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BERNSTEIN POLYNOMIALS AND NUMERICAL INTEGRATION

ERWIN KREYSZIG Department of Mathematics, University of Windsor, Windsor, Ontario, Canada

INTRODUCTION

J. Reinkenhof' has recently proposed an interesting method of numerical integration using Bernstein polynomials, on which we want to comment. (He also considered numerical differen- tiation on a similar basis. We leave this much more delicate process aside.) Using a convenient notation, we can describe Reinkenhof's method as follows.

To integrate a given function g ( y ) from a given a to y ( c b ; b given), consider f(x) = g ( y ) , where x = ( y - a) / (b - a ) , so that the interval a S y C b corresponds to the interval 0 S x S 1. Set

f(x) = B n ( f ; x) + r n ( x )

where

is the n th Bernstein po lynomia l corresponding to f(x). Approximate rn(x) by the corresponding Bernstein polynomial, that is, set

f(X)zBn(f; x ) + B n ( r n ; x). (2)

On the right, develop powers of 1 -x by the binomial theorem. Then integrate:

In terms of the given function g(u) this can be written

where x = ( y - a ) / ( b - a ) . Reinkenhof considered an engineering application with n = 15. The numerical results were found to be in satisfactory agreement with those produced by a composite Newton-Cotes integration in which the integrand was approximated by three fifth order polynomials. A general error analysis was not given. This raises the following problems on which we want to comment.

It is known that Bernstein polynomials, although uniformly convergent, are generally considered rather poor for numerical processes. How comes that, nevertheless, the present process seems to be satisfactory? Is that example typical with respect to accuracy or not? Is the new process similar to known processes?

0029-598 1/79/02 14-0292$01 .OO Received 5 June 1978

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ON BERNSTEIN POLYNOMIALS

Bernstein polynomials were introduced by S. Bernstein in a constructive proof2 of the Weier- strass approximation theorem. This proof also yields error bounds, but the latter are much too conservative to be practically useful. (Example:2 The requirement If(x) - B,(f; x)l < 0.2 for f(x) = ex on [O, 13 gives n > lo4, but actually, If(x) -B2(f; x ) l < 0.2.)

The poor quality of the approximation by Bernstein polynomials can be concluded from the fact that to f ( x ) = x 2 , x 3 , * * there correspond B,(f; x ) given by3

x2+n-'x(1-x),

x 3 + 3n-'x2(1 - x ) + nP2x(1 - x ) ( 1 - 2x), * * *

respectively. This order of approximation is a typical illustration of the Voronowskaja relation3

valid for a twice continuously differentiable function f ( x ) on the interval [0, 13.

APPROXIMATION WITHOUT REMAINDER

This short section will not lead to a new result but will help to motivate our further ideas. We consider the approximation f ( x ) -B,(f; x ) , that is, we disregard rn(x) for a moment. From (1) we then obtain

where

Wx(k + 1, n - k +1) = 1' t k ( l - t ) n - k dt 0

is the incomplete beta f ~ n c t i o n . ~ This function is frequently used in statistics. Numerical values can be obtained from existing tables' of Yx = W,/W, which also discuss the difficulties of computing 9,. Here, W is the beta function. In our case, k and n are integers, so that

k ! ( n - k)! W(k + 1, n - k + 1) = 1' t k ( l - t)n-k d t =

0 (n + l)!

Hence (4) with x = 1 becomes simply

Our first result, which is known,6 is as follows. If the integrand f ( t ) is approximated by B,cf; t), one obtains the 'modified rectangular rule' (5). One should note that (5) was obtained from an approximating rather than an interpolating

polynomial (as incorrectly stated in Reference 6) and that the idea of Sheppard corrections7 does not entail a substantial improvement of the quality of the formula.

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APPROXIMATION WITH REMAINDER

We now proceed to the better approximation (2) and corresponding integration formulas (3) and (3*), taking x = 1 , that is, y = 6, as in ( 5 ) . From (1) we obtain, for k = 0, . * * , n,

r n ( 3 =f (! ) - i=o f f(;)(n)(y( 3 n 1 - y j .

We sum these values from k = 0 to k = n. Then on the right, the coefficient of f ( k / n ) is the sum of the ( k + 1)th row of the matrix R , = I , - P,, where I , is the ( n + 1)-rowed unit matrix and P, = (plkn)); here

1 n I

pj.kn) = -p ( .) k'(n - k)"-'.

These are the values of the probability function of the binomial di~tr ibut ion,~ so that P, has column sums 1, whereas the row sums depend on n. Hence P, is a stochastic matrix. Formula (3) with x = 1 can now be written

or, in terms of the given function g ( u ) defined above,

where a t ) is the ( k + 1)th row sum of Z, + R , +21, -P,. Note that a c ? k = a t ) . For instance, these coefficients are

4,2,4 ( n =2)

3 9 3 , 3 3 3 ( n = 3)

3 3 3

2 4 4 2

0~5086,1~1713,1~0939,1~0900,1~0908,1~0909, - * * (n = 10)

0~4279,1~0894,1~0136,1~0088,1~0098,1~0099, * - - ( n = 100)

We see that for n = 2 this is a composite trapezoidal rule for two adjacent intervals. For n = 3,4 , * * these rules are not of a familiar type. However, as n i, co, these coefficients approach limits, a t ) = a:) + a. close to (conjectured 2 -&r = 0.429 204) and a t ) = Cy?k + ak(k 2 1) close to 1. Practically speaking, this gives the following result.

The integration process suggested by J. Reinkenhof is approximately a composite trapezoidal rule.

It is now clear that the accuracy of this process is practically equal to that of the composite trapezoidal rule. This is perhaps better than one would expect from Bernstein polynomials, taking into account the known fact that convergence, even uniform convergence, may not mean much for practical numerical work (whereas, on the other hand, divergent series, such as asymptotic expansions, may be numerically useful in certain ranges).

Error estimates for (7) and (7*) may be obtained from those for the trapezoidal rule and error bounds for the deviation of the a?) from 1 (or, respectively, t if k = 0 and n ) .

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Numerical examples would not produce any surprises but would give results typical of

i 0.746 824 (exact, 6 decimals)

These values are typical of the outcome of various numerical experiments that were conducted in connection with (7) and (7*). Indeed, it was found that the general picture does not change if one chooses other integrands or intervals of integration, so that in that fashion, one would not gain additional insight into the present process. For further examples, see References 1 and 9. For a study of general convergence properties of Bernstein polynomials we refer to the book by Lorentz” and an article by de Leeuw.”

trapezoidal rule integration. For instance, for n = 10,

0.746 21 (composite trapezoidal rule)

0-746 21 (present method (7))

0.746 825 (composite Simpson rule’)

1 I0 e-X2dx=

REFERENCES

1. J. Reinkenhof, ‘Differentiation and integration using Bernstein’s polynomials’, Int. J. num. Meth. Engng, 11,

2. A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1962, p. 28-31. 3. J. Todd (Ed.), Survey of Numerical Analysis, McGraw-Hill, New York, 1962, p. 125, 126. 4. A. ErdClyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New

5. K. Pearson, Tables of the Incomplete Beta-Function, 2nd edn, Cambridge University Press, Cambridge, 1968. 6. W. Squire, Infegration for Engineers and Scientists, American Elsevier, New York, 1970. 152. 7. E. Kreyszig, Introductory Mathematical Statistics, Wiley, New York, 1970. 8. E. Kreyszig, Adoanced Engineering Mathematics, 3rd edn, Wiley, New York, 1972, p. 657,658. 9. J. Reinkenhof, ‘Darstellung der Bewegung eines Fluggerats bei zeitaquidistanter Diskretisierung’, Zeits. Flugwiss.

1627-1630 (1977).

York, 1953, Vol. I, p. 87.

u. Weltraumforsch, 2, 115-120 (1978). 10. G. G. Lorentz, Bernstein Polynomials, Toronto Press, Toronto, 1953. 11. K. de Leeuw, ‘On the degree of approximation by Bernstein polynomials’, J. d’Analyse Math. 7,89-104 (1959).