# An Improved Diagonal Jacobian Approximation via a New Quasi-Cauchy Condition for Solving Large-Scale Systems of Nonlinear Equations

Authors: Mohammed Yusuf Waziri; Zanariah Abdul Majid
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We present a new diagonal quasi-Newton update with an improved diagonal Jacobian approximation for solving large-scale systems of nonlinear equations. In this approach, the Jacobian approximation is derived based on the quasi-Cauchy condition. The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as to carry some additional information from the functions. The effectiveness of our proposed scheme is appraised through numerical comparison with some well-known Newton-like methods.


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## Body

## 1. Introduction

Let us consider the systems of nonlinear equations(1)F(x)=0,
where F:Rn→Rn is a nonlinear mapping. Often, the mapping F is assumed to be satisfying the following assumptions:(A1)
there exists anx*∈Rn s.t F(x*)=0;(A2)
F is a continuously differentiable mapping in a neighborhood of x*;(A3)
F′(x*) is invertible.The well-known method for finding the solution to (1) is the classical Newton’s method which generates a sequence of iterates {xk} from a given initial point x0 via
(2)xk+1=xk-(F′(xk))-1F(xk),
where k=0,1,2…. The attractive features of this method are rapid convergence and being easy to implement. Nevertheless, Newton’s method requires the computation of the matrix entails the first-order derivatives of the systems. In practice, computations of some functions derivatives are quite costly, and sometimes they are not available or could not be done precisely. In this case, Newton’s method cannot be applied directly.Moreover, some substantial efforts have been made by numerous researchers in order to eliminate the well-known shortcomings of Newton’s method for solving systems of nonlinear equations, particularly large-scale systems (see, e.g., [1, 2]). Notwithstanding, most of these modifications of Newton’s method still have some shortfalls as Newton’s counterpart. For example, Broyden’s method and Chord Newton’s method need to store an n×n matrix, and their floating points operations, are O(n2), respectively.To tackle these disadvantages, a diagonally Newton’s method has been suggested by Leong et al. [3] and showed that their updating formula is significantly cheaper than Newton’s method and some of its variants. Based on this fact, it is pleasing to present an approach which will improve further the diagonal Jacobian approximation, as well as reducing the computational cost, floating points operations and number of iterations. This is what leads to the idea of this paper. The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as to carry some additional information from the functions. We organized the paper as follows. In the next section, we present the details of the proposed method. Convergence results are present in Section 3. Some numerical results are reported in Section 4. Finally, conclusions are made in Section 5.


## 2. Derivation Process

This section presents a new diagonal quasi-Newton-like method for solving large-scale systems of nonlinear equations. The quasi-Newton method is an iterative method that generates a sequence of points{xk} from a given initial guess x0 via the following form:(3)xk+1=xk-αkBkF(xk)k=0,1,2…,
where αk is a step length and Bk is an approximation to the Jacobian inverse which can be updated at each iteration for k=0,1,2…; the updated matrix Bk+1 is chosen in such a way that it satisfies the secant equation, that is,
(4)Bk+1sk=yk.It is clear that the only Jacobian information we have isy, and this is only approximation information. To this end, we incorporate more information from sk and Fk to y in order to present a better approximation to the Jacobian matrix. We consider the modification on y presented by Li and Fukushima [4]:
(5)y˘=yk+vk∥Fk∥sk,
where vk=1+max{-skTyk/∥sk∥2,0}.Our aim here is to build a square matrix, sayB, using diagonal updating scheme which is an approximation to the Jacobian inverse, and we let Bk+1 satisfy the quasi-Cauchy equation, that is,(6)y˘ky˘Tsk=y˘ky˘TBk+1y˘k.In addition, the deviation betweenBk+1 and Bk is minimized under some norms; hence, in the following theorem, we state the resulting update formula for Bk.Theorem 1.
Assume thatBk+1 be the diagonal update of a diagonal matrix Bk. Let us denote the deviation between Bk and Bk+1 as Ψk=Bk+1-Bk. Suppose that y˘k≠0 which is defined by (5). Consider the following problem:(7)min12∥Ψk∥F2s.ty˘ky˘T(Bk+Ψk)y˘k=y˘ky˘Tsk,
where ∥·∥F denotes the Frobenius norm. Hence, the optimal solution of (7) is given by
(8)Ψk=(y˘ky˘Tsk-y˘ky˘TBky˘k)tr(Vk2)Vk,
where Vk=diag((y˘ky˘(1))2,(y˘ky˘(2))2,…,(y˘ky˘(n))2), ∑i=1n‍(y˘ky˘(i))4=tr(Vk2), and Tr is the trace operation.Proof.
Consider the Lagrangian function of (7):
(9)L(Ψk,α)=12∥Ψk∥F2+α(y˘ky˘TΨky˘k-y˘ky˘Tsk+y˘ky˘TBky˘k),
where α is the corresponding Lagrangian multiplier. By differentiating L with respect to each Ψk(1),Ψk(2),…,Ψk(n), and setting them all equal to zero, we obtain
(10)Ψk(i)=-α(y˘ky˘(i))2∀i=1,2,…,n.
Multiplying both sides of (10) by (y˘ky˘(i))2 and summing them all give
(11)∑i=1n‍(y˘ky˘(i))2Ψk(i)=-α∑i=1n‍(y˘ky˘(i))4foreveryi=1,2,…,n.
DifferentiatingL with respect to α, and since y˘ky˘TΨky˘k=∑i=1n‍(y˘ky˘(i))2Ψ(i), then we have
(12)∑i=1n‍((y˘ky˘(i))2Ψk(i))=y˘ky˘Tsk-y˘ky˘TBky˘k.
Equating (11) and (12) and substituting the relation into (10), finally we have
(13)Ψk(i)=y˘ky˘Tsk-y˘ky˘TBky˘k∑i=1n(y˘ky˘(i))4(y˘ky˘(i))2∀i=1,2,…,n.
SinceBk(i) is a diagonal component of Bk, y˘k(i) is the ith component of vector y˘k, then Vk=diag((y˘ky˘(1))2,(y˘ky˘(2))2,…,(y˘ky˘(n))2) and ∑i=1n‍(yk(i))4=tr(Vk2). We further rewrite (13) as
(14)Ψk=(y˘ky˘Tsk-y˘ky˘TBky˘k)tr(Vk2)Vk,
which completes the proof.Hence, the best possible updating formula for diagonal matrixBk+1 is given by
(15)Bk+1=Bk+(y˘ky˘Tsk-y˘ky˘TBky˘k)tr(Vk2)Vk.Now, we can describe the algorithm for our proposed method as follows.Algorithm IDJAStep 1.
Choose an initial guessx0, σ∈(0,1), γ>1, B0=In, α0>0, and let k:=0.Step 2.
ComputeF(xk), and If ∥F(xk)∥≤10-8 stop.Step 3.
Computed=-F(xk)Bk.Step 4.
If∥F(xk+αkdk)∥≤σ∥F(xk)∥, retain αk and go to Step 5. Otherwise set αk+1=αk/2 and repeat Step 4.Step 5.
If∥F(xk+αkdk)-F(xk)∥≥∥F(xk+αkdk)∥-∥F(xk)∥, retain αk and go to Step 6. Otherwise set αk+1:=αk×γ and repeat Step 5.Step 6.
Letxk+1=xk+αkdk.Step 7.
If∥xk+1-xk∥2+∥F(xk)∥2≤10-8 stop. Also go to Step 8.Step 8.
If∥ΔFk∥2≥ϵ1 where ϵ1=10-4, compute Bk+1 as defined by (15); if not, Bk+1=Bk.Step 9.
Setk:=k+1 and go to Step 2.


## 3. Convergence Result

This section presents local convergence results of the IDJA methods. To analyze the convergence of these methods, we will make the following assumptions on nonlinear systemsF.Assumption 2.
(
i
)
F is differentiable in an open convex set E in ℜn.
(
ii
) There exists x*∈E such that F(x*)=0; F′(x) is continuous for all x.
(
iii
)
F
′
(
x
) satisfies the Lipschitz condition of order one that is there exists a positive constant μ such that
(16)∥F′(x)-F′(y)∥≤μ∥x-y∥,
for all x,y∈ℜn.
(
iv
) There exist constants c1≤c2 such that c1∥ω∥2≤ωTF′(x)ω≤c2∥ω∥2 for all x∈E and ω∈ℜn.We can state the following result on the boundedness of{∥Ψk∥F} by assuming that, without loss of generality, the updating matrix (15) is always used, then we have the following.Theorem 3.
Suppose that{xk} is generated by Algorithm IDJA where Bk is defined by (15). Assume that Assumption 2 holds. There exists β>0, δ>0, α>0 and γ>0, such that if x0∈E and B0 satisfies ∥I-B0F′(x*)∥F<δ for all xk∈E then
(17)∥I-BkF′(x*)∥F<δk,
for some constant δk>0, k≥0.Proof.
Since∥Bk+1∥F=∥Bk+Ψk∥F, it follows that
(18)∥Bk+1∥F≤∥Bk∥F+∥Ψk∥F.
Fork=0 and assuming B0=I, we have
(19)|Ψ0(i)|=|y˘0y˘Ts0-y˘0y˘TB0y˘0tr(V02)(y˘0y˘(i))2|≤|y˘0y˘Ts0-y˘0y˘TB0y˘0|tr(V02)(y˘0y˘(max))2,
where (y˘0y˘(max))2 is the largest element among (y˘0y˘(i))2,i=1,2,…,n.
After multiplying (19) by (y˘0y˘(max))2/(y˘0y˘(max))2 and substituting tr(V02)=∑i=1n‍(y˘0y˘(i))4, we have
(20)|Ψ0(i)|≤|y˘0y˘Ts0-y˘0y˘TB0y˘0|(y˘0y˘(max))2∑i=1n(y˘0y˘(i))4(y˘0y˘(max))4.
Since(y˘0y˘(max))4/∑i=1n‍(y˘0y˘(i))4≤1, then (20) turns into
(21)|Ψ0(i)|≤|y˘0y˘TF′(x)y˘0-y˘0y˘TB0y˘0|(y˘0y˘(max))2.
From Assumption2 and B0=I, (21) becomes
(22)|Ψ0(i)|≤|c-1|(y˘0y˘Ty˘0)(y˘0y˘(max))2,
where c=max{|c1|,|c2|}.
Since(y˘0y˘(i))2≤(y˘0y˘(max))2 for i=1,…,n, it follows that
(23)|Ψ0(i)|≤n|c-1|(y˘0y˘(max))2(y˘0y˘(max))2.
Hence, we obtain(24)∥Ψ0∥F≤n3/2|c-1|.
Supposeα=n3/2|c-1|, then
(25)∥Ψ0∥F≤α.
From the fact that∥B0∥F=n, it follows that
(26)∥B1∥F≤β,
where β=n+α>0.
Therefore, if we assume that∥I-B0F′(x*)∥F<δ, then
(27)∥I-B1F′(x*)∥F=∥I-(B0+Ψ0)F′(x*)∥F≤∥I-B0F′(x*)∥F+∥Ψ0F′(x*)∥F≤∥I-B0F′(x*)∥F+∥Ψ0∥F∥F′(x*)∥F;
therefore, ∥I-B1F′(x*)∥F<δ+αϕ=δ1.
Hence, by induction,∥I-BkF′(x*)∥F<δk for all k.


## 4. Numerical Results

In this section, the performance of IDJA method has been presented, when compared with Broyden’s method (BM), Chord Newton’s method (CN), Newton’s method (NM), and (DQNM) method proposed by [3], respectively. The codes are written in MATLAB 7.4 with a double precision computer; the stopping condition used is
(28)∥sk∥+∥F(xk)∥≤10-8.The identity matrix has been chosen as an initial approximate Jacobian inverse.We further design the codes to terminates whenever one of the following happens:(i)
the number of iteration is at least 200 but no point ofxk that satisfies (28) is obtained;(ii)
CPU time in seconds reaches 200;(iii)
Insufficient memory to initial the run.The performance of these methods are compared in terms of number of iterations and CPU time in seconds. In the following, some details on the benchmarks test problems are presented.Problem 1.
Spares 1 function of Shin et al. [5]:(29)fi(x)=xi2-1i=1,2,…,n,x0=(5,5,…,5).Problem 2.
Trigonometric function of Spedicato [6]
(30)fi(x)=n-∑j=1n‍cosxj+i(1-cosxi)-sinxi,i=1,…,n,x0=(1n,1n,…,1n).Problem 3.
System ofn nonlinear equations
(31)fi(x)=sin(1-xi)×∑i=1n‍xi2+2xn-1-3xn-2-12xn-4+12xn-5-xiln(9+xi)-92exp(1-xn)+2i=1,2,…,n,x0=(0,0,…,0)T.Problem 4.
System ofn nonlinear equations
(32)fi(x)=xi2-4exp(sin(4-xi2))+sin(4-xi)2+i(xn-xi)2+2n-∑i=1nxicosxii=1,…,n,x0=(2.8,2.8,2.8,…,2.8).Problem 5.
System ofn nonlinear equations
(33)fi(x)=(∑i=1n‍xi)(xi-2)+(cosxi-2)-1i=1,…,n,x0=(1,1,1,…,1).Problem 6.
System ofn nonlinear equations
(34)fi(x)=∑i=1n‍xi2-(sin(xi)-xi4+sinxi2)i=1,…,n,x0=(.5,.5,.5,…,.5).Problem 7.
System ofn nonlinear equations
(35)fj(x)=(∑i=1n‍xi2-1)(xj-1)+xj(∑i=1n‍(xi-1))-n+1fn(x)=(∑i=1n‍xi2-1)(xn-1)+(cosxn-1)-1j=2,…,n-1,x0=(.5,.5,.5,…).Problem 8.
System ofn nonlinear equations
(36)fi(x)=(1-xi2)+xi+xi2xn-2xn-1xn-2i=1,2,…,n,x0=(.5,.5,…,.5).The numerical results presented in Tables1, 2, 3, 4, and 5 demonstrate clearly the proposed method (IDJA) shows good improvements, when compared with NM, CN, BM, and DQNM, respectively. In addition, it is worth mentioning, the IDJA method does not require more storage locations than classic diagonal quasi-Newton’s methods. One can observe from the tables that the proposed method (IDJA) is faster than DQNM methods and required little time to solve the problems when compared to the other Newton-like methods and still keeping memory requirement and CPU time in seconds to only O(n).Table 1
Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
NM
CN
BM
DQNM
IDJA
NI
CPU
NI
CPU
NI
CPU
NI
CPU
NI
CPU
1
50
7
0.046
55
0.031
15
0.031
14
0.016
2
0.011
2
50
9
0.078
344
0.062
15
0.031
15
0.031
13
0.031
3
50
10
0.062
—
—
—
—
20
0.016
10
0.016
4
50
—
—
—
—
—
—
19
0.031
9
0.031
5
50
12
0.078
—
—
42
0.031
16
0.016
8
0.015
6
50
8
0.064
—
—
16
0.032
14
0.031
7
0.014
7
50
8
0.094
—
—
—
—
25
0.031
14
0.010
8
50
11
0.064
—
—
11
0.0312
11
0.016
9
0.016Table 2
Numerical Results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
NM
CN
BM
DQNM
IDJA
NI
CPU
NI
CPU
NI
CPU
NI
CPU
NI
CPU
1
100
7
0.156
98
0.094
15
0.043
14
0.016
2
0.011
2
100
10
0.187
—
—
18
0.062
16
0.032
13
0.032
3
100
7
0.203
—
—
24
0.140
15
0.031
7
0.015
4
100
—
—
—
—
—
—
13
0.031
10
0.030
5
100
13
0.265
—
—
53
0.109
17
0.031
12
0.031
6
100
8
0.203
—
—
16
0.047
14
0.031
7
0.017
7
100
8
0.185
—
—
—
—
26
0.031
16
0.030
8
100
11
0.234
—
—
11
0.094
11
0.032
10
0.016Table 3
Numerical Results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
NM
CN
BM
DQNM
IDJA
NI
CPU
NI
CPU
NI
CPU
NI
CPU
NI
CPU
1
250
7
0.359
100
0.109
15
0.101
14
0.034
2
0.032
2
250
11
0.640
—
—
21
0.218
18
0.032
8
0.031
3
250
8
0.499
—
—
29
0.250
16
0.016
9
0.016
4
250
—
—
—
—
—
—
15
0.031
10
0.032
5
250
14
0.827
—
—
—
—
19
0.031
8
0.016
6
250
8
0.686
—
—
24
0.250
14
0.031
10
0.031
7
250
8
0.499
—
—
—
—
27
0.031
14
0.031
8
250
11
0.484
—
—
11
0.125
11
0.031
10
0.016Table 4
Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
NM
CN
BM
DQNM
IDJA
NI
CPU
NI
CPU
NI
CPU
NI
CPU
NI
CPU
1
500
7
0.796
101
0.702
15
0.671
14
0.016
2
0.011
2
500
13
1.997
—
—
23
0.972
19
0.031
9
0.032
3
500
7
1.4352
—
—
—
—
17
0.031
9
0.031
4
500
—
—
—
—
—
—
12
0.030
10
0.031
5
500
15
2.449
—
—
—
—
21
0.031
9
0.031
6
500
8
2.184
—
—
23
0.998
14
0.032
10
0.045
7
500
8
1.498
—
—
—
—
32
0.047
15
0.047
8
500
11
1.451
—
—
11
0.515
11
0.031
9
0.031Table 5
Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
NM
CN
BM
DQNM
IDJA
NI
CPU
NI
CPU
NI
CPU
NI
CPU
NI
CPU
1
1000
7
2.730
103
3.167
38
9.438
14
0.016
2
0.011
2
1000
—
—
—
—
31
7.722
20
0.032
8
0.043
3
1000
9
5.819
—
—
—
—
17
0.031
9
0.031
4
1000
—
—
—
—
—
—
11
0.064
10
0.064
5
1000
16
8.705
—
—
—
—
22
0.031
10
0.031
6
1000
8
6.474
—
—
—
—
14
0.062
11
0.061
7
1000
8
4.321
—
—
—
—
38
0.062
31
0.047
8
1000
11
4.882
—
—
11
2.418
11
0.032
10
0.031


## 5. Conclusions

In this paper, we present an improved diagonal quasi-Newton update via new quasi-Cauchy condition for solving large-scale Systems of nonlinear equations (IDJA). The Jacobian inverse approximation is derived based on the quasi-Cauchy condition. The anticipation has been to further improve the diagonal Jacobian, by modifying the quasi-Cauchy relation so as to carry some additional information from the functions. It is also worth mentioning that the method is capable of significantly reducing the execution time (CPU time), as compared to NM, CN, BM, and DQNM methods while maintaining good accuracy of the numerical solution to some extent. Another fact that makes the IDJA method appealing is that throughout the numerical experiments it never fails to converge. Hence, we can claim that our method (IDJA) is a good alternative to Newton-type methods for solving large-scale systems of nonlinear equations.



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