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 |  موضوع: كتاب Orthogonal Polynomials in MATLAB - Exercises and Solutions السبت 13 أغسطس 2022, 10:04 am | |
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أخواني في الله
أحضرت لكم كتاب
Orthogonal Polynomials in MATLAB - Exercises and Solutions
Walter Gautschi
Purdue University
West Lafayette, Indiana
و المحتوى كما يلي :
Contents
Preface vii
1 A Guide to the Software Packages OPQ and SOPQ 1
1.1 Recurrence relation 1
1.2 Chebyshev and modified Chebyshev algorithm 3
1.3 Discretization procedures 5
1.4 Modification algorithms . 19
1.5 Sobolev orthogonal polynomials . 27
1.6 Quadrature formulae . 31
2 Answers to Exercises on Orthogonal Polynomials 35
2.1 Elementary properties 36
2.2 Symmetry 44
2.3 Orthogonality on two separate (symmetric) intervals 48
2.4 Modified and classical Chebyshev algorithm 58
2.5 Discretization method 120
2.6 Modification algorithms . 135
3 Answers to Exercises on Sobolev Orthogonal Polynomials 157
3.1 Elementary properties 158
3.2 Symmetry 161
3.3 Zeros . 162
4 Answers to Exercises on Quadrature 181
4.1 Elementary properties 183
4.2 Cauchy and Cauchy principal value integrals 188
4.3 Gauss–Radau and Gauss–Lobatto quadrature . 195
4.4 Gauss–Kronrod quadrature . 213
4.5 Gauss–Turán quadrature . 214
4.6 Polynomial/rational quadrature formulae of Gaussian type 215
4.7 Quadrature estimates of matrix functionals . 216
5 Answers to Exercises on Approximation 235
5.1 Special functions 236
5.2 Polynomial least squares approximation . 259
5.3 Moment-preserving approximation 272
5.4 Summation by integration 275
5.5 The spiral of Theodorus . 288
vvi Contents
A The Software Package OPQ (Orthogonal Polynomials and Quadrature) 293
A.1 Orthogonal polynomials . 293
A.2 Quadrature . 303
A.3 Examples and tests . 308
A.4 Tables and figures . 310
A.5 Utility . 312
B The Software Package SOPQ (Symbolic Orthogonal Polynomials and
Quadrature) 317
B.1 Orthogonal polynomials . 317
B.2 Quadrature . 320
B.3 Utility . 321
Bibliography 323
Software Index 327
Subject Index 33
Subject Index
Airy differential equation
inhomogeneous, 246
Airy function, 119
complex, 119, 244, 252, 253
Gauss quadrature
approximation of, 247
inhomogeneous, 246
in the complex domain, 120
inhomogeneous, 117, 252
Althammer polynomials, 162
Altheimer’s polynomials, 28, 30
asymptotic expansion, 81
Poincaré-type, 255
atomic measure, 166
one-point, 166
Bessel function, 265, 280
modified, 117, 123, 278
Gaussian quadrature relative
to, 117
bisection
method of, 289
bisection routine, 290
Boltzmann constant, 215
Bose–Einstein measure, 104, 278
moments of, 104
square root, 104, 277, 281, 289
squared, 104
cancellation
degree of, 248
Cauchy integral, 147, 148, 150,
168, 181, 189, 190, 282
Cauchy integrals
ratios of, 23
Cauchy principal value integral,
181, 188, 190, 191
quadrature rule in the strict
sense for, 194
Cauchy transform, 23
of a measure, 188
of the monic nth-degree
orthogonal polynomial,
35, 139, 189
Chebyshev algorithm, 3, 36, 89
classical, viii, 4, 58, 66, 68, 94,
117, 119
in high precision, 94, 104,
107
in high-precision arithmetic,
88
in high precision, 110
modified, viii, 3, 36, 59, 60, 62,
88, 94, 97, 158, 162
Chebyshev orthogonal
polynomials
of a discrete variable, 260
Chebyshev polynomials
discrete on [0,1), 6, 6
recurrence coefficients of, 6
discrete on [0,N], 2
elliptic, 4
monic, 53
monic discrete
recurrence relation of, 261
recurrence relation of the
second kind (in
MATLAB), 40
Christoffel’s theorem, 136
generalized, 138, 141, 142, 145
circle theorem, 186
for algebraic/logarithmic
weight functions
plots of, 188
for Gauss–Jacobi quadrature,
185, 187
plots of, 187
for Gauss–Kronrod
quadrature, 186
with Gauss–Jacobi weight
function, 187
for Gauss–Kronrod quadrature
and algebraic/logarithmic
weight functions
plots of, 188
for Gauss–Kronrod quadrature
and Jacobi weight
functions
plots of, 188
classical Chebyshev algorithm,
66, 68, 119
Clenshaw’s algorithm, 262
complementary error function,
238
complex Airy function, 244,
252, 253
complex error function, 238
complex inhomogeneous Airy
function, 246
condition numbers, 95
estimates of the, 89
for E
m
plot of, 90
connection formula, 255
constraint polynomial, 264
continued fraction, 42
infinite, 42
continued fractions, viii
theory of, 42
curves C
γ
plot of, 172
Darboux formulae
for Sobolev orthogonal
polynomials, 29
Darboux’s formula, 37
for recurrence coefficients, 36
Dawson’s integral, 277, 289
degree of cancellation, 248
Demo 1.15, 30
Demo 1.14, 28
Demo 1.5, 11
331332 Subject Index
Demo 1.3, 6
Demo 1.4, 7
Demo 1.2, 4
Demo 1.17, 31
Demo 1.9, 18
Demo 1.10, 19
Demo 1.7, 14
Demo 1.6, 12, 14
Demo 1.1, 3
Demo 1.8, 16
Demo 1.16, 31
Demo 1.13, 26
Demo 1.11, 21
Demo 1.12, 24
Dirac delta function, 159, 174,
235, 272
discretization
1-component, 12, 129
2-component, 10, 64
Fejér, 134
Gauss–Freud, 134
m-component, 9
multicomponent, 132
general-purpose, 17
discretization methods, 8, 120
discretization procedure, 5, 61,
65, 107, 110
3-component, 21
multicomponent, 36, 124
discretization routine
multicomponent, 9
discretization scheme
2-component, 60
distribution
half-range Hermite, 274
elliptic Chebyshev polynomials,
4
Erdos weight function, 124 ˝
error function
complementary, 114, 238, 265
first and second derivatives
of, 265
complex, 238
Euler’s formula, 278
Example 1.2, 10
Example 1.3, 12
Example 1.4, 16
Example 1.5, 30
Faddeeva function, 238
Fejér nodes and weights, 10
Fejér quadrature, 17, 123, 133
Fermi measure, 107
moments of, 107
square root, 107
moments of, 107
squared, 107
Fermi–Dirac integral
generalized, 215
five-term recurrence relation
advantage and disadvantage of,
161
for special Sobolev
polynomials, 159, 161
merits of the, 159
fixed point
attracting, 49
repelling, 49, 58
fixed point iteration, 49
Fokker–Planck equation, 129
Fourier coefficients, 235, 259
alternative form of, 260
Freud conjecture, 81
for half-range Freud
polynomials, 81
Freud polynomials, 68, 79
half-range, 74
moments of, 80
positive zeros of, 80
plots of, 80
subrange, 68
lower symmetric, 75
upper symmetric, 77
zeros of, 68
Freud weight function, 129
functions T
2(·, b) and T3(·, b)
plot of the, 288
gamma function, 114
reflection formula for, 120
Gauss formula, 44, 115
Gauss quadrature, 44, 63, 119,
153
error of, 122
for a logarithmic weight
function, 186
relative error of, 245
relative to the half-range
Hermite weight function,
239
Gauss quadrature approximation
of the complex Airy function,
247
of the inhomogeneous Airy
function Gi(z), 248
Gauss quadrature error
plot of, 122
Gauss quadrature formula, 31
Gauss quadrature rule, 116
Gauss–Chebyshev quadrature,
10
Gauss–Freud quadrature, 129
Gauss–Freud quadrature points,
131
Gauss–Hermite quadrature, 121,
239
Gauss–Hermite quadrature rule
half-range, 237
Gauss–Jacobi quadrature, 12, 64,
92, 170
Gauss–Jacobi quadrature rule, 64
Gauss–Kronrod formula
for a logarithmic weight
function, 186
Gauss–Kronrod quadrature, 33
Gauss–Kronrod quadrature rule,
181
Gauss–Laguerre quadrature, 16,
92, 125
Gauss–Legendre quadrature, 10,
63
Gauss–Legendre quadrature rule,
62
Gauss–Lobatto formula, 199
generalized, 201, 202
Gauss–Lobatto quadrature, 33,
195, 223, 228, 232
generalized, 208
Gauss–Lobatto quadrature rule,
181
Gauss–Lobatto rule
generalized, 212
ordinary, 212
Gauss–Radau formula, 230
generalized, 200
positivity of, 208
Gauss–Radau quadrature, 33,
195, 223, 228
generalized, 203
Gauss–Radau quadrature rule,
181
Gauss–Radau rule
left- and right-hand, 195
Gauss–Turán quadrature, 214
Gauss–Turán quadrature rule,
182, 214
Gauss-type quadrature
applied to a discrete measure,
182
involving derivatives, 33
Gaussian quadrature, viii, 9, 113,
121, 236Subject Index 333
relative to a Freud-type weight
function, 247
relative to a modified Bessel
weight function, 244
Riemann–Hilbert approach
to, 120
Gaussian quadrature formula,
35, 181
n-point
weights of, 189
Gaussian quadrature rule, 110
Gaussian quadrature sum
expressed in terms of the
Jacobi
matrix, 185
Gaussian weights, 184
Gautschi–Stieltjes procedure, 5
generalized Laguerre
polynomial, 86
Gram–Schmidt
orthogonalization, 37
Gram–Schmidt procedure, 36
ground measure, 166
Hahn polynomial, 6
half-range Freud polynomials,
74, 86
Freud conjecture for, 81
moments of, 75
zeros of, 75
plots of, 75
half-range Gauss–Hermite
quadrature rule, 237
half-range Hermite distribution,
272
half-range Hermite measure, ix,
18
half-range Hermite polynomials,
66
half-range Hermite weight
function, 239
step function approximation
of
plot of, 275
Hardy–Littlewood function,
278, 279
plot of, 279
hat function, 60
Heaviside step function, 274
Hermite measure
generalized, 166
half-range, ix
Hermite polynomial, 274
Hermite–Sobolev polynomials,
166
generalized, 166
histogram for the zeros of the
symmetric Laguerre
measure, 126
histograms for the zeros of πn,
125
homogeneous Airy differential
equation, 253
incomplete gamma function, 94
lower, 241
symbolic routines for, 95
upper, 241
inhomogeneous Airy differential
equation, 246
inhomogeneous Airy functions,
252
integral representations of, 247
inner product, 1, 36
discrete, 44
Gegenbauer-like, 168
positive definite, 36
shift property of, 158
interpolation polynomial, 264
Jacobi continued fraction, 35, 42
numerator in the nth
convergent of the, 43
Jacobi matrix, 2, 35, 39, 184, 185,
223, 228, 232
eigenvalues of, 184
eigenvectors of, 184
extended, 6
spectral resolution of, 185
Jacobi measure, 197, 199
normalized
plus a discrete measure, 12
plus a one-point atomic
measure, 12
Jacobi polynomials
monic, 200
Jacobi–(left-hand)Radau matrix,
197
Jacobi–Lobatto matrix, 199
Jacobi–Radau matrix, 196, 196
Krylov space, 216
Lagrange interpolation formula,
192
Lagrange interpolation
polynomial
elementary, 184, 193
sum of, 193
Laguerre measure, 166
generalized, 197
plus an atomic measure, 166
Laguerre polynomial, 92
generalized, 86, 197
monic generalized, 197
Lanczos algorithm, 182, 216,
217, 223, 228
Lanczos algorithm routine, 219,
223
correctness of, 220
Lanczos vectors, 216–218, 220
orthogonality of, 218
Lanczos’s method, 6
Laplace transform, 275, 282
convolution theorem for, 277
least squares approximation
equally weighted, 260
polynomial, 235, 259
least squares error, 259, 260
least squares problem, 235
constrained, 236, 264, 265
Legendre measure, 47
Legendre polynomials, 45, 61
shifted, 97
loading multiprecision arrays, ix
logistic measure, 16
long recurrence relation
for Sobolev orthogonal
polynomials, 159
lower incomplete gamma
function, 241
in Tricomi’s form, 241
Macdonald functions, 123
MATLAB Symbolic Toolbox, ix
matrix functionals
quadrature estimates of, 216
Maxwell velocity distribution,
274
measure
absolutely continuous, vii
positive definite, 37
atomic, 166
discrete, 5
discrete N-point
positive definite, 37
discretization of, viii
half-range Hermite, 18
logistic, 16
modification of, viii
modification of the
by m linear divisors, 146
by a linear divisor, 139
by a polynomial divisor,
138, 144334 Subject Index
by a quadratic divisor, 141,
149
by a quadratic factor, 136,
264
modified, viii, 19
multipoint
atomic, 173
successive modifications of a
by linear divisors, 145
symmetric, 44, 45, 61
Meijer’s polynomials, 31
minimal solution
of three-term recurrence
relation, 43, 44
mixed moments, 58
modification algorithms, viii, 19,
135
modified measure, 19
modified moment routines, 100
modified moments, viii, 28, 36,
58, 60, 61, 64, 92, 94, 97,
99, 162
of exponential integral weight
function, 92
relative to the monic
Chebyshev polynomials, 4
moment map
condition of, 103
moment-preserving
approximation, 235, 272
moments, viii, 3, 35, 37, 66, 68,
72, 75, 77, 104, 110, 111,
114, 119, 190, 236, 274
mixed, 58
modified, 3, 28, 36, 58, 60–62,
64, 92, 94, 97, 99, 162
of a weight function, vii
of the exponential integral
weight function, 89
ordinary, 58, 94
multiatomic Sobolev orthogonal
polynomials, 173
multiatomic Sobolev polynomial
special, 173
multicomponent discretization,
8
multipoint atomic measure, 173
Newton’s method, 243
ordinary moments, 58
orthogonal polynomials
classical, 2
routines for, 2
constructive theory of, viii
discrete, viii, viii
with respect to Fejér nodes
and weight, generated by
stieltjes.m,
lanczos.m, 7
formal, 19
formal properties of, 35
monic, vii
relative to the exponential
integral weight function,
88
relative to the hat function, 60
orthonormal polynomials, 38
leading coefficient of, 50
partition
4-component, 132
5-component, 132
plasma dispersion function, 238
Poincaré-type asymptotic
expansion, 255
power orthogonality, 182
quadrature formula
polynomial/rational
of Gaussian type, 215
recurrence matrix
of atomic Sobolev orthogonal
polynomials, 166
recurrence relation
for monic Sobolev orthogonal
polynomials, 157, 158
repeated integral of the coerror
function, 236
represented as a single integral,
236
Riemann zeta function, 104, 107
s-orthogonal polynomial, 182,
214
Scorer functions, 246
series
slowly convergent, 97
summation of, 235
Sobolev Fourier sums
and their first two derivatives
evaluating, 268
Sobolev inner product, 157, 158
symmetric, 161
Sobolev least squares
approximation
to the complementary error
function, 265
Sobolev least squares error, 265
Sobolev orthogonal
polynomials, 27, 157, 158,
159
atomic, 166
discrete, 236
multiatomic, 173
recurrence coefficients of, 266
relative to a Gegenbauer-like
inner product, 168
zeros of, 168
special multiatomic, 177
with only a few real zeros, 30
special functions, 236
evaluation of, 235
spectral resolution, 218
spiral of Theodorus, 236, 288
and its twin spiral
plot of, 291
spline approximation
moment-preserving, 236
Stieltjes, 36
Stieltjes algorithm, 158
discretized
for Sobolev orthogonal
polynomials, 29
Stieltjes method, 18
Stieltjes polynomial, 181
monic, 213
Stieltjes procedure, 5, 7, 30, 36
discretized, 5, 36
instability of, 168
typical difficulty with, 6
Stieltjes transform, 56
of the Chebyshev weight
function, 55
subrange Freud polynomials, 68
lower one-sided, 68
moments of, 69
zeros of, 68, 70, 71
lower symmetric, 75
moments of, 76
positive zeros of, 77
upper one-sided, 71
moments of, 72
zeros of, 74
upper symmetric, 77
moments of, 78
positive zeros of, 79
summation
by integration, 275
Szego–Bernstein measure, 145 ˝
Theodorus constant, 276, 277
three-term recurrence relation,
1, 35, 37Subject Index 335
coefficients of, 1
for the orthonormal
polynomials, 38
minimal solution of, 43, 44
twin-spiral of Theodorus, 289
upper incomplete gamma
function, 241
weight function
double exponential, 123
half-range Freud, 87
close relative of, 87
half-range Hermite, 239
symmetric, 35, 48
zeros
of Freud polynomials, 68
of Gegenbauer–Sobolev
polynomials, 165
of symmetric Sobolev
orthogonal polynomials,
162
of the nth-degree orthogonal
polynomial, 35
of the lower one-sided
subrange Freud
polynomials, 68
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