Orthogonal Polynomials: Computation and Approximation

312 Pages·2004·1.61 MB·English

by  Walter Gautschi

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W. Gautschi: Orthogonal polynomials: computation and approximation
Orthogonal Polynomials
Computation and Approximation
Walter Gautschi
Purdue University
1
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CONTENTS
Preface viii
1 Basic Theory 1
1.1 Orthogonalpolynomials 1
1.1.1 Definition and existence 1
1.1.2 Examples 4
1.2 Properties of orthogonalpolynomials 6
1.2.1 Symmetry 6
1.2.2 Zeros 7
1.2.3 Discrete orthogonality 8
1.2.4 Extremal properties 8
1.3 Three-term recurrence relation 10
1.3.1 Monic orthogonalpolynomials 10
1.3.2 Orthonormalpolynomials 12
1.3.3 Christoffel–Darboux formulae 14
1.3.4 Continued fractions 15
1.3.5 The recurrence relation outside the support
interval 17
1.4 Quadrature rules 20
1.4.1 Interpolatory quadrature rules and beyond 21
1.4.2 Gauss-type quadrature rules 22
1.5 Classical orthogonalpolynomials 26
1.5.1 Classical orthogonalpolynomials of a
continuous variable 27
1.5.2 Classical orthogonalpolynomials of a
discrete variable 32
1.6 Kernel polynomials 35
1.6.1 Existence and elementary properties 36
1.6.2 Recurrence relation 38
1.7 Sobolev orthogonalpolynomials 40
1.7.1 Definition and properties 41
1.7.2 Recurrence relation and zeros 41
1.8 Orthogonalpolynomials on the semicircle 43
1.8.1 Definition, existence, and representation 43
1.8.2 Recurrence relation 45
1.8.3 Zeros 47
1.9 Notes to Chapter 1 49
v
vi CONTENTS
2 Computational Methods 52
2.1 Moment-based methods 52
2.1.1 Classical approach via moment determinants 52
2.1.2 Condition of nonlinear maps 55
2.1.3 The moment maps G and K 57
n n
2.1.4 Condition of G : µ γ 59
n
7→
2.1.5 Condition of G : m γ 64
n
7→
2.1.6 Condition of K : m ρ 70
n
7→
2.1.7 Modified Chebyshev algorithm 76
2.1.8 Finite expansions in orthogonalpolynomials 78
2.1.9 Examples 82
2.2 Discretization methods 90
2.2.1 Convergenceofdiscreteorthogonalpolynomials
to continuous ones 90
2.2.2 A general-purpose discretization procedure 93
2.2.3 Computing the recursion coefficients of a
discrete measure 95
2.2.4 A multiple-component discretization method 99
2.2.5 Examples 101
2.2.6 Discretized modified Chebyshev algorithm 111
2.3 Computing Cauchy integrals of orthogonal
polynomials 112
2.3.1 Characterizationin terms of minimal solutions 112
2.3.2 A continued fraction algorithm 113
2.3.3 Examples 116
2.4 Modification algorithms 121
2.4.1 ChristoffelandgeneralizedChristoffeltheorems 122
2.4.2 Linear factors 124
2.4.3 Quadratic factors 125
2.4.4 Linear divisors 128
2.4.5 Quadratic divisors 130
2.4.6 Examples 133
2.5 Computing Sobolev orthogonalpolynomials 138
2.5.1 Algorithm based on moment information 139
2.5.2 Stieltjes-type algorithm 141
2.5.3 Zeros 143
2.5.4 Finite expansions in Sobolev orthogonal
polynomials 146
2.6 Notes to Chapter 2 148
3 Applications 152
3.1 Quadrature 152
3.1.1 Computation of Gauss-type quadrature
formulae 152
CONTENTS vii
3.1.2 Gauss–Kronrod quadrature formulae and their
computation 165
3.1.3 Gauss–Tur´anquadrature formulae and their
computation 172
3.1.4 Quadrature formulae based on rational
functions 180
3.1.5 Cauchy principal value integrals 202
3.1.6 Polynomials orthogonalon several intervals 207
3.1.7 Quadrature estimation of matrix functionals 211
3.2 Least squares approximation 216
3.2.1 Classical least squares approximation 217
3.2.2 Constrained least squares approximation 221
3.2.3 Least squares approximationin Sobolev spaces 225
3.3 Moment-preserving spline approximation 227
3.3.1 Approximation on the positive real line 228
3.3.2 Approximation on a compact interval 237
3.4 Slowly convergent series 239
3.4.1 Series generated by a Laplace transform 240
3.4.2 “Alternating” series generated by a Laplace
transform 245
3.4.3 Series generated by the derivative of a Laplace
transform 246
3.4.4 “Alternating”seriesgeneratedbythederivative
of a Laplace transform 248
3.4.5 Slowly convergentseries occurring in plate
contact problems 249
3.5 Notes to Chapter 3 253
Bibliography 261
Index 283
PREFACE
The main purpose of this book is to present an account of computational meth-
ods for generating orthogonal polynomials on the real line (or part thereof), to
discuss a number of related applications, and to provide software necessary for
implementing all methods and applications.
The choice of topics, admittedly, is influenced by the author’s own past in-
volvement in this area, but it is hoped that the treatment given, and especially
the software provided, will be useful to a large segment of the readership.
In Chapter 1, a brief, but essentially self-contained, review of the theory of
orthogonal polynomials is presented, which emphasizes those parts of the the-
ory that are most relevant to computation. The computational methods them-
selves are treated in Chapter 2. They are basically of two kinds, those based
on moment information and those based on discretization. Other miscellaneous
methods also receive attention, such as the computation of Cauchy integrals,
modificationalgorithmsforthe underlyingmeasures,andcomputingorthogonal
polynomials of Sobolev type. Chapter 3 is devoted to applications, specifically
numerical quadrature,discrete least squares approximation,moment-preserving
spline approximation,and the summation of slowly convergentseries.Historical
comments and references to related topics not treated in this book are given in
separate Notes to each chapter.
All software are in the form of Matlab scripts, which are collected in a suite
of Matlab programs called OPQ, and which are downloadable individually from
the Web Site
http://www.cs.purdue.edu/archives/2002/wxg/codes/
Occasionally, there will be a need to refer to a quadruple-precision version
ORTHPOLq of the Fortran package in Gautschi (1994). This can be found on the
Web Site
http://www.cs.purdue.edu/archives/2001/wxg/codes/
Manytablesthroughoutthebookreportonnumericalresultsofvariousalgo-
rithms. All numbers in these tables are displayed in floating-point format m(e),
wherem,1 m<10,isthemantissaofthenumber,andethesignedexponent.
≤
The author is grateful to Gene H. Golub for his encouragement to write
this book. He also acknowledges helpful comments by Olav Nj˚astad and Lothar
Reichel.
I dedicate this book to my wife, Erika, in appreciation of her patience and
unwavering support.
December, 2003 Walter Gautschi
viii
To
Erika