td>234787</td><td>2-17:13:07</td><td>5621872305</td><td>41.76e-6</td><td>527.05e-3</td><td>83.88</td><td>83.88</td><td>19.50</td></tr><tr><td>BSplines</td><td>56.03</td><td>1525850</td><td>1525850</td><td>17-15:50:50</td><td>37268701884</td><td>40.94e-6</td><td>531.29e-3</td><td>545.1</td><td>545.1</td><td>129.3</td></tr><tr><td>MinNorm</td><td>66.44</td><td>215741</td><td>215741</td><td>2-11:55:41</td><td>5073628293</td><td>42.52e-6</td><td>526.90e-3</td><td>77.07</td><td>77.07</td><td>17.60</td></tr><tr><td>MinVolume</td><td>68.35</td><td>296081</td><td>296081</td><td>3-10:14:41</td><td>6915560218</td><td>42.81e-6</td><td>527.38e-3</td><td>105.8</td><td>105.8</td><td>23.99</td></tr><tr><td>ApproxMinVolume</td><td>65.48</td><td>288926</td><td>288927</td><td>3-08:15:27</td><td>6949340844</td><td>41.58e-6</td><td>527.39e-3</td><td>103.2</td><td>103.2</td><td>24.11</td></tr><tr><td>Recursive</td><td>60.67</td><td>197947</td><td>197947</td><td>2-06:59:07</td><td>4766507865</td><td>41.53e-6</td><td>526.69e-3</td><td>70.72</td><td>70.72</td><td>16.53</td></tr><tr><td>Recursive2</td><td>61.72</td><td>220353</td><td>220353</td><td>2-13:12:33</td><td>5287820395</td><td>41.67e-6</td><td>526.69e-3</td><td>78.72</td><td>78.72</td><td>18.34</td></tr><tr><td rowspan="20">0.01</td><td rowspan="10">2</td><td>Interval</td><td>34.09e-6</td><td>9.222e-3</td><td>9.256e-3</td><td>0.009</td><td>899</td><td>10.26e-6</td><td>1.1306</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.066</td><td>4.494e-3</td><td>1.070</td><td>1.070</td><td>361</td><td>12.45e-6</td><td>1.0927</td><td>48.73</td><td>11560.2</td><td>40.16</td></tr><tr><td>MinVariance</td><td>1.145</td><td>4.969e-3</td><td>1.150</td><td>1.150</td><td>385</td><td>12.91e-6</td><td>1.0933</td><td>53.88</td><td>12422.6</td><td>42.83</td></tr><tr><td>BSplines</td><td>1.066</td><td>7.612e-3</td><td>1.073</td><td>1.073</td><td>913</td><td>8.337e-6</td><td>1.1115</td><td>82.54</td><td>11595.7</td><td>101.6</td></tr><tr><td>MinNorm</td><td>1.124</td><td>4.219e-3</td><td>1.128</td><td>1.128</td><td>375</td><td>11.25e-6</td><td>1.0930</td><td>45.75</td><td>12186.5</td><td>41.71</td></tr><tr><td>MinVolume</td><td>1.108</td><td>4.839e-3</td><td>1.113</td><td>1.113</td><td>425</td><td>11.39e-6</td><td>1.0940</td><td>52.47</td><td>12027.7</td><td>47.27</td></tr><tr><td>ApproxMinVolume</td><td>1.088</td><td>5.074e-3</td><td>1.093</td><td>1.093</td><td>425</td><td>11.94e-6</td><td>1.0940</td><td>55.02</td><td>11812.5</td><td>47.27</td></tr><tr><td>Recursive</td><td>1.066</td><td>5.791e-3</td><td>1.071</td><td>1.071</td><td>559</td><td>10.36e-6</td><td>1.0927</td><td>62.79</td><td>11574.8</td><td>62.18</td></tr><tr><td>Recursive2</td><td>1.106</td><td>7.776e-3</td><td>1.114</td><td>1.114</td><td>577</td><td>13.48e-6</td><td>1.0927</td><td>84.32</td><td>12036.7</td><td>64.18</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>33.14e-6</td><td>762.0e-3</td><td>762.0e-3</td><td>0.762</td><td>106721</td><td>7.140e-6</td><td>794.69e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.911</td><td>342.0e-3</td><td>3.253</td><td>3.253</td><td>30803</td><td>11.10e-6</td><td>766.06e-3</td><td>44.89</td><td>426.9</td><td>28.86</td></tr><tr><td>MinVariance</td><td>4.297</td><td>434.5e-3</td><td>4.731</td><td>4.731</td><td>35577</td><td>12.21e-6</td><td>768.90e-3</td><td>57.03</td><td>620.9</td><td>33.34</td></tr><tr><td>BSplines</td><td>3.210</td><td>1.439</td><td>4.650</td><td>4.650</td><td>130221</td><td>11.05e-6</td><td>797.06e-3</td><td>188.9</td><td>610.2</td><td>122.0</td></tr><tr><td>MinNorm</td><td>4.267</td><td>393.1e-3</td><td>4.660</td><td>4.660</td><td>33395</td><td>11.77e-6</td><td>768.21e-3</td><td>51.60</td><td>611.6</td><td>31.29</td></tr><tr><td>MinVolume</td><td>3.654</td><td>487.5e-3</td><td>4.141</td><td>4.141</td><td>41623</td><td>11.71e-6</td><td>770.54e-3</td><td>63.98</td><td>543.4</td><td>39.00</td></tr><tr><td>ApproxMinVolume</td><td>3.627</td><td>497.5e-3</td><td>4.125</td><td>4.125</td><td>41803</td><td>11.90e-6</td><td>770.68e-3</td><td>65.29</td><td>541.3</td><td>39.17</td></tr><tr><td>Recursive</td><td>3.513</td><td>499.7e-3</td><td>4.013</td><td>4.013</td><td>42361</td><td>11.80e-6</td><td>766.06e-3</td><td>65.58</td><td>526.7</td><td>39.69</td></tr><tr><td>Recursive2</td><td>3.741</td><td>500.4e-3</td><td>4.241</td><td>4.241</td><td>39681</td><td>12.61e-6</td><td>766.06e-3</td><td>65.67</td><td>556.6</td><td>37.18</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>22.17e-6</td><td>67.97</td><td>67.97</td><td>01:07</td><td>7790067</td><td>8.725e-6</td><td>678.03e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.987</td><td>23.07</td><td>27.06</td><td>27.058</td><td>1369727</td><td>16.84e-6</td><td>647.64e-3</td><td>33.95</td><td>39.81</td><td>17.58</td></tr><tr><td>MinVariance</td><td>5.235</td><td>27.03</td><td>32.27</td><td>32.269</td><td>1577077</td><td>17.14e-6</td><td>649.64e-3</td><td>39.78</td><td>47.48</td><td>20.24</td></tr><tr><td>BSplines</td><td>4.078</td><td>152.7</td><td>156.8</td><td>02:36</td><td>9051187</td><td>16.87e-6</td><td>678.47e-3</td><td>224.7</td><td>230.7</td><td>116.2</td></tr><tr><td>MinNorm</td><td>5.032</td><td>24.99</td><td>30.02</td><td>30.019</td><td>1448057</td><td>17.26e-6</td><td>648.77e-3</td><td>36.76</td><td>44.17</td><td>18.59</td></tr><tr><td>MinVolume</td><td>4.597</td><td>33.28</td><td>37.88</td><td>37.876</td><td>1913031</td><td>17.40e-6</td><td>651.67e-3</td><td>48.96</td><td>55.73</td><td>24.56</td></tr><tr><td>ApproxMinVolume</td><td>4.735</td><td>33.69</td><td>38.43</td><td>38.428</td><td>1921565</td><td>17.53e-6</td><td>651.69e-3</td><td>49.57</td><td>56.54</td><td>24.67</td></tr><tr><td>Recursive</td><td>4.172</td><td>43.24</td><td>47.41</td><td>47.408</td><td>2509615</td><td>17.23e-6</td><td>647.64e-3</td><td>63.62</td><td>69.75</td><td>32.22</td></tr><tr><td>Recursive2</td><td>4.083</td><td>59.19</td><td>63.27</td><td>01:03</td><td>3398545</td><td>17.42e-6</td><td>647.64e-3</td><td>87.08</td><td>93.09</td><td>43.63</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>24.08e-6</td><td>6106</td><td>6106</td><td>01:41:46</td><td>653254743</td><td>9.348e-6</td><td>587.12e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>54.65</td><td>4499</td><td>4553</td><td>01:15:53</td><td>106824699</td><td>42.12e-6</td><td>559.75e-3</td><td>73.68</td><td>74.57</td><td>16.35</td></tr><tr><td>MinVariance</td><td>60.05</td><td>5599</td><td>5660</td><td>01:34:20</td><td>131642925</td><td>42.54e-6</td><td>562.22e-3</td><td>91.71</td><td>92.69</td><td>20.15</td></tr><tr><td>BSplines</td><td>55.14</td><td>49263</td><td>49319</td><td>13:41:59</td><td>1145719257</td><td>43.00e-6</td><td>595.66e-3</td><td>806.8</td><td>807.7</td><td>175.4</td></tr><tr><td>MinNorm</td><td>59.58</td><td>4969</td><td>5029</td><td>01:23:49</td><td>116862765</td><td>42.53e-6</td><td>561.16e-3</td><td>81.38</td><td>82.36</td><td>17.89</td></tr><tr><td>MinVolume</td><td>57.21</td><td>7161</td><td>7218</td><td>02:00:18</td><td>167099243</td><td>42.86e-6</td><td>564.56e-3</td><td>117.3</td><td>118.2</td><td>25.58</td></tr><tr><td>ApproxMinVolume</td><td>57.63</td><td>7515</td><td>7573</td><td>02:06:13</td><td>168217769</td><td>44.68e-6</td><td>564.77e-3</td><td>123.1</td><td>124.0</td><td>25.75</td></tr><tr><td>Recursive</td><td>55.21</td><td>8476</td><td>8531</td><td>02:22:11</td><td>203131201</td><td>41.73e-6</td><td>559.75e-3</td><td>138.8</td><td>139.7</td><td>31.10</td></tr><tr><td>Recursive2</td><td>55.43</td><td>12781</td><td>12837</td><td>03:33:57</td><td>301335357</td><td>42.42e-6</td><td>559.75e-3</td><td>209.3</td><td>210.2</td><td>46.13</td></tr></tbody></table>
<table><caption>Table 16: Bisection of problem 2D with problem: 5, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S,ms)</th><th>nb.iter</th><th>time_per_iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>26.94e-6</td><td>18.85e-3</td><td>18.88e-3</td><td>0.019</td><td>2661</td><td>7.085e-6</td><td>2.7566</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>847.4e-3</td><td>7.550e-3</td><td>854.9e-3</td><td>0.855</td><td>1229</td><td>6.143e-6</td><td>2.7533</td><td>40.04</td><td>4528.1</td><td>46.19</td></tr><tr><td>MinVariance</td><td>948.1e-3</td><td>8.439e-3</td><td>956.6e-3</td><td>0.957</td><td>1351</td><td>6.247e-6</td><td>2.7535</td><td>44.76</td><td>5066.3</td><td>50.77</td></tr><tr><td>BSplines</td><td>863.5e-3</td><td>15.00e-3</td><td>878.6e-3</td><td>0.879</td><td>3251</td><td>4.614e-6</td><td>2.7564</td><td>79.55</td><td>4653.1</td><td>122.2</td></tr><tr><td>MinNorm</td><td>925.9e-3</td><td>8.227e-3</td><td>934.2e-3</td><td>0.934</td><td>1295</td><td>6.353e-6</td><td>2.7534</td><td>43.64</td><td>4917.6</td><td>48.67</td></tr><tr><td>MinVolume</td><td>907.5e-3</td><td>9.963e-3</td><td>917.4e-3</td><td>0.917</td><td>1851</td><td>5.382e-6</td><td>2.7536</td><td>52.84</td><td>4859.0</td><td>69.56</td></tr><tr><td>ApproxMinVolume</td><td>900.4e-3</td><td>9.286e-3</td><td>909.7e-3</td><td>0.910</td><td>1857</td><td>5.001e-6</td><td>2.7536</td><td>49.25</td><td>4817.9</td><td>69.79</td></tr><tr><td>Recursive</td><td>857.2e-3</td><td>8.117e-3</td><td>865.3e-3</td><td>0.865</td><td>1377</td><td>5.895e-6</td><td>2.7533</td><td>43.05</td><td>4583.1</td><td>51.75</td></tr><tr><td>Recursive2</td><td>858.7e-3</td><td>8.092e-3</td><td>866.8e-3</td><td>0.867</td><td>1351</td><td>5.990e-6</td><td>2.7533</td><td>42.92</td><td>4590.8</td><td>50.77</td></tr><tr><td>Interval</td><td>27.89e-6</td><td>3.576</td><td>3.576</td><td>3.576</td><td>493957</td><td>7.240e-6</td><td>2.4706</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.025</td><td>1.149</td><td>4.174</td><td>4.174</td><td>115307</td><td>9.961e-6</td><td>2.4646</td><td>32.12</td><td>116.7</td><td>23.34</td></tr><tr><td>MinVariance</td><td>4.468</td><td>1.433</td><td>5.901</td><td>5.901</td><td>133685</td><td>10.72e-6</td><td>2.4649</td><td>40.07</td><td>165.0</td><td>27.06</td></tr><tr><td>BSplines</td><td>3.221</td><td>4.467</td><td>7.688</td><td>7.688</td><td>439985</td><td>10.15e-6</td><td>2.4685</td><td>124.9</td><td>215.0</td><td>89.07</td></tr><tr><td>MinNorm</td><td>4.232</td><td>1.339</td><td>5.570</td><td>5.570</td><td>125473</td><td>10.67e-6</td><td>2.4647</td><td>37.43</td><td>155.8</td><td>25.40</td></tr><tr><td>MinVolume</td><td>3.620</td><td>1.543</td><td>5.163</td><td>5.163</td><td>151441</td><td>10.19e-6</td><td>2.4652</td><td>43.15</td><td>144.4</td><td>30.66</td></tr><tr><td>ApproxMinVolume</td><td>3.601</td><td>1.571</td><td>5.172</td><td>5.172</td><td>153139</td><td>10.26e-6</td><td>2.4652</td><td>43.92</td><td>144.6</td><td>31.00</td></tr><tr><td>Recursive</td><td>3.498</td><td>1.276</td><td>4.773</td><td>4.773</td><td>123205</td><td>10.35e-6</td><td>2.4646</td><td>35.67</td><td>133.5</td><td>24.94</td></tr><tr><td>Recursive2</td><td>3.725</td><td>1.310</td><td>5.034</td><td>5.034</td><td>122811</td><td>10.66e-6</td><td>2.4646</td><td>36.62</td><td>140.8</td><td>24.86</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>27.89e-6</td><td>1506</td><td>1506</td><td>25.06</td><td>189085251</td><td>7.968e-6</td><td>1.4003</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>4.009</td><td>707.1</td><td>711.2</td><td>11.51</td><td>41367965</td><td>17.09e-6</td><td>1.3955</td><td>46.93</td><td>47.20</td><td>21.88</td></tr><tr><td>MinVariance</td><td>5.156</td><td>865.4</td><td>870.5</td><td>14.30</td><td>50285639</td><td>17.21e-6</td><td>1.3959</td><td>57.43</td><td>57.78</td><td>26.59</td></tr><tr><td>BSplines</td><td>3.960</td><td>4073</td><td>4077</td><td>01:07:57</td><td>236383013</td><td>17.23e-6</td><td>1.4017</td><td>270.4</td><td>270.6</td><td>125.0</td></tr><tr><td>MinNorm</td><td>5.080</td><td>797.4</td><td>802.5</td><td>13:22</td><td>46103895</td><td>17.30e-6</td><td>1.3957</td><td>52.92</td><td>53.26</td><td>24.38</td></tr><tr><td>MinVolume</td><td>4.497</td><td>1029</td><td>1033</td><td>17:13</td><td>59826583</td><td>17.21e-6</td><td>1.3964</td><td>68.32</td><td>68.62</td><td>31.64</td></tr><tr><td>ApproxMinVolume</td><td>4.653</td><td>1053</td><td>1057</td><td>17:37</td><td>60059399</td><td>17.54e-6</td><td>1.3964</td><td>69.90</td><td>70.21</td><td>31.76</td></tr><tr><td>Recursive</td><td>4.082</td><td>717.4</td><td>721.5</td><td>12:01</td><td>42155919</td><td>17.02e-6</td><td>1.3955</td><td>47.62</td><td>47.89</td><td>22.29</td></tr><tr><td>Recursive2</td><td>4.030</td><td>791.0</td><td>795.0</td><td>13:15</td><td>45781889</td><td>17.28e-6</td><td>1.3955</td><td>52.50</td><td>52.77</td><td>24.21</td></tr><tr><td>Interval</td><td>26.00e-6</td><td>666585</td><td>666585</td><td>7-17:09:45</td><td>68829667707</td><td>9.685e-6</td><td>1.1860</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>60.23</td><td>542708</td><td>542708</td><td>6-06:45:08</td><td>13295526250</td><td>40.82e-6</td><td>1.1815</td><td>81.42</td><td>81.42</td><td>19.32</td></tr><tr><td>MinVariance</td><td>71.97</td><td>691783</td><td>691783</td><td>8-00:09:43</td><td>16640937217</td><td>41.57e-6</td><td>1.1821</td><td>103.8</td><td>103.8</td><td>24.18</td></tr><tr><td>BSplines</td><td>64.29</td><td>4257390</td><td>4257390</td><td>49:06:36:30</td><td>104557258199</td><td>40.72e-6</td><td>1.1886</td><td>638.7</td><td>638.7</td><td>151.9</td></tr><tr><td>MinNorm</td><td>71.34</td><td>626854</td><td>626854</td><td>7-06:07:34</td><td>15059663173</td><td>41.62e-6</td><td>1.1818</td><td>94.04</td><td>94.04</td><td>21.88</td></tr><tr><td>MinVolume</td><td>65.11</td><td>842860</td><td>842860</td><td>9-18:07:40</td><td>20385478051</td><td>41.35e-6</td><td>1.1826</td><td>126.4</td><td>126.4</td><td>29.62</td></tr><tr><td>ApproxMinVolume</td><td>65.09</td><td>847141</td><td>847141</td><td>9-19:19:01</td><td>20481565927</td><td>41.36e-6</td><td>1.1826</td><td>127.1</td><td>127.1</td><td>29.76</td></tr><tr><td>Recursive</td><td>62.05</td><td>553289</td><td>553290</td><td>6-09:41:30</td><td>13624167152</td><td>40.61e-6</td><td>1.1815</td><td>83.00</td><td>83.00</td><td>19.79</td></tr><tr><td>Recursive2</td><td>59.29</td><td>611016</td><td>611016</td><td>7-01:43:36</td><td>14983085410</td><td>40.78e-6</td><td>1.1815</td><td>91.66</td><td>91.66</td><td>21.77</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>31.95e-6</td><td>6.617e-3</td><td>6.649e-3</td><td>0.007</td><td>661</td><td>10.01e-6</td><td>2.8082</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.086</td><td>5.092e-3</td><td>1.091</td><td>1.091</td><td>405</td><td>12.57e-6</td><td>2.7809</td><td>76.95</td><td>16414.9</td><td>61.27</td></tr><tr><td>MinVariance</td><td>1.175</td><td>4.338e-3</td><td>1.180</td><td>1.180</td><td>425</td><td>10.21e-6</td><td>2.7820</td><td>65.56</td><td>17741.4</td><td>64.30</td></tr><tr><td>BSplines</td><td>1.100</td><td>7.231e-3</td><td>1.107</td><td>1.107</td><td>773</td><td>9.354e-6</td><td>2.8057</td><td>109.3</td><td>16645.9</td><td>116.9</td></tr><tr><td>MinNorm</td><td>1.156</td><td>4.606e-3</td><td>1.161</td><td>1.161</td><td>409</td><td>11.26e-6</td><td>2.7815</td><td>69.61</td><td>17457.3</td><td>61.88</td></tr><tr><td>MinVolume</td><td>1.147</td><td>4.408e-3</td><td>1.151</td><td>1.151</td><td>469</td><td>9.399e-6</td><td>2.7830</td><td>66.62</td><td>17313.7</td><td>70.95</td></tr><tr><td>ApproxMinVolume</td><td>1.142</td><td>5.727e-3</td><td>1.148</td><td>1.148</td><td>469</td><td>12.21e-6</td><td>2.7830</td><td>86.55</td><td>17264.6</td><td>70.95</td></tr><tr><td>Recursive</td><td>1.097</td><td>5.886e-3</td><td>1.103</td><td>1.103</td><td>523</td><td>11.25e-6</td><td>2.7809</td><td>88.95</td><td>16581.7</td><td>79.12</td></tr><tr><td>Recursive2</td><td>1.097</td><td>6.848e-3</td><td>1.104</td><td>1.104</td><td>513</td><td>13.35e-6</td><td>2.7809</td><td>103.5</td><td>16601.1</td><td>77.61</td></tr><tr><td>Interval</td><td>29.09e-6</td><td>5039e-3</td><td>5039e-3</td><td>0.504</td><td>69249</td><td>7.277e-6</td><td>2.5447</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.992</td><td>215.7e-3</td><td>3.208</td><td>3.208</td><td>19403</td><td>11.12e-6</td><td>2.4963</td><td>42.80</td><td>636.5</td><td>28.02</td></tr><tr><td>MinVariance</td><td>4.437</td><td>266.3e-3</td><td>4.703</td><td>4.703</td><td>22085</td><td>12.06e-6</td><td>2.4994</td><td>52.86</td><td>933.3</td><td>31.89</td></tr><tr><td>BSplines</td><td>3.219</td><td>868.4e-3</td><td>4.087</td><td>4.087</td><td>76557</td><td>11.34e-6</td><td>2.5347</td><td>172.3</td><td>811.0</td><td>110.6</td></tr><tr><td>MinNorm</td><td>4.267</td><td>248.5e-3</td><td>4.516</td><td>4.516</td><td>20789</td><td>11.95e-6</td><td>2.4977</td><td>49.31</td><td>896.0</td><td>30.02</td></tr><tr><td>MinVolume</td><td>3.619</td><td>301.9e-3</td><td>3.921</td><td>3.921</td><td>25765</td><td>11.72e-6</td><td>2.5005</td><td>59.91</td><td>778.1</td><td>37.21</td></tr><tr><td>ApproxMinVolume</td><td>3.588</td><td>296.0e-3</td><td>3.884</td><td>3.884</td><td>25839</td><td>11.45e-6</td><td>2.5005</td><td>58.73</td><td>770.7</td><td>37.31</td></tr><tr><td>Recursive</td><td>3.512</td><td>286.0e-3</td><td>3.798</td><td>3.798</td><td>25157</td><td>11.37e-6</td><td>2.4963</td><td>56.75</td><td>753.7</td><td>36.33</td></tr><tr><td>Recursive2</td><td>3.714</td><td>316.1e-3</td><td>4.030</td><td>4.030</td><td>26221</td><td>12.06e-6</td><td>2.4963</td><td>62.73</td><td>799.7</td><td>37.86</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>34.09e-6</td><td>95.20</td><td>95.20</td><td>01:35</td><td>10621131</td><td>8.963e-6</td><td>1.4800</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>4.083</td><td>37.65</td><td>41.74</td><td>41.735</td><td>2137275</td><td>17.62e-6</td><td>1.4400</td><td>39.55</td><td>43.84</td><td>20.12</td></tr><tr><td>MinVariance</td><td>5.235</td><td>46.05</td><td>51.28</td><td>51.281</td><td>2563541</td><td>17.96e-6</td><td>1.4436</td><td>48.37</td><td>53.87</td><td>24.14</td></tr><tr><td>BSplines</td><td>4.097</td><td>239.8</td><td>243.9</td><td>04:03</td><td>13596675</td><td>17.64e-6</td><td>1.4884</td><td>251.9</td><td>256.2</td><td>128.0</td></tr><tr><td>MinNorm</td><td>5.067</td><td>40.68</td><td>45.75</td><td>45.750</td><td>2341173</td><td>17.38e-6</td><td>1.4422</td><td>42.73</td><td>48.06</td><td>22.04</td></tr><tr><td>MinVolume</td><td>4.540</td><td>54.99</td><td>59.53</td><td>59.528</td><td>3113329</td><td>17.66e-6</td><td>1.4475</td><td>57.76</td><td>62.53</td><td>29.31</td></tr><tr><td>ApproxMinVolume</td><td>4.727</td><td>55.92</td><td>60.65</td><td>01:00</td><td>3128003</td><td>17.88e-6</td><td>1.4475</td><td>58.74</td><td>63.70</td><td>29.45</td></tr><tr><td>Recursive</td><td>4.117</td><td>47.86</td><td>51.97</td><td>51.974</td><td>2762643</td><td>17.32e-6</td><td>1.4400</td><td>50.27</td><td>54.59</td><td>26.01</td></tr><tr><td>Recursive2</td><td>4.150</td><td>72.52</td><td>76.67</td><td>01:16</td><td>4081563</td><td>17.77e-6</td><td>1.4400</td><td>76.18</td><td>80.54</td><td>38.43</td></tr><tr><td>Interval</td><td>20.98e-6</td><td>12162</td><td>12162</td><td>03:22:42</td><td>1253315413</td><td>9.704e-6</td><td>1.2690</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>54.84</td><td>10305</td><td>10360</td><td>02:52:40</td><td>240612025</td><td>42.83e-6</td><td>1.2322</td><td>84.73</td><td>85.18</td><td>19.20</td></tr><tr><td>MinVariance</td><td>59.96</td><td>13151</td><td>13211</td><td>03:40:11</td><td>301916707</td><td>43.56e-6</td><td>1.2361</td><td>108.1</td><td>108.6</td><td>24.09</td></tr><tr><td>BSplines</td><td>53.41</td><td>93187</td><td>93240</td><td>1-01:54:00</td><td>2218459037</td><td>42.01e-6</td><td>1.2853</td><td>766.2</td><td>766.6</td><td>177.0</td></tr><tr><td>MinNorm</td><td>59.29</td><td>11602</td><td>11661</td><td>03:14:21</td><td>271019129</td><td>42.81e-6</td><td>1.2342</td><td>95.40</td><td>95.88</td><td>21.62</td></tr><tr><td>MinVolume</td><td>56.84</td><td>16282</td><td>16339</td><td>04:32:19</td><td>380170441</td><td>42.83e-6</td><td>1.2396</td><td>133.9</td><td>134.3</td><td>30.33</td></tr><tr><td>ApproxMinVolume</td><td>57.40</td><td>16292</td><td>16349</td><td>04:32:29</td><td>381984097</td><td>42.65e-6</td><td>1.2397</td><td>134.0</td><td>134.4</td><td>30.48</td></tr><tr><td>Recursive</td><td>54.73</td><td>13669</td><td>13724</td><td>03:48:44</td><td>319837953</td><td>42.74e-6</td><td>1.2322</td><td>112.4</td><td>112.8</td><td>25.52</td></tr><tr><td>Recursive2</td><td>55.35</td><td>22787</td><td>22843</td><td>06:20:43</td><td>538867919</td><td>42.29e-6</td><td>1.2322</td><td>187.4</td><td>187.8</td><td>43.00</td></tr></tbody></table>
<table><caption>Table 17: Bissection of problem 2D with problem: 6, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H-M-S.ms)</th><th>nb.iter</th><th>time_per_iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>26.94e-6</td><td>3.154e-3</td><td>3.181e-3</td><td>0.003</td><td>89</td><td>35.44e-6</td><td>infeasible</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>819.9e-3</td><td>3.422e-3</td><td>823.4e-3</td><td>0.823</td><td>65</td><td>52.65e-6</td><td>infeasible</td><td>108.5</td><td>25884.0</td><td>73.03</td></tr><tr><td>MinVariance</td><td>898.9e-3</td><td>3.175e-3</td><td>902.1e-3</td><td>0.902</td><td>61</td><td>52.05e-6</td><td>infeasible</td><td>100.7</td><td>28358.5</td><td>68.54</td></tr><tr><td>BSplines</td><td>823.9e-3</td><td>3.510e-3</td><td>827.4e-3</td><td>0.827</td><td>111</td><td>31.62e-6</td><td>infeasible</td><td>111.3</td><td>26011.4</td><td>124.7</td></tr><tr><td>MinNorm</td><td>974.2e-3</td><td>2.921e-3</td><td>977.1e-3</td><td>0.977</td><td>59</td><td>49.51e-6</td><td>infeasible</td><td>92.61</td><td>30718.0</td><td>66.29</td></tr><tr><td>MinVolume</td><td>955.8e-3</td><td>3.215e-3</td><td>959.0e-3</td><td>0.959</td><td>69</td><td>46.60e-6</td><td>infeasible</td><td>101.9</td><td>30147.5</td><td>77.53</td></tr><tr><td>ApproxMinVolume</td><td>942.6e-3</td><td>2.806e-3</td><td>945.5e-3</td><td>0.945</td><td>69</td><td>40.67e-6</td><td>infeasible</td><td>88.97</td><td>29722.0</td><td>77.53</td></tr><tr><td>Recursive</td><td>889.0e-3</td><td>3.743e-3</td><td>892.7e-3</td><td>0.893</td><td>91</td><td>41.13e-6</td><td>infeasible</td><td>118.7</td><td>28064.3</td><td>102.2</td></tr><tr><td>Recursive2</td><td>880.0e-3</td><td>3.288e-3</td><td>883.3e-3</td><td>0.883</td><td>119</td><td>27.63e-6</td><td>infeasible</td><td>104.2</td><td>27767.1</td><td>133.7</td></tr><tr><td>Interval</td><td>27.89e-6</td><td>1.662</td><td>1.662</td><td>1.662</td><td>229895</td><td>7.231e-6</td><td>4.0146</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.005</td><td>743.2e-3</td><td>3.748</td><td>3.748</td><td>67617</td><td>10.99e-6</td><td>4.0092</td><td>44.71</td><td>225.5</td><td>29.41</td></tr><tr><td>MinVariance</td><td>4.455</td><td>912.8e-3</td><td>5.367</td><td>5.367</td><td>77275</td><td>11.81e-6</td><td>4.0097</td><td>54.91</td><td>322.9</td><td>33.61</td></tr><tr><td>BSplines</td><td>3.210</td><td>2.946</td><td>6.156</td><td>6.156</td><td>273987</td><td>10.75e-6</td><td>4.0148</td><td>177.2</td><td>370.3</td><td>119.2</td></tr><tr><td>MinNorm</td><td>4.235</td><td>817.5e-3</td><td>5.053</td><td>5.053</td><td>72515</td><td>11.27e-6</td><td>4.0093</td><td>49.18</td><td>304.0</td><td>31.54</td></tr><tr><td>MinVolume</td><td>3.594</td><td>963.4e-3</td><td>4.558</td><td>4.558</td><td>88901</td><td>10.84e-6</td><td>4.0102</td><td>57.96</td><td>274.2</td><td>38.67</td></tr><tr><td>ApproxMinVolume</td><td>3.582</td><td>966.8e-3</td><td>4.548</td><td>4.548</td><td>89173</td><td>10.84e-6</td><td>4.0102</td><td>58.16</td><td>273.6</td><td>38.79</td></tr><tr><td>Recursive</td><td>3.497</td><td>761.8e-3</td><td>4.259</td><td>4.259</td><td>71131</td><td>10.71e-6</td><td>4.0092</td><td>45.83</td><td>256.2</td><td>30.94</td></tr><tr><td>Recursive2</td><td>3.710</td><td>760.7e-3</td><td>4.471</td><td>4.471</td><td>70305</td><td>10.82e-6</td><td>4.0092</td><td>45.76</td><td>269.0</td><td>30.58</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>20.98e-6</td><td>2088</td><td>2088</td><td>34.48</td><td>247582197</td><td>8.435e-6</td><td>2.3843</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.986</td><td>446.5</td><td>450.4</td><td>07.30</td><td>25986905</td><td>17.18e-6</td><td>2.3649</td><td>21.38</td><td>21.57</td><td>10.50</td></tr><tr><td>MinVariance</td><td>5.106</td><td>527.3</td><td>532.4</td><td>08.52</td><td>31011033</td><td>17.00e-6</td><td>2.3654</td><td>25.25</td><td>25.50</td><td>12.53</td></tr><tr><td>BSplines</td><td>3.969</td><td>2150</td><td>2154</td><td>35.54</td><td>132011683</td><td>16.29e-6</td><td>2.3716</td><td>103.0</td><td>103.2</td><td>53.32</td></tr><tr><td>MinNorm</td><td>4.994</td><td>510.1</td><td>515.1</td><td>08.35</td><td>28753529</td><td>17.74e-6</td><td>2.3653</td><td>24.43</td><td>24.67</td><td>11.61</td></tr><tr><td>MinVolume</td><td>4.452</td><td>613.7</td><td>618.2</td><td>10.18</td><td>36155657</td><td>16.97e-6</td><td>2.3659</td><td>29.39</td><td>29.60</td><td>14.60</td></tr><tr><td>ApproxMinVolume</td><td>4.677</td><td>627.9</td><td>632.5</td><td>10.32</td><td>36282015</td><td>17.31e-6</td><td>2.3659</td><td>30.07</td><td>30.29</td><td>14.65</td></tr><tr><td>Recursive</td><td>4.059</td><td>450.7</td><td>454.8</td><td>07.34</td><td>26604143</td><td>16.94e-6</td><td>2.3649</td><td>21.58</td><td>21.78</td><td>10.75</td></tr><tr><td>Recursive2</td><td>4.003</td><td>469.3</td><td>473.3</td><td>07.53</td><td>28214921</td><td>16.63e-6</td><td>2.3649</td><td>22.47</td><td>22.66</td><td>11.40</td></tr><tr><td>Interval</td><td>29.00e-6</td><td>437576</td><td>437576</td><td>5-01:32:56</td><td>44237030546</td><td>9.892e-6</td><td>1.6158</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>60.27</td><td>489912</td><td>489912</td><td>5-16:05:12</td><td>11937538665</td><td>41.04e-6</td><td>1.6106</td><td>112.0</td><td>112.0</td><td>26.99</td></tr><tr><td>MinVariance</td><td>65.25</td><td>622799</td><td>622799</td><td>7-04:59:59</td><td>14924760996</td><td>41.73e-6</td><td>1.6112</td><td>142.3</td><td>142.3</td><td>33.74</td></tr><tr><td>BSplines</td><td>68.47</td><td>3654280</td><td>3654280</td><td>42-07:04:40</td><td>89111525282</td><td>41.01e-6</td><td>1.6186</td><td>835.1</td><td>835.1</td><td>201.4</td></tr><tr><td>MinNorm</td><td>74.44</td><td>564243</td><td>564243</td><td>6-12:44:03</td><td>13505303842</td><td>41.78e-6</td><td>1.6109</td><td>128.9</td><td>128.9</td><td>30.53</td></tr><tr><td>MinVolume</td><td>71.17</td><td>750205</td><td>750206</td><td>8-16:23:26</td><td>18201508396</td><td>41.22e-6</td><td>1.6118</td><td>171.4</td><td>171.4</td><td>41.15</td></tr><tr><td>ApproxMinVolume</td><td>70.25</td><td>763912</td><td>763912</td><td>8-20:11:52</td><td>18301096962</td><td>41.74e-6</td><td>1.6118</td><td>174.6</td><td>174.6</td><td>41.37</td></tr><tr><td>Recursive</td><td>63.53</td><td>497551</td><td>497551</td><td>5-18:12:31</td><td>11956885773</td><td>41.61e-6</td><td>1.6106</td><td>113.7</td><td>113.7</td><td>27.03</td></tr><tr><td>Recursive2</td><td>64.05</td><td>546542</td><td>546542</td><td>6-07:49:02</td><td>13103385566</td><td>41.71e-6</td><td>1.6106</td><td>124.9</td><td>124.9</td><td>29.62</td></tr><tr><td rowspan="20">5</td><td rowspan="10">4</td><td>Interval</td><td>30.04e-6</td><td>3.302e-3</td><td>3.332e-3</td><td>0.003</td><td>89</td><td>37.10e-6</td><td>infeasible</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.082</td><td>3.956e-3</td><td>1.086</td><td>1.086</td><td>65</td><td>60.86e-6</td><td>infeasible</td><td>119.8</td><td>326001</td><td>73.03</td></tr><tr><td>MinVariance</td><td>1.165</td><td>3.361e-3</td><td>1.168</td><td>1.168</td><td>61</td><td>55.10e-6</td><td>infeasible</td><td>101.8</td><td>35061.3</td><td>68.54</td></tr><tr><td>BSplines</td><td>1.105</td><td>4.584e-3</td><td>1.110</td><td>1.110</td><td>111</td><td>41.30e-6</td><td>infeasible</td><td>138.8</td><td>33313.4</td><td>124.7</td></tr><tr><td>MinNorm</td><td>1.147</td><td>3.654e-3</td><td>1.150</td><td>1.150</td><td>59</td><td>61.93e-6</td><td>infeasible</td><td>110.7</td><td>34517.8</td><td>66.29</td></tr><tr><td>MinVolume</td><td>1.165</td><td>3.434e-3</td><td>1.168</td><td>1.168</td><td>69</td><td>49.77e-6</td><td>infeasible</td><td>104.0</td><td>35062.8</td><td>77.53</td></tr><tr><td>ApproxMinVolume</td><td>1.892</td><td>3.181e-3</td><td>1.895</td><td>1.895</td><td>69</td><td>46.10e-6</td><td>infeasible</td><td>96.33</td><td>56872.8</td><td>77.53</td></tr><tr><td>Recursive</td><td>1.892</td><td>7.014e-3</td><td>1.899</td><td>1.899</td><td>91</td><td>77.08e-6</td><td>infeasible</td><td>212.4</td><td>56999.7</td><td>102.2</td></tr><tr><td>Recursive2</td><td>1.892</td><td>8.812e-3</td><td>1.901</td><td>1.901</td><td>119</td><td>74.05e-6</td><td>infeasible</td><td>266.9</td><td>57055.2</td><td>133.7</td></tr><tr><td>Interval</td><td>30.04e-6</td><td>191.8e-3</td><td>191.8e-3</td><td>0.192</td><td>26219</td><td>7.314e-6</td><td>4.0953</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.024</td><td>103.0e-3</td><td>3.127</td><td>3.127</td><td>9415</td><td>10.94e-6</td><td>0.4529</td><td>53.71</td><td>1634.6</td><td>35.91</td></tr><tr><td>MinVariance</td><td>4.469</td><td>124.5e-3</td><td>4.593</td><td>4.593</td><td>10443</td><td>11.92e-6</td><td>0.4544</td><td>64.90</td><td>2394.9</td><td>39.83</td></tr><tr><td>BSplines</td><td>3.212</td><td>357.5e-3</td><td>3.570</td><td>3.570</td><td>32731</td><td>10.92e-6</td><td>0.4987</td><td>186.5</td><td>1861.2</td><td>124.8</td></tr><tr><td>MinNorm</td><td>4.231</td><td>116.2e-3</td><td>4.347</td><td>4.347</td><td>9975</td><td>11.65e-6</td><td>0.4537</td><td>60.62</td><td>2266.8</td><td>38.04</td></tr><tr><td>MinVolume</td><td>3.621</td><td>138.2e-3</td><td>3.759</td><td>3.759</td><td>12107</td><td>11.42e-6</td><td>0.4599</td><td>72.08</td><td>1960.1</td><td>46.18</td></tr><tr><td>ApproxMinVolume</td><td>3.565</td><td>136.4e-3</td><td>3.701</td><td>3.701</td><td>12163</td><td>11.21e-6</td><td>0.4599</td><td>71.11</td><td>1929.7</td><td>46.39</td></tr><tr><td>Recursive</td><td>3.501</td><td>137.1e-3</td><td>3.638</td><td>3.638</td><td>11729</td><td>11.69e-6</td><td>0.4529</td><td>71.51</td><td>1896.9</td><td>44.73</td></tr><tr><td>Recursive2</td><td>3.741</td><td>136.3e-3</td><td>3.877</td><td>3.877</td><td>11957</td><td>11.40e-6</td><td>0.4529</td><td>71.08</td><td>2021.4</td><td>45.60</td></tr><tr><td>Interval</td><td>30.04e-6</td><td>152.3</td><td>152.3</td><td>02:32</td><td>18472325</td><td>8.242e-6</td><td>2.6074</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>4.047</td><td>31.96</td><td>36.01</td><td>36.006</td><td>1787729</td><td>17.88e-6</td><td>2.4438</td><td>20.99</td><td>23.65</td><td>9.678</td></tr><tr><td>MinVariance</td><td>5.216</td><td>37.37</td><td>42.59</td><td>42.586</td><td>2092077</td><td>17.86e-6</td><td>2.4487</td><td>24.54</td><td>27.97</td><td>11.33</td></tr><tr><td>BSplines</td><td>4.077</td><td>155.0</td><td>159.1</td><td>02:39</td><td>8743233</td><td>17.73e-6</td><td>2.4946</td><td>101.8</td><td>104.5</td><td>47.33</td></tr><tr><td>MinNorm</td><td>5.104</td><td>35.58</td><td>40.69</td><td>40.686</td><td>1935949</td><td>18.38e-6</td><td>2.4463</td><td>23.37</td><td>26.72</td><td>10.48</td></tr><tr><td>MinVolume</td><td>4.564</td><td>42.21</td><td>46.77</td><td>46.771</td><td>2416379</td><td>17.47e-6</td><td>2.4509</td><td>27.72</td><td>30.72</td><td>13.08</td></tr><tr><td>ApproxMinVolume</td><td>4.707</td><td>43.08</td><td>47.79</td><td>47.791</td><td>2424401</td><td>17.77e-6</td><td>2.4510</td><td>28.30</td><td>31.39</td><td>13.12</td></tr><tr><td>Recursive</td><td>4.126</td><td>38.99</td><td>43.12</td><td>43.121</td><td>2228407</td><td>17.50e-6</td><td>2.4438</td><td>25.61</td><td>28.32</td><td>12.06</td></tr><tr><td>Recursive2</td><td>4.118</td><td>48.81</td><td>52.92</td><td>52.925</td><td>2756589</td><td>17.71e-6</td><td>2.4438</td><td>32.06</td><td>34.76</td><td>14.92</td></tr><tr><td rowspan="10">0.01</td><td rowspan="10">2</td><td>Interval</td><td>26.94e-6</td><td>6081</td><td>6081</td><td>01:41:21</td><td>625471221</td><td>9.723e-6</td><td>1.7103</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>56.13</td><td>7016</td><td>7072</td><td>01:57:52</td><td>165009075</td><td>42.52e-6</td><td>1.6674</td><td>115.4</td><td>116.3</td><td>26.38</td></tr><tr><td>MinVariance</td><td>60.17</td><td>9070</td><td>9130</td><td>02:32:10</td><td>209533509</td><td>43.29e-6</td><td>1.6722</td><td>149.1</td><td>150.1</td><td>33.50</td></tr><tr><td>BSplines</td><td>55.16</td><td>61689</td><td>61744</td><td>17:09:04</td><td>1459619547</td><td>42.26e-6</td><td>1.7288</td><td>1014.4</td><td>1015.3</td><td>233.4</td></tr><tr><td>MinNorm</td><td>59.41</td><td>7995</td><td>8055</td><td>02:14:15</td><td>187988639</td><td>42.53e-6</td><td>1.6702</td><td>131.5</td><td>132.5</td><td>30.06</td></tr><tr><td>MinVolume</td><td>57.28</td><td>11383</td><td>11440</td><td>03:10:40</td><td>261910657</td><td>43.46e-6</td><td>1.6761</td><td>187.2</td><td>188.1</td><td>41.87</td></tr><tr><td>ApproxMinVolume</td><td>55.50</td><td>11065</td><td>11121</td><td>03:05:21</td><td>263267293</td><td>42.03e-6</td><td>1.6762</td><td>182.0</td><td>182.9</td><td>42.09</td></tr><tr><td>Recursive</td><td>56.42</td><td>7498</td><td>7554</td><td>02:05:54</td><td>177834483</td><td>42.16e-6</td><td>1.6674</td><td>123.3</td><td>124.2</td><td>28.43</td></tr><tr><td>Recursive2</td><td>55.16</td><td>15240</td><td>15295</td><td>04:14:55</td><td>352701941</td><td>43.21e-6</td><td>1.6674</td><td>250.6</td><td>251.5</td><td>56.39</td></tr></tbody></table>
<table><caption>Table 18: Bissection of problem 2D with problem: 7, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time.per.iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>31.95e-6</td><td>29.86e-3</td><td>29.89e-3</td><td>0.030</td><td>3901</td><td>7.654e-6</td><td>755.10e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>875.2e-3</td><td>6.288e-3</td><td>881.5e-3</td><td>0.882</td><td>839</td><td>7.495e-6</td><td>746.14e-3</td><td>21.06</td><td>2949.1</td><td>21.51</td></tr><tr><td>MinVariance</td><td>981.9e-3</td><td>6.032e-3</td><td>987.9e-3</td><td>0.988</td><td>887</td><td>6.801e-6</td><td>750.68e-3</td><td>20.20</td><td>3305.0</td><td>22.74</td></tr><tr><td>BSplines</td><td>909.9e-3</td><td>10.08e-3</td><td>920.0e-3</td><td>0.920</td><td>1861</td><td>5.419e-6</td><td>755.93e-3</td><td>33.77</td><td>3077.6</td><td>47.71</td></tr><tr><td>MinNorm</td><td>976.3e-3</td><td>5.548e-3</td><td>981.9e-3</td><td>0.982</td><td>851</td><td>6.519e-6</td><td>750.65e-3</td><td>18.58</td><td>3284.7</td><td>21.81</td></tr><tr><td>MinVolume</td><td>942.7e-3</td><td>6.550e-3</td><td>949.3e-3</td><td>0.949</td><td>979</td><td>6.690e-6</td><td>750.74e-3</td><td>21.94</td><td>3175.7</td><td>25.10</td></tr><tr><td>ApproxMinVolume</td><td>928.7e-3</td><td>6.837e-3</td><td>935.5e-3</td><td>0.936</td><td>983</td><td>6.955e-6</td><td>750.74e-3</td><td>22.90</td><td>3129.6</td><td>25.20</td></tr><tr><td>Recursive</td><td>892.3e-3</td><td>6.743e-3</td><td>899.1e-3</td><td>0.899</td><td>989</td><td>6.818e-6</td><td>746.14e-3</td><td>22.58</td><td>3007.8</td><td>25.35</td></tr><tr><td>Recursive2</td><td>892.4e-3</td><td>6.848e-3</td><td>899.3e-3</td><td>0.899</td><td>1021</td><td>6.707e-6</td><td>746.14e-3</td><td>22.93</td><td>3008.5</td><td>26.17</td></tr><tr><td>Interval</td><td>27.89e-6</td><td>6.309</td><td>6.309</td><td>6.309</td><td>881625</td><td>7.156e-6</td><td>531.59e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.045</td><td>2.613</td><td>5.658</td><td>5.658</td><td>243299</td><td>10.74e-6</td><td>528.78e-3</td><td>41.43</td><td>89.69</td><td>27.60</td></tr><tr><td>MinVariance</td><td>4.496</td><td>3.277</td><td>7.773</td><td>7.773</td><td>283695</td><td>11.55e-6</td><td>529.10e-3</td><td>51.94</td><td>123.2</td><td>32.18</td></tr><tr><td>BSplines</td><td>3.242</td><td>10.60</td><td>13.84</td><td>13.838</td><td>939967</td><td>11.27e-6</td><td>532.31e-3</td><td>168.0</td><td>219.3</td><td>106.6</td></tr><tr><td>MinNorm</td><td>4.278</td><td>3.022</td><td>7.300</td><td>7.300</td><td>265135</td><td>11.40e-6</td><td>528.95e-3</td><td>47.90</td><td>115.7</td><td>30.07</td></tr><tr><td>MinVolume</td><td>3.646</td><td>3.585</td><td>7.231</td><td>7.231</td><td>322223</td><td>11.13e-6</td><td>529.32e-3</td><td>56.83</td><td>114.6</td><td>36.55</td></tr><tr><td>ApproxMinVolume</td><td>3.607</td><td>3.635</td><td>7.242</td><td>7.242</td><td>323159</td><td>11.25e-6</td><td>529.34e-3</td><td>57.62</td><td>114.8</td><td>36.65</td></tr><tr><td>Recursive</td><td>3.513</td><td>2.864</td><td>6.377</td><td>6.377</td><td>258469</td><td>11.08e-6</td><td>528.78e-3</td><td>45.40</td><td>101.1</td><td>29.32</td></tr><tr><td>Recursive2</td><td>3.743</td><td>2.828</td><td>6.571</td><td>6.571</td><td>251577</td><td>11.24e-6</td><td>528.78e-3</td><td>44.82</td><td>104.2</td><td>28.54</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>28.85e-6</td><td>1094</td><td>1094</td><td>18.14</td><td>131412411</td><td>8.331e-6</td><td>444.32e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.998</td><td>394.7</td><td>398.7</td><td>06:38</td><td>22173355</td><td>17.80e-6</td><td>441.41e-3</td><td>36.05</td><td>36.42</td><td>16.87</td></tr><tr><td>MinVariance</td><td>5.113</td><td>463.3</td><td>468.4</td><td>07:48</td><td>26813445</td><td>17.28e-6</td><td>441.73e-3</td><td>42.32</td><td>42.78</td><td>20.40</td></tr><tr><td>BSplines</td><td>3.989</td><td>2223</td><td>2227</td><td>37:07</td><td>129187203</td><td>17.21e-6</td><td>445.18e-3</td><td>203.1</td><td>203.4</td><td>98.31</td></tr><tr><td>MinNorm</td><td>5.067</td><td>418.4</td><td>423.4</td><td>07:03</td><td>24612139</td><td>17.00e-6</td><td>441.58e-3</td><td>38.21</td><td>38.68</td><td>18.73</td></tr><tr><td>MinVolume</td><td>4.496</td><td>550.6</td><td>555.1</td><td>09:15</td><td>31996365</td><td>17.21e-6</td><td>441.99e-3</td><td>50.29</td><td>50.70</td><td>24.35</td></tr><tr><td>ApproxMinVolume</td><td>4.592</td><td>557.8</td><td>562.4</td><td>09:22</td><td>32131023</td><td>17.36e-6</td><td>441.99e-3</td><td>50.95</td><td>51.37</td><td>24.45</td></tr><tr><td>Recursive</td><td>4.102</td><td>404.3</td><td>408.4</td><td>06:48</td><td>23361233</td><td>17.30e-6</td><td>441.41e-3</td><td>36.93</td><td>37.30</td><td>17.78</td></tr><tr><td>Recursive2</td><td>4.044</td><td>438.4</td><td>442.4</td><td>07:22</td><td>25613173</td><td>17.12e-6</td><td>441.41e-3</td><td>40.04</td><td>40.41</td><td>19.49</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>29.00e-6</td><td>288665</td><td>288665</td><td>3-08:11:05</td><td>29200962790</td><td>9.885e-6</td><td>378.44e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>55.05</td><td>189537</td><td>189537</td><td>2-04:38:57</td><td>4578807797</td><td>41.39e-6</td><td>375.83e-3</td><td>65.66</td><td>65.66</td><td>15.68</td></tr><tr><td>MinVariance</td><td>60.33</td><td>245741</td><td>245741</td><td>2-20:15:41</td><td>5814608735</td><td>42.26e-6</td><td>376.15e-3</td><td>85.13</td><td>85.13</td><td>19.91</td></tr><tr><td>BSplines</td><td>55.12</td><td>1601720</td><td>1601720</td><td>18-12:55:20</td><td>39354884660</td><td>40.70e-6</td><td>379.89e-3</td><td>554.9</td><td>554.9</td><td>134.8</td></tr><tr><td>MinNorm</td><td>59.13</td><td>219163</td><td>219163</td><td>2-12:52:43</td><td>5219983231</td><td>41.99e-6</td><td>376.00e-3</td><td>75.92</td><td>75.92</td><td>17.88</td></tr><tr><td>MinVolume</td><td>56.79</td><td>296661</td><td>296661</td><td>3-10:24:21</td><td>7165627846</td><td>41.40e-6</td><td>376.45e-3</td><td>102.8</td><td>102.8</td><td>24.54</td></tr><tr><td>ApproxMinVolume</td><td>57.26</td><td>286122</td><td>286122</td><td>3-07:28:42</td><td>7197590487</td><td>39.75e-6</td><td>376.46e-3</td><td>99.12</td><td>99.12</td><td>24.65</td></tr><tr><td>Recursive</td><td>54.88</td><td>200772</td><td>200772</td><td>2-07:46:12</td><td>4788156051</td><td>41.93e-6</td><td>375.83e-3</td><td>69.55</td><td>69.55</td><td>16.40</td></tr><tr><td>Recursive2</td><td>55.19</td><td>225061</td><td>225061</td><td>2-14:31:01</td><td>5238902215</td><td>42.96e-6</td><td>375.83e-3</td><td>77.97</td><td>77.97</td><td>17.94</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>27.89e-6</td><td>9.792e-3</td><td>9.820e-3</td><td>0.010</td><td>1173</td><td>8.348e-6</td><td>846.26e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.892</td><td>8.748e-3</td><td>1.901</td><td>1.901</td><td>551</td><td>15.88e-6</td><td>813.57e-3</td><td>89.34</td><td>19354.0</td><td>46.97</td></tr><tr><td>MinVariance</td><td>1.892</td><td>10.20e-3</td><td>1.902</td><td>1.902</td><td>583</td><td>17.50e-6</td><td>814.15e-3</td><td>104.2</td><td>19372.2</td><td>49.70</td></tr><tr><td>BSplines</td><td>1.892</td><td>14.34e-3</td><td>1.907</td><td>1.907</td><td>1231</td><td>11.65e-6</td><td>829.81e-3</td><td>146.5</td><td>19417.2</td><td>104.9</td></tr><tr><td>MinNorm</td><td>1.891</td><td>6.668e-3</td><td>1.898</td><td>1.898</td><td>561</td><td>11.89e-6</td><td>813.89e-3</td><td>68.10</td><td>19329.7</td><td>47.83</td></tr><tr><td>MinVolume</td><td>1.893</td><td>13.70e-3</td><td>1.906</td><td>1.906</td><td>645</td><td>21.24e-6</td><td>814.70e-3</td><td>139.9</td><td>19412.0</td><td>54.99</td></tr><tr><td>ApproxMinVolume</td><td>1.893</td><td>14.43e-3</td><td>1.907</td><td>1.907</td><td>651</td><td>22.17e-6</td><td>814.71e-3</td><td>147.4</td><td>19422.3</td><td>55.50</td></tr><tr><td>Recursive</td><td>1.893</td><td>17.23e-3</td><td>1.910</td><td>1.910</td><td>725</td><td>23.77e-6</td><td>813.57e-3</td><td>176.0</td><td>19453.7</td><td>61.81</td></tr><tr><td>Recursive2</td><td>1.893</td><td>16.31e-3</td><td>1.909</td><td>1.909</td><td>735</td><td>22.20e-6</td><td>813.57e-3</td><td>166.6</td><td>19441.0</td><td>62.66</td></tr><tr><td>Interval</td><td>24.80e-6</td><td>826.8e-3</td><td>826.8e-3</td><td>0.827</td><td>112547</td><td>7.346e-6</td><td>577.23e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td rowspan="10">0.01</td><td rowspan="10">2</td><td>Interval</td><td>2.980</td><td>342.1e-3</td><td>3.322</td><td>3.322</td><td>32889</td><td>10.40e-6</td><td>553.92e-3</td><td>41.38</td><td>401.8</td><td>29.22</td></tr><tr><td>Bernstein</td><td>4.418</td><td>435.2e-3</td><td>4.853</td><td>4.853</td><td>38459</td><td>11.32e-6</td><td>556.24e-3</td><td>52.63</td><td>586.9</td><td>34.17</td></tr><tr><td>MinVariance</td><td>3.186</td><td>1.463</td><td>4.649</td><td>4.649</td><td>140001</td><td>10.45e-6</td><td>581.31e-3</td><td>176.9</td><td>562.2</td><td>124.4</td></tr><tr><td>BSplines</td><td>4.204</td><td>394.7e-3</td><td>4.598</td><td>4.598</td><td>35501</td><td>11.12e-6</td><td>555.33e-3</td><td>47.73</td><td>556.2</td><td>31.54</td></tr><tr><td>MinNorm</td><td>3.583</td><td>482.3e-3</td><td>4.066</td><td>4.066</td><td>44633</td><td>10.81e-6</td><td>558.27e-3</td><td>58.33</td><td>491.7</td><td>39.66</td></tr><tr><td>MinVolume</td><td>3.527</td><td>506.5e-3</td><td>4.033</td><td>4.033</td><td>44815</td><td>11.30e-6</td><td>558.29e-3</td><td>61.26</td><td>487.8</td><td>39.82</td></tr><tr><td>ApproxMinVolume</td><td>3.438</td><td>483.1e-3</td><td>3.921</td><td>3.921</td><td>43007</td><td>11.23e-6</td><td>553.92e-3</td><td>58.42</td><td>474.2</td><td>38.21</td></tr><tr><td>Recursive</td><td>3.649</td><td>454.0e-3</td><td>4.103</td><td>4.103</td><td>40095</td><td>11.32e-6</td><td>553.92e-3</td><td>54.91</td><td>496.2</td><td>35.63</td></tr><tr><td>Recursive2</td><td>40.05e-6</td><td>66.38</td><td>66.38</td><td>01:06</td><td>7999761</td><td>8.298e-6</td><td>493.46e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td rowspan="10">3</td><td rowspan="10">3</td><td>Interval</td><td>4.007</td><td>24.01</td><td>28.01</td><td>28:012</td><td>1395687</td><td>17.20e-6</td><td>468.22e-3</td><td>36.16</td><td>42.20</td><td>17.45</td></tr><tr><td>Bernstein</td><td>5.299</td><td>30.00</td><td>35.30</td><td>35:297</td><td>1650545</td><td>18.17e-6</td><td>470.39e-3</td><td>45.19</td><td>53.17</td><td>20.63</td></tr><tr><td>MinVariance</td><td>4.100</td><td>157.3</td><td>161.4</td><td>02:41</td><td>9124755</td><td>17.24e-6</td><td>496.65e-3</td><td>237.0</td><td>243.1</td><td>114.1</td></tr><tr><td>BSplines</td><td>5.131</td><td>26.27</td><td>31.41</td><td>31.406</td><td>1502657</td><td>17.49e-6</td><td>469.51e-3</td><td>39.58</td><td>47.31</td><td>18.78</td></tr><tr><td>MinNorm</td><td>4.592</td><td>35.12</td><td>39.72</td><td>39.716</td><td>1998391</td><td>17.58e-6</td><td>472.38e-3</td><td>52.91</td><td>59.83</td><td>24.98</td></tr><tr><td>MinVolume</td><td>4.708</td><td>34.67</td><td>39.38</td><td>39.383</td><td>2006989</td><td>17.28e-6</td><td>472.40e-3</td><td>52.24</td><td>59.33</td><td>25.09</td></tr><tr><td>ApproxMinVolume</td><td>4.186</td><td>38.16</td><td>42.35</td><td>42.347</td><td>2226589</td><td>17.14e-6</td><td>468.22e-3</td><td>57.49</td><td>63.80</td><td>27.83</td></tr><tr><td>Recursive</td><td>4.088</td><td>53.97</td><td>58.06</td><td>58.055</td><td>3055941</td><td>17.66e-6</td><td>468.22e-3</td><td>81.30</td><td>87.46</td><td>38.20</td></tr><tr><td>Recursive2</td><td>27.89e-6</td><td>6163</td><td>6163</td><td>01:42:43</td><td>622244871</td><td>9.905e-6</td><td>427.47e-3</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td rowspan="10">4</td><td rowspan="10">4</td><td>Interval</td><td>55.35</td><td>4499</td><td>4555</td><td>01:15:55</td><td>105941253</td><td>42.48e-6</td><td>405.09e-3</td><td>73.01</td><td>73.91</td><td>17.03</td></tr><tr><td>Bernstein</td><td>59.48</td><td>5579</td><td>5638</td><td>01:33:58</td><td>131851157</td><td>42.31e-6</td><td>407.22e-3</td><td>90.52</td><td>91.49</td><td>21.19</td></tr><tr><td>MinVariance</td><td>54.81</td><td>45141</td><td>45196</td><td>12:33:16</td><td>1089856989</td><td>41.42e-6</td><td>436.23e-3</td><td>73.24</td><td>73.33</td><td>175.1</td></tr><tr><td>BSplines</td><td>59.23</td><td>5021</td><td>5081</td><td>01:24:41</td><td>117150599</td><td>42.87e-6</td><td>405.95e-3</td><td>81.48</td><td>82.44</td><td>18.83</td></tr><tr><td>MinNorm</td><td>56.83</td><td>7061</td><td>7118</td><td>01:58:38</td><td>166600157</td><td>42.39e-6</td><td>409.21e-3</td><td>114.6</td><td>115.5</td><td>26.77</td></tr><tr><td>MinVolume</td><td>57.22</td><td>7116</td><td>7173</td><td>01:59:33</td><td>167540789</td><td>42.47e-6</td><td>409.22e-3</td><td>115.5</td><td>116.4</td><td>26.93</td></tr><tr><td>ApproxMinVolume</td><td>53.52</td><td>7134</td><td>7187</td><td>01:59:47</td><td>168848047</td><td>42.25e-6</td><td>405.09e-3</td><td>115.8</td><td>116.6</td><td>27.14</td></tr><tr><td>Recursive</td><td>55.31</td><td>10895</td><td>10950</td><td>03:02:30</td><td>250882225</td><td>43.43e-6</td><td>405.09e-3</td><td>176.8</td><td>177.7</td><td>40.32</td></tr></tbody></table>
<table><caption>Table 19: Bissection of problem 2D with problem: 8, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time_per_iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>35.05e-6</td><td>21.99e-3</td><td>22.02e-3</td><td>0.022</td><td>2865</td><td>7.675e-6</td><td>2.4606</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>878.1e-3</td><td>7.775e-3</td><td>885.9e-3</td><td>0.886</td><td>1363</td><td>5.704e-6</td><td>2.4563</td><td>35.36</td><td>4022.6</td><td>47.57</td></tr><tr><td>MinVariance</td><td>951.0e-3</td><td>8.664e-3</td><td>959.7e-3</td><td>0.960</td><td>1469</td><td>5.898e-6</td><td>2.4564</td><td>39.40</td><td>4357.6</td><td>51.27</td></tr><tr><td>BSplines</td><td>868.8e-3</td><td>13.89e-3</td><td>882.7e-3</td><td>0.883</td><td>2817</td><td>4.930e-6</td><td>2.4598</td><td>63.15</td><td>4008.0</td><td>98.32</td></tr><tr><td>MinNorm</td><td>941.1e-3</td><td>8.407e-3</td><td>949.6e-3</td><td>0.950</td><td>1415</td><td>5.941e-6</td><td>2.4564</td><td>38.23</td><td>4311.5</td><td>49.39</td></tr><tr><td>MinVolume</td><td>911.1e-3</td><td>8.794e-3</td><td>919.9e-3</td><td>0.920</td><td>1623</td><td>5.418e-6</td><td>2.4571</td><td>39.99</td><td>4176.7</td><td>56.65</td></tr><tr><td>ApproxMinVolume</td><td>898.9e-3</td><td>8.709e-3</td><td>907.7e-3</td><td>0.908</td><td>1623</td><td>5.366e-6</td><td>2.4571</td><td>39.61</td><td>4121.2</td><td>56.65</td></tr><tr><td>Recursive</td><td>873.7e-3</td><td>8.270e-3</td><td>881.9e-3</td><td>0.882</td><td>1467</td><td>5.637e-6</td><td>2.4563</td><td>37.61</td><td>4004.5</td><td>51.20</td></tr><tr><td>Recursive2</td><td>878.8e-3</td><td>8.374e-3</td><td>887.2e-3</td><td>0.887</td><td>1479</td><td>5.662e-6</td><td>2.4563</td><td>38.08</td><td>4028.2</td><td>51.62</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>32.90e-6</td><td>4.437</td><td>4.437</td><td>4.437</td><td>624303</td><td>7.108e-6</td><td>2.2083</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.050</td><td>1.389</td><td>4.439</td><td>4.439</td><td>135837</td><td>10.22e-6</td><td>2.2022</td><td>31.30</td><td>100.0</td><td>21.76</td></tr><tr><td>MinVariance</td><td>4.476</td><td>1.774</td><td>6.251</td><td>6.251</td><td>158425</td><td>11.20e-6</td><td>2.2026</td><td>39.98</td><td>140.9</td><td>25.38</td></tr><tr><td>BSplines</td><td>3.219</td><td>5.687</td><td>8.906</td><td>8.906</td><td>541987</td><td>10.49e-6</td><td>2.2060</td><td>128.2</td><td>200.7</td><td>86.81</td></tr><tr><td>MinNorm</td><td>4.249</td><td>1.666</td><td>5.916</td><td>5.916</td><td>148295</td><td>11.24e-6</td><td>2.2024</td><td>37.55</td><td>133.3</td><td>23.75</td></tr><tr><td>MinVolume</td><td>3.631</td><td>1.998</td><td>5.628</td><td>5.628</td><td>182409</td><td>10.95e-6</td><td>2.2028</td><td>45.02</td><td>126.8</td><td>29.22</td></tr><tr><td>ApproxMinVolume</td><td>3.575</td><td>1.931</td><td>5.506</td><td>5.506</td><td>182933</td><td>10.55e-6</td><td>2.2028</td><td>43.51</td><td>124.1</td><td>29.30</td></tr><tr><td>Recursive</td><td>3.496</td><td>1.525</td><td>5.020</td><td>5.020</td><td>142967</td><td>10.66e-6</td><td>2.2022</td><td>34.36</td><td>113.1</td><td>22.90</td></tr><tr><td>Recursive2</td><td>3.721</td><td>1.557</td><td>5.278</td><td>5.278</td><td>142927</td><td>10.89e-6</td><td>2.2022</td><td>35.09</td><td>119.0</td><td>22.89</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>29.09e-6</td><td>1724</td><td>1724</td><td>28.44</td><td>213360751</td><td>8.081e-6</td><td>1.2199</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.937</td><td>831.2</td><td>835.1</td><td>13.55</td><td>50454883</td><td>16.47e-6</td><td>1.2155</td><td>48.21</td><td>48.44</td><td>23.65</td></tr><tr><td>MinVariance</td><td>5.187</td><td>1042</td><td>1047</td><td>17.27</td><td>61013369</td><td>17.09e-6</td><td>1.2160</td><td>60.47</td><td>60.77</td><td>28.60</td></tr><tr><td>BSplines</td><td>4.039</td><td>4407</td><td>4411</td><td>01:13:31</td><td>277497315</td><td>15.88e-6</td><td>1.2216</td><td>255.6</td><td>255.9</td><td>130.1</td></tr><tr><td>MinNorm</td><td>5.032</td><td>967.2</td><td>972.2</td><td>16:12</td><td>56014997</td><td>17.27e-6</td><td>1.2157</td><td>56.10</td><td>56.39</td><td>26.25</td></tr><tr><td>MinVolume</td><td>4.518</td><td>1232</td><td>1236</td><td>20:36</td><td>72210907</td><td>17.07e-6</td><td>1.2164</td><td>71.47</td><td>71.74</td><td>33.84</td></tr><tr><td>ApproxMinVolume</td><td>4.615</td><td>1246</td><td>1250</td><td>20:50</td><td>72439199</td><td>17.20e-6</td><td>1.2164</td><td>72.29</td><td>72.55</td><td>33.95</td></tr><tr><td>Recursive</td><td>4.021</td><td>864.4</td><td>868.4</td><td>14:28</td><td>51056239</td><td>16.93e-6</td><td>1.2155</td><td>50.14</td><td>50.37</td><td>23.93</td></tr><tr><td>Recursive2</td><td>4.075</td><td>928.0</td><td>932.1</td><td>15:32</td><td>54123815</td><td>17.15e-6</td><td>1.2155</td><td>53.83</td><td>54.06</td><td>25.37</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>28.00e-6</td><td>756744</td><td>756744</td><td>8-18:12:24</td><td>77832279864</td><td>9.723e-6</td><td>1.0322</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>54.95</td><td>603329</td><td>603330</td><td>6-23:35:30</td><td>15121569445</td><td>39.90e-6</td><td>1.0281</td><td>79.73</td><td>79.73</td><td>19.43</td></tr><tr><td>MinVariance</td><td>59.79</td><td>776113</td><td>776113</td><td>8-23:35:13</td><td>19026489275</td><td>40.79e-6</td><td>1.0287</td><td>102.6</td><td>102.6</td><td>24.45</td></tr><tr><td>BSplines</td><td>54.96</td><td>4920300</td><td>4920300</td><td>56-22:45:00</td><td>121021741505</td><td>40.66e-6</td><td>1.0350</td><td>650.2</td><td>650.2</td><td>155.5</td></tr><tr><td>MinNorm</td><td>59.30</td><td>711755</td><td>711755</td><td>8-05:42:35</td><td>17136869562</td><td>41.53e-6</td><td>1.0284</td><td>94.05</td><td>94.05</td><td>22.02</td></tr><tr><td>MinVolume</td><td>57.06</td><td>959119</td><td>959119</td><td>11-02:25:19</td><td>23305249196</td><td>41.15e-6</td><td>1.0292</td><td>126.7</td><td>126.7</td><td>29.94</td></tr><tr><td>ApproxMinVolume</td><td>57.50</td><td>976462</td><td>976463</td><td>11-07:14:23</td><td>23435498886</td><td>41.67e-6</td><td>1.0292</td><td>129.0</td><td>129.0</td><td>30.11</td></tr><tr><td>Recursive</td><td>55.27</td><td>637856</td><td>637856</td><td>7-09:10:56</td><td>15336253901</td><td>41.60e-6</td><td>1.0281</td><td>84.29</td><td>84.29</td><td>19.70</td></tr><tr><td>Recursive2</td><td>55.29</td><td>675568</td><td>675568</td><td>7-19:39:28</td><td>16431248947</td><td>41.11e-6</td><td>1.0281</td><td>89.27</td><td>89.27</td><td>21.11</td></tr><tr><td rowspan="10">0.01</td><td rowspan="10">2</td><td>Interval</td><td>29.09e-6</td><td>7.103e-3</td><td>7.132e-3</td><td>0.007</td><td>685</td><td>10.37e-6</td><td>2.5169</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.893</td><td>4.445e-3</td><td>1.897</td><td>1.897</td><td>337</td><td>13.19e-6</td><td>2.4865</td><td>62.58</td><td>26605.3</td><td>49.20</td></tr><tr><td>MinVariance</td><td>1.889</td><td>14.20e-3</td><td>1.903</td><td>1.903</td><td>343</td><td>41.41e-6</td><td>2.4875</td><td>200.0</td><td>26687.7</td><td>50.07</td></tr><tr><td>BSplines</td><td>1.892</td><td>13.75e-3</td><td>1.906</td><td>1.906</td><td>703</td><td>19.56e-6</td><td>2.5101</td><td>193.6</td><td>26725.0</td><td>102.6</td></tr><tr><td>MinNorm</td><td>2.166</td><td>6.107e-3</td><td>2.172</td><td>2.172</td><td>331</td><td>18.45e-6</td><td>2.4871</td><td>85.98</td><td>30453.0</td><td>48.32</td></tr><tr><td>MinVolume</td><td>2.167</td><td>11.74e-3</td><td>2.179</td><td>2.179</td><td>383</td><td>30.66e-6</td><td>2.4885</td><td>165.3</td><td>30546.9</td><td>55.91</td></tr><tr><td>ApproxMinVolume</td><td>2.167</td><td>7.666e-3</td><td>2.175</td><td>2.175</td><td>383</td><td>20.02e-6</td><td>2.4885</td><td>107.9</td><td>30494.0</td><td>55.91</td></tr><tr><td>Recursive</td><td>2.167</td><td>6.831e-3</td><td>2.174</td><td>2.174</td><td>411</td><td>16.62e-6</td><td>2.4865</td><td>96.17</td><td>30476.1</td><td>60.00</td></tr><tr><td>Recursive2</td><td>2.166</td><td>4.908e-3</td><td>2.171</td><td>2.171</td><td>449</td><td>10.93e-6</td><td>2.4865</td><td>69.10</td><td>30442.0</td><td>65.55</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>26.94e-6</td><td>657.8e-3</td><td>657.8e-3</td><td>0.658</td><td>98143</td><td>6.703e-6</td><td>2.2797</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.904</td><td>296.7e-3</td><td>3.201</td><td>3.201</td><td>26149</td><td>11.35e-6</td><td>2.2314</td><td>45.10</td><td>486.6</td><td>26.64</td></tr><tr><td>MinVariance</td><td>4.409</td><td>357.4e-3</td><td>4.766</td><td>4.766</td><td>29819</td><td>11.98e-6</td><td>2.2340</td><td>54.33</td><td>724.5</td><td>30.38</td></tr><tr><td>BSplines</td><td>3.196</td><td>1.159</td><td>4.356</td><td>4.356</td><td>100765</td><td>11.50e-6</td><td>2.2655</td><td>176.2</td><td>662.1</td><td>102.7</td></tr><tr><td>MinNorm</td><td>4.253</td><td>343.8e-3</td><td>4.597</td><td>4.597</td><td>28013</td><td>12.27e-6</td><td>2.2334</td><td>52.27</td><td>698.8</td><td>28.54</td></tr><tr><td>MinVolume</td><td>3.645</td><td>388.9e-3</td><td>4.034</td><td>4.034</td><td>33971</td><td>11.45e-6</td><td>2.2350</td><td>59.11</td><td>613.2</td><td>34.61</td></tr><tr><td>ApproxMinVolume</td><td>3.589</td><td>400.0e-3</td><td>3.989</td><td>3.989</td><td>34081</td><td>11.74e-6</td><td>2.2351</td><td>60.81</td><td>606.4</td><td>34.73</td></tr><tr><td>Recursive</td><td>3.515</td><td>363.4e-3</td><td>3.878</td><td>3.878</td><td>31559</td><td>11.51e-6</td><td>2.2314</td><td>55.24</td><td>589.5</td><td>32.16</td></tr><tr><td>Recursive2</td><td>3.744</td><td>379.0e-3</td><td>4.123</td><td>4.123</td><td>32767</td><td>11.57e-6</td><td>2.2314</td><td>57.62</td><td>626.7</td><td>33.39</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>25.99e-6</td><td>93.35</td><td>93.35</td><td>01:33</td><td>11048427</td><td>8.449e-6</td><td>1.2952</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>4.059</td><td>43.40</td><td>47.46</td><td>47.462</td><td>2422819</td><td>17.91e-6</td><td>1.2587</td><td>46.49</td><td>50.84</td><td>21.93</td></tr><tr><td>MinVariance</td><td>5.260</td><td>53.05</td><td>58.31</td><td>58.310</td><td>2939667</td><td>18.05e-6</td><td>1.2628</td><td>56.83</td><td>62.46</td><td>26.61</td></tr><tr><td>BSplines</td><td>4.070</td><td>243.0</td><td>247.0</td><td>04:07</td><td>13938915</td><td>17.43e-6</td><td>1.3068</td><td>260.3</td><td>264.6</td><td>126.2</td></tr><tr><td>MinNorm</td><td>5.107</td><td>47.16</td><td>52.27</td><td>52.269</td><td>2691281</td><td>17.52e-6</td><td>1.2610</td><td>50.52</td><td>55.99</td><td>24.36</td></tr><tr><td>MinVolume</td><td>4.548</td><td>62.00</td><td>66.55</td><td>01:06</td><td>3517435</td><td>17.63e-6</td><td>1.2665</td><td>66.41</td><td>71.28</td><td>31.84</td></tr><tr><td>ApproxMinVolume</td><td>4.775</td><td>63.06</td><td>67.84</td><td>01:07</td><td>3530765</td><td>17.86e-6</td><td>1.2665</td><td>67.55</td><td>72.67</td><td>31.96</td></tr><tr><td>Recursive</td><td>4.120</td><td>50.63</td><td>54.75</td><td>54.752</td><td>2868969</td><td>17.65e-6</td><td>1.2587</td><td>54.24</td><td>58.65</td><td>25.97</td></tr><tr><td>Recursive2</td><td>4.191</td><td>69.51</td><td>73.70</td><td>01:13</td><td>3940107</td><td>17.64e-6</td><td>1.2587</td><td>74.46</td><td>78.95</td><td>35.66</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>20.03e-6</td><td>13449</td><td>13449</td><td>03:44:09</td><td>1366693269</td><td>9.841e-6</td><td>1.1113</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>54.58</td><td>11390</td><td>11444</td><td>03:10:44</td><td>272881425</td><td>41.74e-6</td><td>1.0771</td><td>84.69</td><td>85.10</td><td>19.97</td></tr><tr><td>MinVariance</td><td>60.09</td><td>14933</td><td>14993</td><td>04:09:53</td><td>345680561</td><td>43.20e-6</td><td>1.0810</td><td>111.0</td><td>111.5</td><td>25.29</td></tr><tr><td>BSplines</td><td>55.34</td><td>103622</td><td>103677</td><td>1-04:47:57</td><td>2444878999</td><td>42.38e-6</td><td>1.1296</td><td>770.5</td><td>770.9</td><td>178.9</td></tr><tr><td>MinNorm</td><td>59.25</td><td>13373</td><td>13432</td><td>03:43:52</td><td>309751173</td><td>43.17e-6</td><td>1.0789</td><td>99.44</td><td>99.88</td><td>22.66</td></tr><tr><td>MinVolume</td><td>57.14</td><td>18399</td><td>18457</td><td>05:07:37</td><td>431370277</td><td>42.65e-6</td><td>1.0845</td><td>136.8</td><td>137.2</td><td>31.56</td></tr><tr><td>ApproxMinVolume</td><td>56.87</td><td>17898</td><td>17954</td><td>04:59:14</td><td>433301867</td><td>41.31e-6</td><td>1.0845</td><td>133.1</td><td>133.5</td><td>31.70</td></tr><tr><td>Recursive</td><td>55.41</td><td>13724</td><td>13780</td><td>03:49:40</td><td>326904643</td><td>41.98e-6</td><td>1.0771</td><td>102.0</td><td>102.5</td><td>23.92</td></tr><tr><td>Recursive2</td><td>55.55</td><td>21385</td><td>21441</td><td>05:57:21</td><td>504445255</td><td>42.39e-6</td><td>1.0771</td><td>159.0</td><td>159.4</td><td>36.91</td></tr></tbody></table>
<table><caption>Table 20: Bissection of problem 2D with problem: 9, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S,ms)</th><th>nb iter</th><th>time per iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>30.04e-6</td><td>3.881e-3</td><td>3.911e-3</td><td>0.004</td><td>107</td><td>36.27e-6</td><td>infeasible</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>846.8e-3</td><td>3.931e-3</td><td>850.7e-3</td><td>0.851</td><td>69</td><td>56.97e-6</td><td>infeasible</td><td>101.3</td><td>21752.4</td><td>64.49</td></tr><tr><td>MinVariance</td><td>951.6e-3</td><td>3.598e-3</td><td>955.2e-3</td><td>0.955</td><td>63</td><td>57.11e-6</td><td>infeasible</td><td>92.71</td><td>24423.7</td><td>58.88</td></tr><tr><td>BSplines</td><td>815.2e-3</td><td>3.912e-3</td><td>819.2e-3</td><td>0.819</td><td>113</td><td>34.62e-6</td><td>infeasible</td><td>100.8</td><td>20945.4</td><td>105.6</td></tr><tr><td>MinNorm</td><td>876.2e-3</td><td>3.897e-3</td><td>880.1e-3</td><td>0.880</td><td>61</td><td>63.88e-6</td><td>infeasible</td><td>100.4</td><td>22502.8</td><td>57.01</td></tr><tr><td>MinVolume</td><td>809.4e-3</td><td>3.393e-3</td><td>812.8e-3</td><td>0.813</td><td>69</td><td>49.17e-6</td><td>infeasible</td><td>87.42</td><td>20782.5</td><td>64.49</td></tr><tr><td>ApproxMinVolume</td><td>810.7e-3</td><td>3.030e-3</td><td>813.8e-3</td><td>0.814</td><td>69</td><td>43.91e-6</td><td>infeasible</td><td>78.07</td><td>20807.3</td><td>64.49</td></tr><tr><td>Recursive</td><td>874.4e-3</td><td>3.224e-3</td><td>877.7e-3</td><td>0.878</td><td>93</td><td>34.67e-6</td><td>infeasible</td><td>83.07</td><td>22440.7</td><td>86.92</td></tr><tr><td>Recursive2</td><td>862.8e-3</td><td>3.524e-3</td><td>866.3e-3</td><td>0.866</td><td>123</td><td>28.65e-6</td><td>infeasible</td><td>90.80</td><td>22150.3</td><td>115.0</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>25.99e-6</td><td>2.260</td><td>2.260</td><td>2.260</td><td>320711</td><td>7.046e-6</td><td>3.7270</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.996</td><td>1.118</td><td>4.114</td><td>4.114</td><td>106597</td><td>10.49e-6</td><td>3.7217</td><td>49.47</td><td>182.1</td><td>33.24</td></tr><tr><td>MinVariance</td><td>4.398</td><td>1.360</td><td>5.759</td><td>5.759</td><td>121437</td><td>11.20e-6</td><td>3.7221</td><td>60.20</td><td>254.8</td><td>37.86</td></tr><tr><td>BSplines</td><td>3.173</td><td>3.647</td><td>6.819</td><td>6.819</td><td>350145</td><td>10.41e-6</td><td>3.7270</td><td>161.4</td><td>301.8</td><td>109.2</td></tr><tr><td>MinNorm</td><td>4.188</td><td>1.303</td><td>5.491</td><td>5.491</td><td>114461</td><td>11.38e-6</td><td>3.7220</td><td>57.66</td><td>243.0</td><td>35.69</td></tr><tr><td>MinVolume</td><td>3.574</td><td>1.456</td><td>5.030</td><td>5.030</td><td>135677</td><td>10.73e-6</td><td>3.7225</td><td>64.43</td><td>222.6</td><td>42.31</td></tr><tr><td>ApproxMinVolume</td><td>3.580</td><td>1.424</td><td>5.003</td><td>5.003</td><td>136049</td><td>10.46e-6</td><td>3.7225</td><td>63.00</td><td>221.4</td><td>42.42</td></tr><tr><td>Recursive</td><td>3.503</td><td>1.153</td><td>4.656</td><td>4.656</td><td>109979</td><td>10.48e-6</td><td>3.7217</td><td>51.02</td><td>206.0</td><td>34.29</td></tr><tr><td>Recursive2</td><td>3.716</td><td>1.150</td><td>4.867</td><td>4.867</td><td>109183</td><td>10.53e-6</td><td>3.7217</td><td>50.89</td><td>215.4</td><td>34.04</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>21.93e-6</td><td>2219</td><td>2219</td><td>36.59</td><td>258043271</td><td>8.600e-6</td><td>2.1947</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>4.003</td><td>573.8</td><td>577.8</td><td>09.37</td><td>33494421</td><td>17.13e-6</td><td>2.1783</td><td>25.86</td><td>26.04</td><td>12.98</td></tr><tr><td>MinVariance</td><td>5.150</td><td>710.2</td><td>715.4</td><td>11.55</td><td>39986051</td><td>17.76e-6</td><td>2.1787</td><td>32.00</td><td>32.24</td><td>15.50</td></tr><tr><td>BSplines</td><td>3.984</td><td>2725</td><td>2729</td><td>45.29</td><td>167557199</td><td>16.26e-6</td><td>2.1849</td><td>122.8</td><td>123.0</td><td>64.93</td></tr><tr><td>MinNorm</td><td>4.978</td><td>636.8</td><td>641.8</td><td>10.41</td><td>37060967</td><td>17.18e-6</td><td>2.1785</td><td>28.70</td><td>28.92</td><td>14.36</td></tr><tr><td>MinVolume</td><td>4.483</td><td>789.2</td><td>793.6</td><td>13.13</td><td>46731837</td><td>16.89e-6</td><td>2.1793</td><td>35.56</td><td>35.76</td><td>18.11</td></tr><tr><td>ApproxMinVolume</td><td>4.648</td><td>819.1</td><td>823.7</td><td>13.43</td><td>46906683</td><td>17.46e-6</td><td>2.1793</td><td>36.91</td><td>37.12</td><td>18.18</td></tr><tr><td>Recursive</td><td>4.019</td><td>589.0</td><td>593.0</td><td>09.53</td><td>33952573</td><td>17.35e-6</td><td>2.1783</td><td>26.54</td><td>26.72</td><td>13.16</td></tr><tr><td>Recursive2</td><td>4.011</td><td>598.8</td><td>602.8</td><td>10.02</td><td>35550901</td><td>16.84e-6</td><td>2.1783</td><td>26.98</td><td>27.16</td><td>13.78</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>28.00e-6</td><td>446838</td><td>446838</td><td>5-04:07:18</td><td>46148673782</td><td>9.683e-6</td><td>1.4639</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>54.88</td><td>520018</td><td>520018</td><td>6-00:26:58</td><td>12563231767</td><td>41.39e-6</td><td>1.4591</td><td>116.4</td><td>116.4</td><td>27.22</td></tr><tr><td>MinVariance</td><td>60.26</td><td>665826</td><td>665826</td><td>7-16:57:06</td><td>15807794175</td><td>42.12e-6</td><td>1.4597</td><td>149.0</td><td>149.0</td><td>34.25</td></tr><tr><td>BSplines</td><td>55.54</td><td>4128640</td><td>4128640</td><td>47-18:50:40</td><td>100479468274</td><td>41.09e-6</td><td>1.4670</td><td>924.0</td><td>924.0</td><td>217.7</td></tr><tr><td>MinNorm</td><td>59.51</td><td>593157</td><td>593157</td><td>6-20:45:57</td><td>14262155737</td><td>41.59e-6</td><td>1.4594</td><td>132.7</td><td>132.7</td><td>30.90</td></tr><tr><td>MinVolume</td><td>57.34</td><td>802157</td><td>802157</td><td>9-06:49:17</td><td>19369407825</td><td>41.42e-6</td><td>1.4603</td><td>179.5</td><td>179.5</td><td>41.96</td></tr><tr><td>ApproxMinVolume</td><td>57.53</td><td>812740</td><td>812740</td><td>9-09:45:40</td><td>19473261169</td><td>41.74e-6</td><td>1.4603</td><td>181.9</td><td>181.9</td><td>42.20</td></tr><tr><td>Recursive</td><td>54.82</td><td>520266</td><td>520266</td><td>6-00:31:06</td><td>12581302301</td><td>41.35e-6</td><td>1.4591</td><td>116.4</td><td>116.4</td><td>27.26</td></tr><tr><td>Recursive2</td><td>55.18</td><td>556135</td><td>556136</td><td>6-10:28:56</td><td>13433505549</td><td>41.40e-6</td><td>1.4591</td><td>124.5</td><td>124.5</td><td>29.11</td></tr><tr><td rowspan="20">0.01</td><td rowspan="10">2</td><td>Interval</td><td>26.94e-6</td><td>111.2e-3</td><td>111.2e-3</td><td>0.111</td><td>107</td><td>1.039e-3</td><td>infeasible</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.166</td><td>4.789e-3</td><td>2.171</td><td>2.171</td><td>69</td><td>69.40e-6</td><td>infeasible</td><td>4.308</td><td>1952.4</td><td>64.49</td></tr><tr><td>MinVariance</td><td>2.086</td><td>3.739e-3</td><td>2.089</td><td>2.089</td><td>63</td><td>59.35e-6</td><td>infeasible</td><td>3.364</td><td>1879.2</td><td>58.88</td></tr><tr><td>BSplines</td><td>2.167</td><td>8.552e-3</td><td>2.176</td><td>2.176</td><td>113</td><td>75.68e-6</td><td>infeasible</td><td>7.693</td><td>1957.0</td><td>105.6</td></tr><tr><td>MinNorm</td><td>2.167</td><td>12.75e-3</td><td>2.179</td><td>2.179</td><td>61</td><td>209.0e-6</td><td>infeasible</td><td>11.47</td><td>1960.1</td><td>57.01</td></tr><tr><td>MinVolume</td><td>2.167</td><td>9.690e-3</td><td>2.177</td><td>2.177</td><td>69</td><td>140.4e-6</td><td>infeasible</td><td>8.717</td><td>1957.8</td><td>64.49</td></tr><tr><td>ApproxMinVolume</td><td>2.152</td><td>11.40e-3</td><td>2.164</td><td>2.164</td><td>69</td><td>165.2e-6</td><td>infeasible</td><td>10.26</td><td>1946.0</td><td>64.49</td></tr><tr><td>Recursive</td><td>2.152</td><td>9.382e-3</td><td>2.161</td><td>2.161</td><td>93</td><td>100.9e-6</td><td>infeasible</td><td>8.440</td><td>1944.0</td><td>86.92</td></tr><tr><td>Recursive2</td><td>2.077</td><td>12.47e-3</td><td>2.090</td><td>2.090</td><td>123</td><td>101.4e-6</td><td>infeasible</td><td>11.22</td><td>1879.4</td><td>115.0</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>28.85e-6</td><td>307.9e-3</td><td>307.9e-3</td><td>0.308</td><td>40963</td><td>7.516e-6</td><td>3.8013</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>3.003</td><td>153.6e-3</td><td>3.156</td><td>3.156</td><td>13987</td><td>10.98e-6</td><td>3.7604</td><td>49.89</td><td>1025.2</td><td>34.15</td></tr><tr><td>MinVariance</td><td>4.479</td><td>198.5e-3</td><td>4.677</td><td>4.677</td><td>15869</td><td>12.51e-6</td><td>3.7636</td><td>64.47</td><td>1519.2</td><td>38.74</td></tr><tr><td>BSplines</td><td>3.213</td><td>534.8e-3</td><td>3.748</td><td>3.748</td><td>48657</td><td>10.99e-6</td><td>3.8029</td><td>173.7</td><td>1217.2</td><td>118.8</td></tr><tr><td>MinNorm</td><td>4.249</td><td>189.9e-3</td><td>4.439</td><td>4.439</td><td>14961</td><td>12.69e-6</td><td>3.7612</td><td>61.67</td><td>1441.8</td><td>36.52</td></tr><tr><td>MinVolume</td><td>3.633</td><td>205.4e-3</td><td>3.838</td><td>3.838</td><td>18009</td><td>11.41e-6</td><td>3.7668</td><td>66.72</td><td>1246.6</td><td>43.96</td></tr><tr><td>ApproxMinVolume</td><td>3.600</td><td>202.5e-3</td><td>3.802</td><td>3.802</td><td>18053</td><td>11.22e-6</td><td>3.7668</td><td>65.77</td><td>1234.9</td><td>44.07</td></tr><tr><td>Recursive</td><td>3.502</td><td>185.9e-3</td><td>3.688</td><td>3.688</td><td>16495</td><td>11.27e-6</td><td>3.7604</td><td>60.40</td><td>1198.0</td><td>40.27</td></tr><tr><td>Recursive2</td><td>3.720</td><td>189.2e-3</td><td>3.909</td><td>3.909</td><td>16503</td><td>11.46e-6</td><td>3.7604</td><td>61.46</td><td>1269.7</td><td>40.29</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>25.03e-6</td><td>150.5</td><td>150.5</td><td>02:30</td><td>17689035</td><td>8.507e-6</td><td>2.3987</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>4.091</td><td>36.03</td><td>40.13</td><td>40.126</td><td>20543441</td><td>17.54e-6</td><td>2.2545</td><td>23.95</td><td>26.66</td><td>11.61</td></tr><tr><td>MinVariance</td><td>5.224</td><td>42.35</td><td>47.57</td><td>47.575</td><td>2415005</td><td>17.54e-6</td><td>2.2579</td><td>28.14</td><td>31.61</td><td>13.65</td></tr><tr><td>BSplines</td><td>4.060</td><td>176.6</td><td>180.6</td><td>03:00</td><td>10142423</td><td>17.41e-6</td><td>2.3074</td><td>117.3</td><td>120.0</td><td>57.34</td></tr><tr><td>MinNorm</td><td>4.914</td><td>39.54</td><td>44.46</td><td>44.456</td><td>2238933</td><td>17.66e-6</td><td>2.2569</td><td>26.28</td><td>29.54</td><td>12.66</td></tr><tr><td>MinVolume</td><td>4.603</td><td>56.52</td><td>61.12</td><td>01:01</td><td>2829311</td><td>19.98e-6</td><td>2.2634</td><td>37.56</td><td>40.62</td><td>15.99</td></tr><tr><td>ApproxMinVolume</td><td>4.716</td><td>50.48</td><td>55.20</td><td>55.197</td><td>2838669</td><td>17.78e-6</td><td>2.2634</td><td>33.54</td><td>36.68</td><td>16.05</td></tr><tr><td>Recursive</td><td>4.110</td><td>42.38</td><td>46.49</td><td>46.492</td><td>2418191</td><td>17.53e-6</td><td>2.2545</td><td>28.16</td><td>30.89</td><td>13.67</td></tr><tr><td>Recursive2</td><td>4.111</td><td>51.67</td><td>55.78</td><td>55.781</td><td>2902143</td><td>17.80e-6</td><td>2.2545</td><td>34.33</td><td>37.07</td><td>16.41</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>20.98e-6</td><td>7032</td><td>7032</td><td>01:57:12</td><td>708080401</td><td>9.931e-6</td><td>1.5549</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>55.22</td><td>8640</td><td>8695</td><td>02:24:55</td><td>200204705</td><td>43.16e-6</td><td>1.5150</td><td>122.9</td><td>123.7</td><td>28.27</td></tr><tr><td>MinVariance</td><td>60.16</td><td>11126</td><td>11186</td><td>02:06:26</td><td>254680529</td><td>43.69e-6</td><td>1.5198</td><td>158.2</td><td>159.1</td><td>35.97</td></tr><tr><td>BSplines</td><td>54.90</td><td>72776</td><td>72831</td><td>20:13:51</td><td>1720598853</td><td>42.30e-6</td><td>1.5759</td><td>1034.9</td><td>1035.7</td><td>243.0</td></tr><tr><td>MinNorm</td><td>59.29</td><td>9779</td><td>9838</td><td>02:43:58</td><td>228013011</td><td>42.89e-6</td><td>1.5175</td><td>139.1</td><td>139.9</td><td>32.20</td></tr><tr><td>MinVolume</td><td>57.09</td><td>13670</td><td>13727</td><td>03:48:47</td><td>315310113</td><td>43.36e-6</td><td>1.5237</td><td>194.4</td><td>195.2</td><td>44.53</td></tr><tr><td>ApproxMinVolume</td><td>57.34</td><td>13701</td><td>13759</td><td>03:49:19</td><td>317430217</td><td>43.16e-6</td><td>1.5239</td><td>194.8</td><td>195.7</td><td>44.83</td></tr><tr><td>Recursive</td><td>55.13</td><td>8979</td><td>9035</td><td>02:30:35</td><td>207894885</td><td>43.19e-6</td><td>1.5150</td><td>127.7</td><td>128.5</td><td>29.36</td></tr><tr><td>Recursive2</td><td>54.98</td><td>14663</td><td>14718</td><td>04:05:18</td><td>345337759</td><td>42.46e-6</td><td>1.5150</td><td>208.5</td><td>209.3</td><td>48.77</td></tr></tbody></table>
<table><caption>Table 21: Bissection of problem 2D with problem: 11, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time_per_iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>37.19e-6</td><td>6.781e-3</td><td>6.818e-3</td><td>0.007</td><td>671</td><td>10.11e-6</td><td>80.714</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>716.8e-3</td><td>4.455e-3</td><td>721.3e-3</td><td>0.721</td><td>583</td><td>7.642e-6</td><td>80.593</td><td>65.70</td><td>10578.7</td><td>86.89</td></tr><tr><td>MinVariance</td><td>714.6e-3</td><td>4.524e-3</td><td>719.1e-3</td><td>0.719</td><td>583</td><td>7.760e-6</td><td>80.595</td><td>66.72</td><td>10546.9</td><td>86.89</td></tr><tr><td>BSplines</td><td>710.5e-3</td><td>4.763e-3</td><td>715.3e-3</td><td>0.715</td><td>777</td><td>6.130e-6</td><td>80.623</td><td>70.24</td><td>10491.0</td><td>115.8</td></tr><tr><td>MinNorm</td><td>732.1e-3</td><td>4.216e-3</td><td>736.3e-3</td><td>0.736</td><td>583</td><td>7.231e-6</td><td>80.594</td><td>62.17</td><td>10799.7</td><td>86.89</td></tr><tr><td>MinVolume</td><td>733.5e-3</td><td>4.538e-3</td><td>738.1e-3</td><td>0.738</td><td>597</td><td>7.601e-6</td><td>80.597</td><td>66.92</td><td>10825.0</td><td>88.97</td></tr><tr><td>ApproxMinVolume</td><td>726.7e-3</td><td>4.123e-3</td><td>730.8e-3</td><td>0.731</td><td>597</td><td>6.907e-6</td><td>80.597</td><td>60.81</td><td>10719.3</td><td>88.97</td></tr><tr><td>Recursive</td><td>725.3e-3</td><td>4.248e-3</td><td>729.6e-3</td><td>0.730</td><td>591</td><td>7.188e-6</td><td>80.593</td><td>62.65</td><td>10700.4</td><td>88.08</td></tr><tr><td>Recursive2</td><td>720.7e-3</td><td>4.555e-3</td><td>725.3e-3</td><td>0.725</td><td>589</td><td>7.733e-6</td><td>80.593</td><td>67.17</td><td>10637.8</td><td>87.78</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>30.04e-6</td><td>3.216</td><td>3.216</td><td>3.216</td><td>393723</td><td>8.169e-6</td><td>85.311</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.420</td><td>449.0e-3</td><td>1.869</td><td>1.869</td><td>69973</td><td>6.417e-6</td><td>85.168</td><td>13.96</td><td>58.12</td><td>17.77</td></tr><tr><td>MinVariance</td><td>1.519</td><td>479.7e-3</td><td>1.999</td><td>1.999</td><td>74615</td><td>6.429e-6</td><td>85.172</td><td>14.92</td><td>62.15</td><td>18.95</td></tr><tr><td>BSplines</td><td>1.420</td><td>906.9e-3</td><td>2.327</td><td>2.327</td><td>154947</td><td>5.853e-6</td><td>85.215</td><td>28.20</td><td>72.36</td><td>39.35</td></tr><tr><td>MinNorm</td><td>1.538</td><td>486.7e-3</td><td>2.025</td><td>2.025</td><td>72049</td><td>6.755e-6</td><td>85.170</td><td>15.13</td><td>62.95</td><td>18.30</td></tr><tr><td>MinVolume</td><td>1.483</td><td>526.5e-3</td><td>2.009</td><td>2.009</td><td>80979</td><td>6.502e-6</td><td>85.175</td><td>16.37</td><td>62.47</td><td>20.57</td></tr><tr><td>ApproxMinVolume</td><td>1.474</td><td>535.9e-3</td><td>2.010</td><td>2.010</td><td>81135</td><td>6.605e-6</td><td>85.175</td><td>16.66</td><td>62.51</td><td>20.61</td></tr><tr><td>Recursive</td><td>1.479</td><td>465.4e-3</td><td>1.945</td><td>1.945</td><td>70077</td><td>6.642e-6</td><td>85.168</td><td>14.47</td><td>60.46</td><td>17.80</td></tr><tr><td>Recursive2</td><td>1.485</td><td>468.4e-3</td><td>1.954</td><td>1.954</td><td>70347</td><td>6.658e-6</td><td>85.168</td><td>14.56</td><td>60.75</td><td>17.87</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>28.85e-6</td><td>1922</td><td>1922</td><td>32.02</td><td>198196587</td><td>9.701e-6</td><td>100.24</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>11.05</td><td>294.6</td><td>305.7</td><td>05:05</td><td>12832141</td><td>22.96e-6</td><td>100.11</td><td>15.32</td><td>15.90</td><td>6.474</td></tr><tr><td>MinVariance</td><td>14.42</td><td>358.7</td><td>373.2</td><td>06:13</td><td>14448165</td><td>24.83e-6</td><td>100.12</td><td>18.66</td><td>19.41</td><td>7.290</td></tr><tr><td>BSplines</td><td>11.69</td><td>998.8</td><td>1010</td><td>16:50</td><td>44530227</td><td>22.43e-6</td><td>100.19</td><td>51.94</td><td>52.55</td><td>22.47</td></tr><tr><td>MinNorm</td><td>14.32</td><td>346.5</td><td>360.9</td><td>06:00</td><td>13694269</td><td>25.31e-6</td><td>100.12</td><td>18.02</td><td>18.77</td><td>6.909</td></tr><tr><td>MinVolume</td><td>12.60</td><td>382.4</td><td>395.0</td><td>06:34</td><td>16109081</td><td>23.74e-6</td><td>100.12</td><td>19.89</td><td>20.54</td><td>8.128</td></tr><tr><td>ApproxMinVolume</td><td>12.29</td><td>351.2</td><td>363.5</td><td>06:03</td><td>16152151</td><td>21.74e-6</td><td>100.13</td><td>18.27</td><td>18.91</td><td>8.150</td></tr><tr><td>Recursive</td><td>14.73</td><td>320.8</td><td>335.5</td><td>05:35</td><td>12833477</td><td>25.00e-6</td><td>100.11</td><td>16.68</td><td>17.45</td><td>6.475</td></tr><tr><td>Recursive2</td><td>15.54</td><td>308.9</td><td>324.5</td><td>05:24</td><td>12849603</td><td>24.04e-6</td><td>100.11</td><td>16.07</td><td>16.88</td><td>6.483</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>24.00e-6</td><td>4968410</td><td>4968410</td><td>57-12:06:50</td><td>429100212907</td><td>11.58e-6</td><td>116.26</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>72.10</td><td>576410</td><td>576410</td><td>6-16:06:50</td><td>16067783129</td><td>35.87e-6</td><td>116.06</td><td>11.60</td><td>11.60</td><td>3.745</td></tr><tr><td>MinVariance</td><td>75.38</td><td>657375</td><td>657375</td><td>7-14:36:15</td><td>18084989359</td><td>36.35e-6</td><td>116.06</td><td>13.23</td><td>13.23</td><td>4.215</td></tr><tr><td>BSplines</td><td>72.51</td><td>1938060</td><td>1938060</td><td>22-10:21:00</td><td>57904499645</td><td>33.47e-6</td><td>116.16</td><td>39.01</td><td>39.01</td><td>13.49</td></tr><tr><td>MinNorm</td><td>75.07</td><td>613132</td><td>613132</td><td>7-02:18:52</td><td>17193195041</td><td>35.66e-6</td><td>116.06</td><td>12.34</td><td>12.34</td><td>4.007</td></tr><tr><td>MinVolume</td><td>73.50</td><td>718739</td><td>718739</td><td>8-07:38:59</td><td>20235843440</td><td>35.52e-6</td><td>116.07</td><td>14.47</td><td>14.47</td><td>4.716</td></tr><tr><td>ApproxMinVolume</td><td>73.54</td><td>710916</td><td>710916</td><td>8-05:28:36</td><td>20291967886</td><td>35.03e-6</td><td>116.07</td><td>14.31</td><td>14.31</td><td>4.729</td></tr><tr><td>Recursive</td><td>72.25</td><td>563737</td><td>563738</td><td>6-12:35:38</td><td>16107783885</td><td>35.00e-6</td><td>116.06</td><td>11.35</td><td>11.35</td><td>3.754</td></tr><tr><td>Recursive2</td><td>72.19</td><td>573654</td><td>573654</td><td>6-15:20:54</td><td>16154599175</td><td>35.51e-6</td><td>116.06</td><td>11.55</td><td>11.55</td><td>3.765</td></tr><tr><td rowspan="10">0.01</td><td rowspan="10">2</td><td>Interval</td><td>26.94e-6</td><td>69.50e-3</td><td>69.53e-3</td><td>0.070</td><td>265</td><td>262.3e-6</td><td>82.162</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>2.098</td><td>13.60e-3</td><td>2.112</td><td>2.112</td><td>195</td><td>69.77e-6</td><td>81.681</td><td>19.57</td><td>3036.9</td><td>73.58</td></tr><tr><td>MinVariance</td><td>965.6e-3</td><td>10.43e-3</td><td>976.0e-3</td><td>0.976</td><td>209</td><td>49.90e-6</td><td>81.700</td><td>15.01</td><td>1403.7</td><td>78.87</td></tr><tr><td>BSplines</td><td>965.7e-3</td><td>12.53e-3</td><td>978.3e-3</td><td>0.978</td><td>297</td><td>42.18e-6</td><td>81.923</td><td>18.02</td><td>1406.9</td><td>112.1</td></tr><tr><td>MinNorm</td><td>962.1e-3</td><td>6.176e-3</td><td>968.3e-3</td><td>0.968</td><td>195</td><td>31.67e-6</td><td>81.692</td><td>8.886</td><td>1392.7</td><td>73.58</td></tr><tr><td>MinVolume</td><td>978.0e-3</td><td>14.78e-3</td><td>992.8e-3</td><td>0.993</td><td>209</td><td>70.74e-6</td><td>81.718</td><td>21.27</td><td>1427.8</td><td>78.87</td></tr><tr><td>ApproxMinVolume</td><td>981.4e-3</td><td>4.838e-3</td><td>986.3e-3</td><td>0.986</td><td>209</td><td>23.15e-6</td><td>81.719</td><td>6.961</td><td>1418.5</td><td>78.87</td></tr><tr><td>Recursive</td><td>1.023</td><td>5.526e-3</td><td>1.029</td><td>1.029</td><td>203</td><td>27.22e-6</td><td>81.681</td><td>7.950</td><td>1479.6</td><td>76.60</td></tr><tr><td>Recursive2</td><td>952.5e-3</td><td>4.929e-3</td><td>957.5e-3</td><td>0.957</td><td>201</td><td>24.52e-6</td><td>81.681</td><td>7.091</td><td>1377.1</td><td>75.85</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>24.08e-6</td><td>623.9e-3</td><td>623.9e-3</td><td>0.624</td><td>75517</td><td>8.262e-6</td><td>87.099</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.426</td><td>125.1e-3</td><td>1.551</td><td>1.551</td><td>17463</td><td>7.162e-6</td><td>85.900</td><td>20.05</td><td>248.6</td><td>23.12</td></tr><tr><td>MinVariance</td><td>1.451</td><td>124.9e-3</td><td>1.575</td><td>1.575</td><td>18765</td><td>6.656e-6</td><td>85.929</td><td>20.02</td><td>252.5</td><td>24.85</td></tr><tr><td>BSplines</td><td>1.389</td><td>231.3e-3</td><td>1.620</td><td>1.620</td><td>36935</td><td>6.263e-6</td><td>86.271</td><td>37.08</td><td>259.7</td><td>48.91</td></tr><tr><td>MinNorm</td><td>1.522</td><td>126.1e-3</td><td>1.648</td><td>1.648</td><td>18103</td><td>6.965e-6</td><td>85.916</td><td>20.21</td><td>264.1</td><td>23.97</td></tr><tr><td>MinVolume</td><td>1.522</td><td>136.0e-3</td><td>1.658</td><td>1.658</td><td>19955</td><td>6.814e-6</td><td>85.957</td><td>21.79</td><td>265.7</td><td>26.42</td></tr><tr><td>ApproxMinVolume</td><td>1.515</td><td>136.0e-3</td><td>1.651</td><td>1.651</td><td>19995</td><td>6.802e-6</td><td>85.958</td><td>21.80</td><td>264.6</td><td>26.48</td></tr><tr><td>Recursive</td><td>1.476</td><td>120.3e-3</td><td>1.597</td><td>1.597</td><td>17499</td><td>6.874e-6</td><td>85.900</td><td>19.28</td><td>255.9</td><td>23.17</td></tr><tr><td>Recursive2</td><td>1.503</td><td>124.1e-3</td><td>1.627</td><td>1.627</td><td>17611</td><td>7.048e-6</td><td>85.900</td><td>19.89</td><td>260.8</td><td>23.32</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>29.09e-6</td><td>188.1</td><td>188.1</td><td>03:08</td><td>19105663</td><td>9.845e-6</td><td>102.12</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>11.10</td><td>48.34</td><td>59.44</td><td>59.438</td><td>2005097</td><td>24.11e-6</td><td>101.19</td><td>25.70</td><td>31.60</td><td>10.49</td></tr><tr><td>MinVariance</td><td>14.58</td><td>56.95</td><td>71.53</td><td>01:11</td><td>2217569</td><td>25.68e-6</td><td>101.24</td><td>30.28</td><td>38.03</td><td>11.61</td></tr><tr><td>BSplines</td><td>11.73</td><td>131.2</td><td>142.9</td><td>02:22</td><td>5620563</td><td>23.34e-6</td><td>101.18</td><td>69.75</td><td>75.98</td><td>29.42</td></tr><tr><td>MinNorm</td><td>14.39</td><td>54.10</td><td>68.49</td><td>01:08</td><td>2116625</td><td>25.56e-6</td><td>101.22</td><td>28.76</td><td>36.41</td><td>11.08</td></tr><tr><td>MinVolume</td><td>12.83</td><td>60.24</td><td>73.07</td><td>01:13</td><td>2429159</td><td>24.80e-6</td><td>101.28</td><td>32.03</td><td>38.85</td><td>12.71</td></tr><tr><td>ApproxMinVolume</td><td>12.34</td><td>60.49</td><td>72.83</td><td>01:12</td><td>2434605</td><td>24.85e-6</td><td>101.28</td><td>32.16</td><td>38.72</td><td>12.74</td></tr><tr><td>Recursive</td><td>14.83</td><td>50.82</td><td>65.66</td><td>01:05</td><td>2006849</td><td>25.32e-6</td><td>101.19</td><td>27.02</td><td>34.90</td><td>10.50</td></tr><tr><td>Recursive2</td><td>15.74</td><td>52.36</td><td>68.09</td><td>01:08</td><td>2016491</td><td>25.97e-6</td><td>101.19</td><td>27.84</td><td>36.20</td><td>10.55</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>25.03e-6</td><td>160487</td><td>160487</td><td>1-20:34:47</td><td>13868976571</td><td>11.57e-6</td><td>119.00</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>72.00</td><td>43819</td><td>43891</td><td>12:11:31</td><td>1073020121</td><td>40.84e-6</td><td>117.60</td><td>27.30</td><td>27.35</td><td>7.737</td></tr><tr><td>MinVariance</td><td>75.06</td><td>48345</td><td>48420</td><td>13:27:00</td><td>1178829583</td><td>41.01e-6</td><td>117.67</td><td>30.12</td><td>30.17</td><td>8.500</td></tr><tr><td>BSplines</td><td>72.58</td><td>116949</td><td>117022</td><td>1-08:30:22</td><td>2884178133</td><td>40.55e-6</td><td>118.42</td><td>72.87</td><td>72.92</td><td>20.80</td></tr><tr><td>MinNorm</td><td>75.07</td><td>46753</td><td>46828</td><td>13:00:28</td><td>1129756421</td><td>41.38e-6</td><td>117.64</td><td>29.13</td><td>29.18</td><td>8.146</td></tr><tr><td>MinVolume</td><td>73.23</td><td>51536</td><td>51609</td><td>14:20:09</td><td>1284918461</td><td>40.11e-6</td><td>117.73</td><td>32.11</td><td>32.16</td><td>9.265</td></tr><tr><td>ApproxMinVolume</td><td>73.40</td><td>53663</td><td>53736</td><td>14:55:36</td><td>1287711255</td><td>41.67e-6</td><td>117.73</td><td>33.44</td><td>33.48</td><td>9.285</td></tr><tr><td>Recursive</td><td>69.94</td><td>43679</td><td>43749</td><td>12:09:09</td><td>1073021185</td><td>40.71e-6</td><td>117.60</td><td>27.22</td><td>27.26</td><td>7.737</td></tr><tr><td>Recursive2</td><td>72.37</td><td>46092</td><td>46165</td><td>12:49:25</td><td>1088055199</td><td>42.36e-6</td><td>117.60</td><td>28.72</td><td>28.77</td><td>7.845</td></tr></tbody></table>
<table><caption>Table 22: Bissection of problem 2D with problem: 12, type: MinFirst</caption><thead><tr><th>precision</th><th>ndof</th><th>solver</th><th>prep.time</th><th>comput.time</th><th>total.time</th><th>(D-H:M:S.ms)</th><th>nb.iter</th><th>time-per.iter</th><th>criteria</th><th>comput.time (%)</th><th>total.time (%)</th><th>nb.iter (%)</th></tr></thead><tbody><tr><td rowspan="20">0.001</td><td rowspan="10">2</td><td>Interval</td><td>25.03e-6</td><td>4.878e-3</td><td>4.903e-3</td><td>0.005</td><td>299</td><td>16.31e-6</td><td>56.017</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>715.6e-3</td><td>3.639e-3</td><td>719.3e-3</td><td>0.719</td><td>291</td><td>12.51e-6</td><td>56.017</td><td>74.60</td><td>14669.7</td><td>97.32</td></tr><tr><td>MinVariance</td><td>716.5e-3</td><td>3.320e-3</td><td>719.8e-3</td><td>0.720</td><td>291</td><td>11.41e-6</td><td>56.020</td><td>68.06</td><td>14681.1</td><td>97.32</td></tr><tr><td>BSplines</td><td>719.7e-3</td><td>3.521e-3</td><td>723.3e-3</td><td>0.723</td><td>353</td><td>9.974e-6</td><td>56.047</td><td>72.18</td><td>14751.2</td><td>118.1</td></tr><tr><td>MinNorm</td><td>730.4e-3</td><td>3.550e-3</td><td>734.0e-3</td><td>0.734</td><td>291</td><td>12.20e-6</td><td>56.019</td><td>72.78</td><td>14970.1</td><td>97.32</td></tr><tr><td>MinVolume</td><td>727.2e-3</td><td>3.829e-3</td><td>731.0e-3</td><td>0.731</td><td>291</td><td>13.16e-6</td><td>56.022</td><td>78.49</td><td>14909.0</td><td>97.32</td></tr><tr><td>ApproxMinVolume</td><td>716.8e-3</td><td>3.509e-3</td><td>720.3e-3</td><td>0.720</td><td>291</td><td>12.06e-6</td><td>56.022</td><td>71.94</td><td>14691.4</td><td>97.32</td></tr><tr><td>Recursive</td><td>724.0e-3</td><td>3.945e-3</td><td>728.0e-3</td><td>0.728</td><td>291</td><td>13.56e-6</td><td>56.017</td><td>80.87</td><td>14847.6</td><td>97.32</td></tr><tr><td>Recursive2</td><td>718.8e-3</td><td>3.302e-3</td><td>722.1e-3</td><td>0.722</td><td>291</td><td>11.35e-6</td><td>56.017</td><td>67.69</td><td>14728.4</td><td>97.32</td></tr><tr><td>Interval</td><td>28.85e-6</td><td>356.7e-3</td><td>356.7e-3</td><td>0.357</td><td>43383</td><td>8.221e-6</td><td>66.191</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.432</td><td>165.0e-3</td><td>1.597</td><td>1.597</td><td>24033</td><td>6.867e-6</td><td>66.184</td><td>46.27</td><td>447.7</td><td>55.40</td></tr><tr><td>MinVariance</td><td>1.513</td><td>178.4e-3</td><td>1.692</td><td>1.692</td><td>25667</td><td>6.951e-6</td><td>66.188</td><td>50.03</td><td>474.3</td><td>59.16</td></tr><tr><td>BSplines</td><td>1.445</td><td>315.3e-3</td><td>1.760</td><td>1.760</td><td>50105</td><td>6.293e-6</td><td>66.235</td><td>88.41</td><td>493.5</td><td>115.5</td></tr><tr><td>MinNorm</td><td>1.517</td><td>172.5e-3</td><td>1.689</td><td>1.689</td><td>24875</td><td>6.934e-6</td><td>66.186</td><td>48.36</td><td>473.6</td><td>57.34</td></tr><tr><td>MinVolume</td><td>1.479</td><td>185.6e-3</td><td>1.665</td><td>1.665</td><td>27379</td><td>6.779e-6</td><td>66.192</td><td>52.04</td><td>466.8</td><td>63.11</td></tr><tr><td>ApproxMinVolume</td><td>1.466</td><td>185.3e-3</td><td>1.651</td><td>1.651</td><td>27383</td><td>6.769e-6</td><td>66.192</td><td>51.97</td><td>462.8</td><td>63.12</td></tr><tr><td>Recursive</td><td>1.466</td><td>164.6e-3</td><td>1.630</td><td>1.630</td><td>24047</td><td>6.844e-6</td><td>66.184</td><td>46.15</td><td>457.0</td><td>55.43</td></tr><tr><td>Recursive2</td><td>1.479</td><td>168.4e-3</td><td>1.648</td><td>1.648</td><td>24181</td><td>6.965e-6</td><td>66.184</td><td>47.22</td><td>461.9</td><td>55.74</td></tr><tr><td rowspan="10">3</td><td>Interval</td><td>26.94e-6</td><td>981.7</td><td>981.7</td><td>16:21</td><td>101228629</td><td>9.698e-6</td><td>60.598</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>11.08</td><td>240.0</td><td>251.1</td><td>04:11</td><td>10735221</td><td>22.36e-6</td><td>60.430</td><td>24.45</td><td>25.58</td><td>10.60</td></tr><tr><td>MinVariance</td><td>14.49</td><td>304.3</td><td>318.8</td><td>05:18</td><td>12083445</td><td>25.18e-6</td><td>60.436</td><td>31.00</td><td>32.47</td><td>11.94</td></tr><tr><td>BSplines</td><td>11.63</td><td>785.8</td><td>797.4</td><td>13:17</td><td>35977821</td><td>21.84e-6</td><td>60.512</td><td>80.05</td><td>81.23</td><td>35.54</td></tr><tr><td>MinNorm</td><td>14.34</td><td>280.6</td><td>294.9</td><td>04:54</td><td>11453841</td><td>24.50e-6</td><td>60.433</td><td>28.58</td><td>30.04</td><td>11.31</td></tr><tr><td>MinVolume</td><td>12.62</td><td>294.3</td><td>306.9</td><td>05:06</td><td>13450631</td><td>21.88e-6</td><td>60.443</td><td>29.98</td><td>31.26</td><td>13.29</td></tr><tr><td>ApproxMinVolume</td><td>12.33</td><td>327.7</td><td>340.0</td><td>05:40</td><td>13484069</td><td>24.30e-6</td><td>60.443</td><td>33.38</td><td>34.64</td><td>13.32</td></tr><tr><td>Recursive</td><td>14.83</td><td>267.3</td><td>282.1</td><td>04:42</td><td>10737599</td><td>24.89e-6</td><td>60.430</td><td>27.23</td><td>28.74</td><td>10.61</td></tr><tr><td>Recursive2</td><td>15.62</td><td>261.1</td><td>276.7</td><td>04:36</td><td>10753521</td><td>24.28e-6</td><td>60.430</td><td>26.60</td><td>28.19</td><td>10.62</td></tr><tr><td rowspan="10">4</td><td>Interval</td><td>31.00e-6</td><td>1080000</td><td>1080000</td><td>12-12:00:00</td><td>93548792545</td><td>11.54e-6</td><td>68.184</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>72.30</td><td>200761</td><td>200761</td><td>2:07:46:01</td><td>5438985345</td><td>36.91e-6</td><td>68.008</td><td>18.59</td><td>18.59</td><td>5.814</td></tr><tr><td>MinVariance</td><td>75.46</td><td>230761</td><td>230762</td><td>2-16:06:02</td><td>6217429189</td><td>37.12e-6</td><td>68.018</td><td>21.37</td><td>21.37</td><td>6.646</td></tr><tr><td>BSplines</td><td>71.76</td><td>754119</td><td>754119</td><td>8-17:28:39</td><td>22437050555</td><td>33.61e-6</td><td>68.128</td><td>69.83</td><td>69.83</td><td>23.98</td></tr><tr><td>MinNorm</td><td>74.83</td><td>216449</td><td>216449</td><td>2-12:07:29</td><td>5852510559</td><td>36.98e-6</td><td>68.013</td><td>20.04</td><td>20.04</td><td>6.256</td></tr><tr><td>MinVolume</td><td>73.48</td><td>254547</td><td>254547</td><td>2-22:42:27</td><td>7028985835</td><td>36.21e-6</td><td>68.027</td><td>23.57</td><td>23.57</td><td>7.514</td></tr><tr><td>ApproxMinVolume</td><td>73.27</td><td>255586</td><td>255586</td><td>2-22:59:46</td><td>7040768432</td><td>36.30e-6</td><td>68.027</td><td>23.67</td><td>23.67</td><td>7.526</td></tr><tr><td>Recursive</td><td>72.59</td><td>202257</td><td>202257</td><td>2-08:10:57</td><td>5429930518</td><td>37.25e-6</td><td>68.008</td><td>18.73</td><td>18.73</td><td>5.804</td></tr><tr><td>Recursive2</td><td>71.92</td><td>199032</td><td>199032</td><td>2-07:17:12</td><td>5469998197</td><td>36.39e-6</td><td>68.008</td><td>18.43</td><td>18.43</td><td>5.847</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>27.89e-6</td><td>3.334e-3</td><td>3.362e-3</td><td>0.003</td><td>137</td><td>24.34e-6</td><td>57.100</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>965.8e-3</td><td>6.902e-3</td><td>972.7e-3</td><td>0.973</td><td>109</td><td>63.32e-6</td><td>57.100</td><td>207.0</td><td>28931.7</td><td>79.56</td></tr><tr><td>MinVariance</td><td>966.4e-3</td><td>12.90e-3</td><td>979.4e-3</td><td>0.979</td><td>109</td><td>118.4e-6</td><td>57.119</td><td>387.0</td><td>29130.7</td><td>79.56</td></tr><tr><td>BSplines</td><td>966.9e-3</td><td>15.68e-3</td><td>982.6e-3</td><td>0.983</td><td>149</td><td>105.2e-6</td><td>57.342</td><td>470.2</td><td>29227.8</td><td>108.8</td></tr><tr><td>MinNorm</td><td>965.7e-3</td><td>13.96e-3</td><td>979.7e-3</td><td>0.980</td><td>109</td><td>128.0e-6</td><td>57.110</td><td>418.6</td><td>29140.8</td><td>79.56</td></tr><tr><td>MinVolume</td><td>965.4e-3</td><td>5.669e-3</td><td>971.1e-3</td><td>0.971</td><td>109</td><td>52.01e-6</td><td>57.137</td><td>170.0</td><td>28883.8</td><td>79.56</td></tr><tr><td>ApproxMinVolume</td><td>963.6e-3</td><td>5.885e-3</td><td>969.5e-3</td><td>0.970</td><td>109</td><td>53.99e-6</td><td>57.138</td><td>176.5</td><td>28883.8</td><td>79.56</td></tr><tr><td>Recursive</td><td>960.4e-3</td><td>6.185e-3</td><td>966.6e-3</td><td>0.967</td><td>109</td><td>56.74e-6</td><td>57.100</td><td>185.5</td><td>28751.3</td><td>79.56</td></tr><tr><td>Recursive2</td><td>958.2e-3</td><td>6.852e-3</td><td>965.0e-3</td><td>0.965</td><td>109</td><td>62.86e-6</td><td>57.100</td><td>205.5</td><td>28704.6</td><td>79.56</td></tr><tr><td rowspan="10">0.01</td><td rowspan="10">2</td><td>Interval</td><td>30.04e-6</td><td>172.4e-3</td><td>172.4e-3</td><td>0.172</td><td>19889</td><td>8.667e-6</td><td>67.355</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>1.436</td><td>67.12e-3</td><td>1.503</td><td>1.503</td><td>8905</td><td>7.462e-6</td><td>67.294</td><td>38.94</td><td>871.9</td><td>45.23</td></tr><tr><td>MinVariance</td><td>1.513</td><td>73.50e-3</td><td>1.586</td><td>1.586</td><td>9653</td><td>7.614e-6</td><td>67.327</td><td>42.64</td><td>920.1</td><td>48.53</td></tr><tr><td>BSplines</td><td>1.440</td><td>124.7e-3</td><td>1.565</td><td>1.565</td><td>18429</td><td>6.769e-6</td><td>67.709</td><td>72.37</td><td>907.7</td><td>92.66</td></tr><tr><td>MinNorm</td><td>1.529</td><td>69.39e-3</td><td>1.598</td><td>1.598</td><td>9339</td><td>7.430e-6</td><td>67.312</td><td>40.25</td><td>927.0</td><td>46.96</td></tr><tr><td>MinVolume</td><td>1.469</td><td>74.80e-3</td><td>1.544</td><td>1.544</td><td>10193</td><td>7.339e-6</td><td>67.358</td><td>43.40</td><td>895.5</td><td>51.25</td></tr><tr><td>ApproxMinVolume</td><td>1.459</td><td>75.19e-3</td><td>1.535</td><td>1.535</td><td>10199</td><td>7.372e-6</td><td>67.359</td><td>43.62</td><td>890.2</td><td>51.28</td></tr><tr><td>Recursive</td><td>1.477</td><td>66.86e-3</td><td>1.544</td><td>1.544</td><td>9011</td><td>7.420e-6</td><td>67.294</td><td>38.79</td><td>895.8</td><td>45.31</td></tr><tr><td>Recursive2</td><td>1.471</td><td>68.14e-3</td><td>1.539</td><td>1.539</td><td>9143</td><td>7.453e-6</td><td>67.294</td><td>39.53</td><td>892.6</td><td>45.97</td></tr><tr><td>Interval</td><td>32.19e-6</td><td>92.62</td><td>92.62</td><td>01:32</td><td>9453841</td><td>9.798e-6</td><td>63.021</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>11.22</td><td>32.11</td><td>43.33</td><td>43.332</td><td>1365633</td><td>23.52e-6</td><td>61.646</td><td>34.67</td><td>46.78</td><td>14.45</td></tr><tr><td>MinVariance</td><td>14.55</td><td>38.54</td><td>53.09</td><td>53.091</td><td>1505723</td><td>25.59e-6</td><td>61.697</td><td>41.61</td><td>57.32</td><td>15.93</td></tr><tr><td>BSplines</td><td>11.77</td><td>84.89</td><td>96.66</td><td>01:36</td><td>3652473</td><td>23.24e-6</td><td>62.298</td><td>91.65</td><td>104.4</td><td>38.63</td></tr><tr><td>MinNorm</td><td>14.21</td><td>36.33</td><td>50.54</td><td>50.536</td><td>1440987</td><td>25.21e-6</td><td>61.674</td><td>39.22</td><td>54.56</td><td>15.24</td></tr><tr><td>MinVolume</td><td>12.79</td><td>41.95</td><td>54.74</td><td>54.743</td><td>1643093</td><td>25.53e-6</td><td>61.746</td><td>45.29</td><td>59.10</td><td>17.38</td></tr><tr><td>ApproxMinVolume</td><td>12.49</td><td>40.35</td><td>52.84</td><td>52.836</td><td>1646349</td><td>24.51e-6</td><td>61.747</td><td>43.56</td><td>57.04</td><td>17.41</td></tr><tr><td>Recursive</td><td>14.78</td><td>33.85</td><td>48.63</td><td>48.633</td><td>1367341</td><td>24.76e-6</td><td>61.646</td><td>36.54</td><td>52.51</td><td>14.46</td></tr><tr><td>Recursive2</td><td>15.76</td><td>34.88</td><td>50.64</td><td>50.642</td><td>1377377</td><td>25.33e-6</td><td>61.646</td><td>37.66</td><td>54.68</td><td>14.57</td></tr><tr><td rowspan="10">5</td><td>Interval</td><td>27.89e-6</td><td>59962</td><td>59962</td><td>16:39:22</td><td>5023190703</td><td>11.94e-6</td><td>71.090</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Bernstein</td><td>73.11</td><td>18268</td><td>18341</td><td>05:05:41</td><td>453747671</td><td>40.26e-6</td><td>69.785</td><td>30.47</td><td>30.59</td><td>9.033</td></tr><tr><td>MinVariance</td><td>75.51</td><td>21196</td><td>21272</td><td>05:54:32</td><td>510747573</td><td>41.50e-6</td><td>69.861</td><td>35.35</td><td>35.48</td><td>10.17</td></tr><tr><td>BSplines</td><td>72.34</td><td>61019</td><td>61091</td><td>16:58:11</td><td>1568563887</td><td>38.90e-6</td><td>70.743</td><td>101.8</td><td>101.9</td><td>31.23</td></tr><tr><td>MinNorm</td><td>75.08</td><td>20128</td><td>20203</td><td>05:36:43</td><td>484179977</td><td>41.57e-6</td><td>69.826</td><td>33.57</td><td>33.69</td><td>9.639</td></tr><tr><td>MinVolume</td><td>73.76</td><td>23774</td><td>23848</td><td>06:37:28</td><td>569356941</td><td>41.76e-6</td><td>69.932</td><td>39.65</td><td>39.77</td><td>11.33</td></tr><tr><td>ApproxMinVolume</td><td>73.21</td><td>22885</td><td>22958</td><td>06:22:38</td><td>570913097</td><td>40.09e-6</td><td>69.934</td><td>38.17</td><td>38.29</td><td>11.37</td></tr><tr><td>Recursive</td><td>72.43</td><td>18612</td><td>18684</td><td>05:11:24</td><td>454231491</td><td>40.98e-6</td><td>69.785</td><td>31.04</td><td>31.16</td><td>9.043</td></tr><tr><td>Recursive2</td><td>72.50</td><td>19138</td><td>19211</td><td>05:20:11</td><td>466446525</td><td>41.03e-6</td><td>69.785</td><td>31.92</td><td>32.04</td><td>9.286</td></tr></tbody></table>