NAT L INST. OF STAND & TECH NBS

Reference publications

NBSIR 86-3389

A Near-Optimal Starting Solution

for Polynomial Approximation of a

Continuous Function in the L-j Norm

P. D. Domich

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

QC

100

• U56

86-3389

1986

DEPARTMENT OF COMMERCE, Malcolm Baidrige. Secr etary

GNAI. BUREAU Of STANDARDS, tin- >: A.nbiei Direoto''

3d June 1986

NBS

RESEARCH

INFORMATION CENTER

NBSIR 86-3389

A Near-Optimal Starting Solution

for Polynomial Approximation of a

Continuous Function in the L-i Norm

P. D. Domich

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

U S. DEPARTMENT OF COMMERCE, Malcolm Baldrige. Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director

Issued June 1986

ABSTRACT

This paper presents a method of selecting a near-optimal starting solution for

a large class of discrete polynomial approximation problems in the Lj norm.

While it is possible to prove the optimality of these advanced starting

solutions for only a small class of continuous polynomial approximation problems, empirical evidence indicates the starting bases will be nearly optimal

for a much larger class of discrete problems. This paper presents the method

used to determine the starting basis and a heuristic justification backed by

empirical results supporting its use.

iii

1

I. INTRODUCTION

The investigation of a near-optimal starting solution for discrete polynomial

approximation problems in the norm resulted from a study to develop a

methodology for comparing and evaluating mathematical programming software.

In this study four approximation algorithms^ > 3 ,4 ,5 were tested on a large

set of problems with diverse problem structures and characteristics. Included

were 80 problems used to test code performance in the polynomial approximation

of a continuous function, a specific type of approximation problem. In these

problems, the continuous function being approximated, the degree of the

approximating polynomial, and the number of observation points were

independently varied. In all problems, the observations were equally spaced

over the same interval of approximation, [0,1].

© «

When the interpolation points defining the best solution were tabulated,

with the degree of the polynomial and the number of observations fixed, but

varying the functional form, a recurrent pattern emerged which indicated that

the points of interpolation remained relatively stationary within the normalized interval regardless of the functional form being approximated. This

pattern suggested a method of selecting a starting basis which would be nearly

optimal for a large number of approximation problems.
2

After further study, an analytic expression was identified that can be used to

calculate a feasible starting solution which, when used in the continuous

analog of the discrete approximation problem, is the optimal solution in the

Lj norm. ^ Calculating this starting solution and using it in the discrete

approximation problem also produced impressive empirical results, both in

terms of its relative distance to the optimal solution and also in the marked

reduction in the number of iterations needed to reach the optimal solution.

Furthermore, the integration of this method of selecting the starting basis

into existing algorithms is easy. An explanation of the success of this

method can be found in the underlying approximation theory and is presented in

the next two sections. The final section reports the results of using this

near-optimal starting basis on a large set of test problems.

II. THE DISCRETE POLYNOMIAL APPROXIMATION PROBLEM

The general form of the discrete Lj approximation problem can be formulated

as follows.

Given n sets of observations on a dependent variable y^ and a single independent variable x^

{xj^y*} »

determine m+1 parameters

6 - Bq* ^1* B 2 » • • • » ^m *

Z i-1

m 4 7± ~ l B

j x

,

I j-0 1

which minimize
3

This is the polynomial approximation of a continuous function when y-j_ = f(x^),

where f(x) is a continuous function defined over an approximation interval I.

Introducing variables d and d , corresponding to the positive and negative

deviation, the approximation problem can be posed as the following linear

programming problem.

subject to

n Minimize £ i=l

+

d + d

i i

»

+ raj

d - d + yi - [ 8^ x

i i

1

-i=n ^^ 0 ,

+ —

d _> 0 , d^ _> 0 for i = 1, 2, 3, . . . ,n

i

For this class of problems it is not unreasonable to assume the problems are

of full rank (independent columns) and overdetermined (the number of observations strictly greater than m, the number of 3 parameters in the problem).

A further condition necessary for one to "predict" the best solution to any

problem of this type is that the solution be unique. When the uniqueness condition is satisfied and the number of observations sufficiently large, the

starting solution we suggest is very close to the optimum. Unfortunately,

uniqueness cannot be guaranteed for certain approximation problems.
4

III. FUNCTIONAL ANALYSIS AND PROPERTIES OF Li APPROXIMATION OF A CONTINUOUS

FUNCTION

This section presents the theoretical basis behind a unique best L^ approximation solution for a special class of continuous functions. The section begins

by presenting several properties of the polynomial approximation of a continuous function in the Lj norm. These properties establish the existence,

uniqueness and optimality of a solution. A description of a class of functions which have these properties is developed and it will be shown that the

starting basis we propose will be the optimal Lj solution. An alternate

representation of the optimal L^ solution, which serves as a useful guide in

determining the "goodness" of an approximation is also presented as is a

method for calculating the starting basis easily and efficiently.

Since the best Lj polynomial approximation of degree m to a continuous function, f(x), intersects the function in- at least m+1 points^, we will examine

the additional characteristics a function must possess to insure the uniqueness of the optimal solution. The first assumption on the function requires

the existence of a non-trivial approximation problem, i.e., f(x) is different

from the Lj approximation almost everywhere. Had this not been assumed, any

m+1 observations would have defined the best Lj approximation. By requiring

that f(x) have a m+l st derivative that is continuous and non-constant, a

nontrivial approximation problem is guaranteed. We characterize any such

function as a "higher-order function" than the polynomial of degree m. This

is closely analogous to the term "higher-order polynomials” though more

general
5

The uniqueness of the final solution and ultimately the optimality of the suggested starting basis is established by making one further requirement on the

derivatives of f(x). If the m+l st derivative, fm+ ^(x) , is nocvanishing over

the approximation interval, then it will be shown that the best Lj polynomial

approximation to f(x) will intersect at exactly m+1 observations and, therefore, will be unique. With this condition satisfied, one can then find the

m+1 observations which define the optimal Lj approximation to the continuous

problem, evaluate the function at these points and enter these observations to

the discrete problem basis. Solving the corresponding (m+1)* (m+1) system of

equations defined by this new basis will locate a near-optimal L]_ solution to

the discrete problem.

The following examples and subsequent theorems will solidify the theoretical

underpinnings bounding the number of intersections between the continuous

function and the best approximating polynomial. First consider the problem of

locating all the possible intersections between a second-order polynomial

P 2 (x) and a first-order polynomial p^(x), (shown graphically in figure 1).

FIGURE 1
6

Three possible situations exist: first, p^(x) and P 2 (x) fail to intersect in

[a, b] ; second, pj(x) and P 2 (x) are tangent which will produce only one point

of intersection; or finally, p^(x) and P 2 (x) intersect at two points. The

problem of locating the points of intersection of p^(x) and P 2 (x) is

equivalent to that of finding all possible roots of a residual function defined as their difference in [a, b],

r(x) = p 2 (x) - PjCx) = 0, x e [a, b].

Since r(x) is binomial, it can have at most two real-valued roots. This

argument can easily be extended for any two polynomials, pm

(x) and pra+ ^(x), of

degree m and m+1 respectively, with no loss of generality.

The task of establishing the number of intersections of pm

(x) and a polynomial

of degree m+k, Pni+k^ x ^ » f° r k > 1 , or any function of higher order is slightly

more difficult. Again, the behavior of the derivatives will determine the

number of roots the residual function

r(x) = Ptn+kCx) “ Pm

(x),

will have over [a, b]. By assuming a restriction on p^k, it is again possible to limit the number of roots r(x) may have to at most m+1 and therefore

define a larger class of problems which will have a unique best Lj solution.
7

Lemma 1. Suppose a function f(x) is continuously differentiable at least k

times in an interval [a, b]. Suppose further that f k(x) > 0 (<0) in [a, b].

Then f k”l(x) is strictly monotone increasing (decreasing) in [a, b], and there

is only one possible zero of f k~*(x) in [a, b].

Together with the following theorem it will be possible to establish upper

bounds on the number of real-valued roots for each successively lower-ordered

derivative

Theorem 1 [McCormick^]. Suppose a function f(x) is continuously differentiable

at least k times in an interval [a, b]. Suppose that f k (x) has q zeros in that

interval and that they are known and ordered as a < zj < 22 < Z

3 ... <Zq _< b.

Then fk”*(x) is strictly monotone in [a, zj], [z^, z i+ll> (for i=l , . . . ,q-i ) , and

[z

q

, b]. There are at most q+1 zeros of fk_ 1 (x) in [a, b], Specifically, there

may be one in [a, z^], one in each of [z^, z^+j], (for i=l,...,q-l) and one in

[zq, b] . If f k~^(a) *f k~^(zj) > 0 there is no zero in that interval, otherwise

there is exactly one there. If (for i=»l, 2,...,q-l) f k_ 1 (z.£) *f k~ 1 (zi+i) > 0,

there is no zero in that interval. If f k“^-(Zq) *f k“^(b) > 0, there is no zero in

[Zq, b]. Otherwise there is exactly one.

Proof. Since f k(z^) 3 f k

(z^+ j) * 0 and there are no zeros between z^ and z i+ j then f k(x) > 0 for all x in ( z it z i+1 ), or f k

(x) < 0 in that interval. Thus

the hypotheses of the previous lemma are satisfied and the appropriate conclusion follows for [z^t z i+l 1 • The other cases are identical.
8

Thus, for any function f(x) with a nonvanishing and monotonic (m+l) st derivative Theorem 1 can be applied successively to construct upper bounds to each

successively lower-order derivative until a upper bound of m+1 can be

established for the number of real valued roots to f(x) over the interval

[a,b]. The upper bound on the number of roots of the residual function r(x)

is, therefore, dependent only on the behavior of fm+ ^(x) since the (m+l) st

derivative of the approximating polynomial pm(x) is zero valued over the real

line , thus

rm+1 (x) = fm+1 (x) - pm+1 (x) m

= fm+1 (x).

Under the assumptions above and by requiring that the associated approximation

problem is of full rank, the existence of a unique optimal solution to the

discrete polynomial approximation problem defined by exactly m+1 observations

is guaranteed. The final question remains. How to select the m+1 observations close to the optimal solution?

As shown in R±yil inland Rice there exists a large class of continuous

functions for which the optimal solution in the continuous approximation

problem can be determined a priori whenever it is known a priori that the

optimal solution is unique and that the interpolation of these two functions

occur at m+1 points. The necessary and sufficient requirements on f(x) for

this to occur are 1) f(x) is continuous, 2) the difference of the function

f(x) and the Lj approximating polynomial Pm

(x)is different from zero almost

everywhere on [a,b], 3) the residual function, r(x)-f (x)-Pm

(x) , changes In

sign at a unique set of m+1 points in the interval [a,b]. These points of

interpolation are determined in the following theorem.
9

Theorem 2: If f(x) is continuous and differentiable, and f(x) and pm

(x)

intersect nrt-1 times in the interval [-1, 1], then the least approximation

is the unique polynomial pm

*(x) which satisfies

Pm

i H ill ,

(cos ) = f(cos ), for j=l , 2, m+l. 1- 1 m+2 m+2

til This optimal set of observations are the roots of the m

c degree Chebyshev

polynomial of the second kind, Um(x) . The explicit expression for this family

of polynomials is listed below:

Um( x ) m/2

= l (-1

i=l

(m-i)

i ! (m-2i )

(2x)

m-2i

or equivalently

sin(nH-l)0

Um (cos 0) for x=cos0 ; sin©

which have the following recurrence relation

Urc+iU) - 2xU

m

(x) - U

m_ x

( x ) . 1

The first four of these polynomials are:
10

Uo^x) “ 1»

UjCx) = 2x,

U2 (x) = 4x2-1;

U3 (x) = 8x3_4x#

The series in which these polynomials are used in approximation to a continuous function f(x),

f(x) = lk bjUj (x)

j=0

are called Chebyshev Series and if

lim b-j^ = 0 > ^

->-00

the series is termed a Chebyshev series expansion of the function f(x).

Chebyshev series expansions have long been known to converge very quickly to

the target function compared to other series approximations and can be a useful tool in determining the significance of the neglected term in the polynomial approximation. Generating the equivalent Chebyshev representation to

the best Li approximating polynomial provides the user with information on the

relative significance of the last term of the Chebyshev representation. When

the coefficient of the higher order terms are very close to zero, the user can

assume little accuracy will be gained by increasing the degree of the approximating polynomial. Conversely, if the last term of the Chebyshev approximation are of the same order as the previous coefficients, attempts to Include

more terms may provide a better approximation, or give the user an Indication

that the target function f(x) may not have a converging Chebyshev series

expansion and an alternate approximation method should be used.
11

IV. GENERATING THE STARTING BASIS

The method used in calculating the starting basis involves determining the

zero-valued points of the Chebyshev polynomial. Using the trigonometric

representation of the Chebyshev polynomials,

Sin(m+1 )

Um

( x ) " U

m (cos9) = Sin9

it is clear that the zero values of Um

(x) occur at

kir x.* = cos for k = 1 , 2, . . . , m and -1 < x^ < 1

m-t-1

In our work, the zero-valued points of Um

(x) were translated into the interval

of approximation, evaluated by the continuous function, and entered as constraints to the problem. Results of using these points as a starting basis

are presented in the next section.

V. COMPUTATIONAL RESULTS

In our experiment, 256 approximation problems were used in testing the performance of the advanced starting method described. All problems were produced

by an Li polynomial approximation test-problem generator 1 ^ using eight different continuous functions, varying both in form and difficulty (see Table la of

the appendix). The problems generated ranged in degree from 3 to 10, with

100, 200, 400, and 800 observations over the closed interval [0,1], Each
12

problem was solved twice. One run of all 256 problems did not use the

advanced solution method but included together with the sorted equidistant

observations, the m+1 generated observations {xj_ } used in the new starting

solution method, thereby guaranteeing identical problems in both runs. The

256 problems were then rerun. The observations (xi } now formed the first m+1

constraints to the problem and were selected to enter the starting basis by

the Li code on the first m+1 pivots. The method used to solve all problems

was double-precision L^-approximation algorithm developed by N. N.

Abdelmalek.

In this section, the results from solving the problem with the new starting

solution method are compared to results from the original method. Two performance measures were used in the comparison. One measured the distance between the objective function value Z at the near-optimal starting point, and

the optimum objective function value Z , normalized by the distance from the

g — * starting objective function value, Z (where 3=0), to Z . Or, mathematically,

the normalized distance is

Zi “

D . = i=l,2, ...,256

Z

s - Zi*

A second measure was difference in iteration count between using the nearoptimal "starting basis" and the starting basis defined by 3=0.
13

The results in measuring the performance of the advanced solution method by

the normalized objective function distance are given in Table 2 of the appendix and summarized

0.25

f

0.20

E

|

0.15 o

LLS 0.10 c

<0

</} a

o 0.05

0.00

0 50 100 150 200 250 300

Number of the Problem

FIGURE 2. Relative Distance from the Optimum

Of the total 256 problems, only 12 were more than 1 percent from the optimum.

Closer examination of the outliers reveals that all outliers of magnitude

greater than 1 percent are related to problems produced by two of the eight

continuous functions being approximated. Closer investigation reveals that

the distance D is worst in the approximations made by polynomials of lower degree. The functions (illustrated in Figures 3 and 4) connected with these

outlier values are highly nonlinear and approximations by lesser degree polynomials are very poor.

in Figure 2.

Advanced Starting Solution Results

*The regular pattern of the outliers in Figure 2 illustrate that problems

of a specific function and degree are the same relative distance from the

optimum for 100, 200, 400, and 800 observations.
Observed

Values

Y = EXP((6*X + 1)*SIN(6*X + 1)) Y = (1 50 * X)/(1 + ( 1 5

* X) * *2)

0.0 0.2 0.4 0.6 0.8 1.0

Interval of Approximation

FIGURE 3. FUNCTION 3:

Interval of Approximation

FIGURE 4. FUNCTION 5:

Though selection of a low degree polynomial was unwise by the user, the starting basis did reduce the number of iterations required to solve these problems

in 13 of the 16 cases, as shown in Table 1. Similar data for problems of 200,

400, and 800 observations are given in Table 2a of the appendix.

DEGREE

POLYNOMIAL

OF 10 9876543

STARTING

METHOD NO YES NO YES NO YES NO YES NO YES NO YES NO YES NO YES

FUNCTION #3 28 14 42 27 19 14 24 8 20 23 24 19 13 11 14 15

FUNCTION #5 26 15 19 14 24 8 18 7 14 6 19 17 13 10 23 31

TABLE 1. ITERATIONS NEEDED TO OBTAIN AN OPTIMAL SOLUTION

(NUMBER OF OBSERVATIONS => 100).
15

Increasing the degree of the approximating polynomial, in these outlier cases,

did reduce the values substantially, with 4 of the 16 problems listed above

having an advanced starting objective function value identical to the optimal,

to 8 significant digits.

Over all 256 problems, a total of 62 problems using the advanced method were

— 8

at the optimum, requiring no additional pivots for an accuracy of 10 . A

total of 5558 iterations were necessary to solve all 256 problems using the

standard method. Solving the 256 problems using the starting method reduced

the total number of iterations needed to solve all problems by 2028, a reduction of 36 percent. In all, 241 problems had a reduced iteration count, 2

marked no change, and 13 increased.

In problems where the degree of the polynomial was overspecified, a degenerate

problem much more difficult to solve, is produced. We noted several problems

of this type which had an increased iteration count but also had a very small

normalized distance measure This suggests there are many other solutions

with objective function values very close to the optimal solution. This could

force the problem solver to test a large number of solutions of nearly

identical objective function value before obtaining the optimal objective

function value. Solving these problems with a polynomial of lesser degree

improves iteration count, and in several cases required no additional pivots.

From this we can infer that in the problems where a lesser degree polynomial

fit the function quite well, the Chebyshev expansion converged very quickly,

thereby producing a good approximation to the problem. Inferences drawn from

the results such as those described above may provide the user insight into
16

the problem concerning "goodness of fit” and "sufficiency of degree" of the

approximating polynomial.

The results of this computational study indicate ways in which L} -approximation

codes used for polynomial approximation problems can be improved. They are

collected below.

1. Use the zeroes of the Chebyshev polynomials as the observations in

the starting basis.

2. Convert the Li polynomial coefficients into the coefficients of the

equivalent Chebyshev polynomial representation to provide the user with information on the rate of convergence of the Chebyshev terms and the correctness

of the user's degree specification.
17

VI. REFERENCES

[1] Domich, P. D., Hoffman, K. L. , Jackson, R. H. F., Saunders, P. B., and

Shier, D. R. , Evaluation of Li Codes Using Polynomial Approximation

Problems, unpublished working paper, National Bureau of Standards.

[2] Abdelmalek, N. N. , An Efficient Method for the Discrete Linear

Approximation Problem, Mathematics of Computation, 29_, 844-850 (1975).

[3] Armstrong, R. D. and Frome, E. L., A Comparison of Two Algorithms for

Absolute Deviation Curve Fitting, Journal of the American Statistical

Association, _7_1_, No. 354, 328-330 (1976).

[4] Barrodale, I. and Roberts, F. D. K. , An Improved Algorithm for Discrete

Linear Approximation, SIAM Journal on Numerical Analysis, _10_, No. 5,

839-848 (1973).

[51 3artels, R. H. , Conn, A. R., and Sinclair, J. W. , Minimization

Techniques for Piecewise Differentiable Functions: The L^ Solution to an Overwhelmed Linear System, SIAM Journal on Numerical Analysis, 15 , No. 2, 224-241 (1978).

[6] Rivlin, T. J., An Introduction to the Approximation of Functions

(Blaisdell Publishing. Company , Waltham, MA, pg. 73, 1969).

[7] Barrodale, I. and Roberts, F. D. K. , An Improved Algorithm for Discrete

Linear Approximation, SIAM Journal on Numerical Analysis, _10, No. 5,

pg. 839 (1973).

[8] McCormick, G. P., Finding the Global Minimum of a Function of One

Variable Using the Method of Constant Signed Higher Order Derivatives,

GWU Technical Report T-411, The George Washington University, 1979.

[9] Rivlin, T. J., An Introduction to the Approximation of Functions , (Blaisdell Publishing Company, Waltham, MA, pg. 73, 1969).

[10] Rice, J. R. , The Approximation Functions, _1. (Addison-Wesley , Reading,

MA, page 106, 1964).

[11] Rivlin, T. J. , An Introduction to the Approximation of Functions,

(Blaisdell Publishing Company, Waltham, MA, pg. 73, 1969).

[12] Davis, P. I., Interpolation and Approximation (Blaisdell Publishing

Company, Waltham, MA, pg. 366, 1969).

[13] Domich, P. D. , Lawrence ,J. t and Shier, D.R. Generators for Discrete

Polynomial L^ Approximation Problems, Journal of Research, National

Bureau of Standards, Vol. 84, No. 6, November-December 1979.

[14] Abdelmalek, N. N. , An Efficient Method for the Discrete Linear L^

Approximation Problem, Mathematics of Computation, 29, 844-850 (1975).
18

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19

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Appendix

Table

2a

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w/Starting

w/o

Starting

Basis

Basis

23

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Table

2a

(Cont'd.
M6S-114A (rev. ®-7S)

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