Estimation of errors of numerical integration for analytic functions

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A THESIS ON

ESTIMATION OF ERP..!..)aS OF NMERICAL INTEGRATION FOR i‘tNALYTIC FUNCTIONS

By Chawla

Department of Mathematics Indian Institute of Technology

New Delhi

Submitted to the Indian Institute of Technology, Yel.A1 Delhi for the award of the Degree of Doctor of Philosophy

in Mathematics 1968

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CERTIFICATE

This is to certify that the thesis entitled 'Estimation of Errors of Numerical Integration for Analytic Functions' which is being submitted. by

kr. Man Mohan Char-la for the award of Degree of Doctor of Philosophy (Matheniatics) to the Indi:n institute of Technology, Delhi, is a record of boncride research work.

He has worked for the last three years under my guidance and supervision.

The thesis has reached the standard fulfilling the requirements of the regulations relating to the degre'e. The results obtained in this thesis have not been submitted to any other University or Institute for the award of any degree or diplomat

12.• 1. 4,5- ( Jain)

Head: Department of Mathematics Indian Institute of Technology

Delhi

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ACKNOWLEDGEMENTS

I am profoundly grateful to Professor Jain,

M.A., D. Phil., D.Sc., Head of the Department of Mathematics, Indian Institute of Technology, Delhi, for his valuable

guidance throughout my research work. But for his keen interest, generous encouragement and inspiration to me, this work would not have been possible.

I am particularly grateful to Professor Philip J. Davis, Division of Applied Mathematics, Brown University, for

communicatirg four of ;iv reserch papers to Mathematics of Computation. I also wish to thank Professor L. Collatz,

Director, Institute for Applied Mathematics, University of Hamburg, West Germany, for offering encouraging comments on this work.

I also wish to thank the Director, Professor It. N, Dagra, for providing, necosary facilities for carrying out this

research work.

Thanks are :also due to Mr. Dev Raj Joshi for expert typing of the thesis, and also to Mrs. Padma Narayana Swamy for doing Chapter II.

Department of Mathematics ( Chawla ) Indian Institute of Technology,

Delhi.

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SYNOPSIS

Classical error estimates for the rules of approxi- mate integration using derivatives can be used but they are not of great practical value since the derivatives are not usually available. In this thesis we have

studied the analytic function theory methods for the estimation of errors of numerical integration formulas -

Gauss type quadratures, Gaussian cubature formulas and certain Lagrangian quadratures (e.g., the Clenshaw- Curtis quadrature). Davis' method for the estimation of quadrature errors based on Hilbert space techniques has also been analysed. In contrast to the usual real—

variable theory methods, these methods do not involve the use of the higher derivatives of the function but require instead a knowledge of the size of the integrand in the complex plane. These estimates will therefore be of practical value.

A description of the contents of six chapters included in the thesis follows..

CHAPTER I: We have established simple derivative—

free error estimates for the Gauss—Legendre, the two Gauss—Chebyshev and a Gauss—Jacobi quadrature applied to analytic functions.. A few lemmas giving asymptotic expansions, which ^are useful for the derivation of these estimates, have also been proved. The estimates obtained

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improve upon certain known error estimates for these Gauss type quadra.tures. Contour integral representations obtained for the errors ^E_t^{( f )} have the advantage that,

in specific cases, it may be possible, by a further analysis on the evaluation of the contour integral, to

obtain £n(f), possibly exactior asymptotically (Chapter II), depending upon the nature of the function in the complex plane; as, for example, in the case of functions with poles. This is particularly true of the Chebyshev error formulas which are very simply expressed with contours as certain ellipses.

CHAPTER

II:

:le have studied the asymtotics of the Gaussian quadrature error obtaining estimates according

to the nature of the integrand in the complex plane:

entire functions, functions with poles, function having singularities on the real axis - branch point, singularity at an end-point of the interval of integration, or a

logarithmic singularity. The analsis also brings out the effect of the nature of ^f(z) on the rate of convergence of the Gauss quadrature formula.

CHAPTER continue the study of Chapter I to discuss the estimation of errors and convergence of

Gaussian quadrature formulas of the closed type - Lobatto, Radau and a Causs-Chebyshev quadrature formula of the closed type. A few lemmas required to obtain these estimates have also been established.

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CHAPTER IV: Error estimates have been obtained for the two Lagrangian quLdrature schemes, applied to analytic functions, based respectively on the "classic-a.1 and

"practical" abscissas.

CHAPTER V: Error estimates through Davis' method employing the "double integral" norm as well as the

"line integral" norm are obtained for Gauss type quadratures whose abscissas and weights are simly expressed.

Error—functional norms for these quadrotures, as also for the trapezoidal and Simpson rules, have been texpli- citelyi evaluated. The line integral norm error estimates obtained through IZvist method are essentially the same as those obtained through the analytic function theory in. Chapter I.

CHAPTER VI: Two—term contour integr.1 expressions are obtained for the error of Gaussian cubature formulas, which represent generalization of the known quadm ture error formulas and irmsroves upon the known cubature error formula Derivative—free error estimates are obtained for the Gauss—Legendre and Gauss—Chebyshev cubature formulas.

The thesis is based on the following eight papers,

1. "Error Estimates for Gauss Quadrature Formulas for Anolytic Functions," Mathematics of Computation, January 1968.

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2. "Asymptotic Error Estimates for the Gauss Quadrature Formula," Mathematics of Computation, January 1968.

"Error Estimates for the Clenshaw-Curtis Quadrature,"

Mathematics of Computation, July 1968.

"Error Pounds for the Gauss-Chebyshev Quadrature Formula of the Closed Type," Mathematics of Compu- tation, July 1968.

"On Davis' Method for the Estimation of Errors of Gauss Type Quadratures," Mathematics of Computation communicated by Professor Philip J. Davis.

6. "On the Chebyshev Polynomials of the Second Kind,"

SIAM Review, October 1967.

7. "On the Estimation of Errors of Gaussian Cubature Formulas," SIAIVi Journal on 1\Umerical Analysis, March 1968.

8. "A note on the estimaLion of the coefficients in the Chebyshev series expansion of a function having a logarithmic singularity," Computer Journal, Vol. 9, 1967, ^{p. 413.}

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CCNTENTS

Page Synopsis

CHAPTER I: E95.0. ESTIMATES FOR GAUSS QUADR.-',.-

TURE FORlaiLAS FOR ANALYTIC FUNCTIONS 1 - 38 1.1 Introduction, 1

1.2 Error of Gauss Type Quadratures for Analytic Functions, 2

1.3 The Gauss-Legendre Quadrature Formula, 3 1.3.1 A Lemma for Qm(2), 5

1.3,2 Convergence of' the Gauss- Legendre Quacirature, 7

1.3.3 Error Estimates for the Gauss- Legendre Quadrature, 9

1.3.4 Integr,ind with Poles, 10

1.4 The Gaus-Chebyshev Quadrature Formula (First Kind), 11

1.4.1 Lemmas for cp (.), 12

1,4.2 Convergence of the Gauss-

Chebyshev Qua drature ( Firs t Kind), 16 1.4.3 Error Estimates for the Gauss-

Oh eby sh ev Quadrature (First Ki nd),17 1.5 The Thebyshev Polynomials of the Second Kind,19

1.5.1 Lemma s for c ) 20

1.5,2 Error Estimates for the Gauss- Chebyshev Quadrature Formula

(Korkine-Zolotoreff), 23 1.6 A Gauss-Jacobi Quadrature, 28

1.6.1 Lemmas for S4 (2), 30

1.6.2 Error Estimates for the Gauss- Jacobi Quadrature, 31

1.7 An Alternative Formula for the Error of

the G_uss-Legendre Quadrature, 32

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1,8 Known Error Estimates for Gauss Type Quadratures„ 33

1.. 8 . 1

Error Estimates Through the Theory of Approximation, 33 1.8.2 Error Estimates Through Davis'

Method, 37

CHAPTER II: ASYMPTOTIC E::ROR ESTIMATES FOR THE

GAUSS QUAIPATIRE F:)itYfULA 39 —. 54 2,1 Introduction, 39

2.2 An Asymptotic Formula EG (f), 40 2.2.1 Entire Functions, 741

2.2.2 Function with Pole, 43

2,2.3 Function with Singularity on the Red Axis, 46

2.3 Extension to the Gauss—Jacobi Quadrature,50 2,3.1 Asymptotic Error Estimates for the

Gauss—Chebyshev Quadrature, 51

2.4 Appendix. The estimation of the coefficients in the Chebyshev series expansion of a

function having a logarithmic singularity, 52 CH,..PTER. COIWEitGENCE OF GAUSS TYPE QUAIA<ATURE

r.1.1)R ANALYTIC FU TIO 55 — 73 3.1 Introduction, 55

3.2 The Lobatto Quadrature, 56 3.2.1 The Error EL M, 57 3.2.2 Error Bounds, 59

3.2.3 Convergence of the Lobatto Cd'uadrature, 60

3.3 The Radau Cuadrature, 63 3.3.1 The Error ER (f) 65 3.3,2 Error Bounds, 66

3.3.3 C,'onvergence..- of the Radau Quadratiii-e,67 3.4 Error Bounds for the Gaus.s—C-ihebyShev

Quadrature Formula of the Closed Type, 69 3.4.1 Error Bounds, 70

3.4.2 Exi:mple, 72

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CHAPTER

IV:

E:LIZOR ESTIMATES FOR THE CLEN3HAW-

CURTIS ',',..1ALP.ATURE

74 —

86

4.1

Introduction,

74

4.2

The Clenshaw-Curtis Quadrature Formula,75

4.2.1

The cuadrature Formula based on the

Classical Abscissas, 78 4.3 A Lemma for 1.471(z), 80

4.4 Error Estimates, 83 4,5

Example, 85

CHAPTER

V: 11AVIS' METHOD

FOP, THE ESTIMATION

OF 1.7.PRORS OF GAUSS TYPE QU1\01::',TURE 87 - 117 5.1 Introduction, 87

5,2 Error-Functional Norms, 88 5.2.1 Er:1)r Estimates, 90

5,3 Error-fkurtional Norms for Gauss Type Qu ture I

91

5.3.1

Gauss-Chebyshev Quadra ture(First Kind)92 5.3.2 Gauss-Chebyshev Quadrature(Second Kind),95

5.3.3

Gauss-Chebyshev Quadra to ro based on

the

"Practical"

Abscissas,

99 5.3.4

A Gauss-Jacobi Quadrature, 101

5.4

Error-Functional Norms for the Trapezoidal and Simpson A.ul es,

107

5.4.1 Trapezoidal Rule,

107

5.4.2 Simpson's Rule, 109

5.5 Asymptotic Evaluation of the Error- Functicrul Norms of the Gauss-Legendrc Qua dratui:.e, Ill

5..6 Conclusion,

114 5,7

.appendix, 115

CHAPTER VI: ESTID..t.tTION OF EP ORS OF GAUSSIAN

CUB1`1.-URE a.:Y1,!ULAS 118 -

133

6.1 Introduction, 118

6.2 The Gaussian Cubature Formula, 119

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6.3 The Error E(f) expressed as a double

contour integral, 121

6.3.1 The Gauss-Legendre Case, 121

6.3.2 The Gauss-Chebyshev case, 123 6.4 The Error E(f) expressed as the sum of

two single contour integrals, 126 604.1 Error estimates in the Gauss-

Chebyshev case, 130 605 Example, 131

RE FE kE 134 - 138

'vote. A r:::ference to Then of Chapter 171, is made as Th.rn.n,

same for Lermils, etc. Similarly, Eqn. (men) means Eqn. ntnbereci: n of Chapter in.