Numerical analysis 2000. Orthogonal polynomials

368 Pages·2001·2.479 MB·English

Journal of Computational and Applied Mathematics
Copyright © 2003 Elsevier B.V. All rights reserved

Volume 127, Issues 1-2, Pages 1-369 (15 January 2001)
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Polynomial zerofinders based on Szeg polynomials, Pages 1-16
1. gfedc
G. S. Ammar, D. Calvetti, W. B. Gragg and L. Reichel
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Complex Jacobi matrices, Pages 17-65
2. gfedc
Bernhard Beckermann
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Quadrature and orthogonal rational functions, Pages 67-91
3. gfedc
A. Bultheel, P. González-Vera, E. Hendriksen and Olav Njåstad
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An iterative method with error estimators, Pages 93-119
4. gfedc
D. Calvetti, S. Morigi, L. Reichel and F. Sgallari
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Cubature formulae and orthogonal polynomials, Pages 121-152
5. gfedc
R. Cools, I. P. Mysovskikh and H. J. Schmid
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Stopping functionals for Gaussian quadrature formulas, Pages 153-171
6. gfedc
Sven Ehrich
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Computation of matrix-valued formally orthogonal polynomials and applications, Pages 173-199
7. gfedc
Roland W. Freund
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Computation of Gauss-type quadrature formulas, Pages 201-217
8. gfedc
Dirk P. Laurie
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Sobolev orthogonal polynomials in the complex plane, Pages 219-230
9. gfedc
G. López Lagomasino, H. Pijeira Cabrera and I. Pérez Izquierdo
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On the "Favard theorem" and its extensions, Pages 231-254
10. gfedc
Francisco Marcellán and Renato Álvarez-Nodarse
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Analytic aspects of Sobolev orthogonal polynomials revisited, Pages 255-266
11. gfedc
A. Martínez-Finkelshtein
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Quadratures with multiple nodes, power orthogonality, and moment-preserving spline
12. gfedc
approximation, Pages 267-286
Gradimir V. Milovanovi
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The double-exponential transformation in numerical analysis, Pages 287-296
13. gfedc
Masatake Mori and Masaaki Sugihara
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Orthogonal and L-extremal polynomials on inverse images of polynomial mappings, Pages 297-315
14. gfedc q
Franz Peherstorfer and Robert Steinbauer
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Some classical multiple orthogonal polynomials, Pages 317-347
15. gfedc
Walter Van Assche and Els Coussement
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Orthogonal polynomials and cubature formulae on balls, simplices, and spheres, Pages 349-368
16. gfedc
Yuan Xu
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Author Index Volume 127 (2001), Page 369
17. gfedc

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Contents, Pages vii-viii
18. gfedc

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Preface, Pages ix-xi
19. gfedc
Walter Gautschi, Francisco Marcellán and Lothar Reichel
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Volume 127, Numbers 1I2, 15 January 2001
Contents
Preface ix
G.S.Ammar,D.Calvetti,W.B.GraggandL.Reichel
Polynomialzerofindersbased on Szego? polynomials 1
B.Beckermann
ComplexJacobi matrices 17
A.Bultheel,P.Gonza&lez-Vera,E.HendriksenandO.Njatstad
Quadratureand orthogonalrational functions 67
D.Calvetti,S.Morigi,L.ReichelandF.Sgallari
An iterative methodwith error estimators 93
R.Cools,I.P.MysovskikhandH.J.Schmid
Cubatureformulaeand orthogonalpolynomials 121
S.Ehrich
Stoppingfunctionalsfor Gaussianquadratureformulas 153
R.W.Freund
Computationof matrix-valuedformally orthogonalpolynomialsand applications 173
D.P.Laurie
Computationof Gauss-type quadratureformulas 201
G.LopezLagomasino,H.PijeiraCabreraandI.PerezIzquierdo
& &
Sobolevorthogonalpolynomialsin the complexplane 219
F.MarcellanandR.Alvarez-Nodarse
& !
On the Favard theorem and its extensions 231
‘ a
A.MartOnez-Finkelshtein
&
Analyticaspects of Sobolev orthogonalpolynomialsrevisited 255
0377-0427/01/$-seefrontmatter(2001ElsevierScienceB.V.Allrightsreserved.
PII:S0377-0427(00)00692-0
viii Contents
G.V.Milovanovic
&
Quadratures with multiple nodes, power orthogonality,and moment-preservingspline approxi-
mation 267
M.MoriandM.Sugihara
The double-exponentialtransformationin numericalanalysis 287
F.PeherstorferandR.Steinbauer
OrthogonalandL-extremalpolynomialson inverse imagesof polynomialmappings 297
q
W.VanAsscheandE.Coussement
Some classicalmultipleorthogonalpolynomials 317
Y.Xu
Orthogonalpolynomialsand cubature formulaeon balls, simplices, and spheres 349
Author Index Volume 127 369
Journal of Computational and Applied Mathematics 127 (2001) ix{xi
www.elsevier.nl/locate/cam
Preface
Orthogonal polynomials play a prominent role in pure, applied, and computational mathematics,
as well as in the applied sciences. It is the aim of the present volume in the series \Numerical
Analysis in the 20th Century" to review, and sometimes extend, some of the many known results
and properties of orthogonal polynomials and related quadrature rules. In addition, this volume
discusses techniques available for the analysis of orthogonal polynomials and associated quadrature
rules. Indeed, the design and computation of numerical integration methods is an important area in
numerical analysis, and orthogonal polynomials play a fundamental role in the analysis of many
integration methods.
The 20th century has witnessed a rapid development of orthogonal polynomials and related quadra-
ture rules, and we therefore cannot even attempt to review all signi(cid:12)cant developments within this
volume. We primarily have sought to emphasize results and techniques that have been of signi(cid:12)cance
in computational or applied mathematics, or which we believe may lead to signi(cid:12)cant progress in
these areas in the near future. Unfortunately, we cannot claim completeness even within this limited
scope. Nevertheless, we hope that the readers of this volume will (cid:12)nd the papers of interest and
many references to related work of help.
We outline the contributions in the present volume. Properties of orthogonal polynomials are
the focus of the papers by Marcella(cid:19)n and A(cid:19)lvarez-Nodarse and by Freund. The former contribu-
tion discusses \Favard’s theorem", i.e., the question under which conditions the recurrence coef-
(cid:12)cients of a family of polynomials determine a measure with respect to which the polynomials
in this family are orthogonal. Polynomials that satisfy a three-term recurrence relation as well as
Szego} polynomials are considered. The measure is allowed to be signed, i.e., the moment ma-
trix is allowed to be inde(cid:12)nite. Freund discusses matrix-valued polynomials that are orthogonal
with respect to a measure that de(cid:12)nes a bilinear form. This contribution focuses on breakdowns
of the recurrence relations and discusses techniques for overcoming this di(cid:14)culty. Matrix-valued
orthogonal polynomials form the basis for algorithms for reduced-order modeling. Freund’s con-
tribution to this volume provides references to such algorithms and their application to circuit
simulation.
The contribution by Peherstorfer and Steinbauer analyzes inverse images of polynomial map-
pings in the complex plane and their relevance to extremal properties of polynomials orthogonal
with respect to measures supported on a variety of sets, such as several intervals, lemniscates, or
equipotential lines. Applications include fractal theory and Julia sets.
Orthogonality with respect to Sobolev inner products has attracted the interest of many re-
searchers during the last decade. The paper by Martinez discusses some of the recent developments
0377-0427/01/$-see front matter (cid:13)c 2001 Elsevier Science B.V. All rights reserved.
PII: S0377-0427(00)00490-8
x Preface
in this area. The contribution by Lo(cid:19)pez Lagomasino, Pijeira, and Perez Izquierdo deals with or-
thogonal polynomials associated with measures supported on compact subsets of the complex plane.
The location and asymptotic distribution of the zeros of the orthogonal polynomials, as well as
the nth-root asymptotic behavior of these polynomials is analyzed, using methods of potential
theory.
Investigations based on spectral theory for symmetric operators can provide insight into the analytic
properties of both orthogonal polynomials and the associated Pade(cid:19) approximants. The contribution
by Beckermann surveys these results.
Van Assche and Coussement study multiple orthogonal polynomials. These polynomials arise
in simultaneous rational approximation; in particular, they form the foundation for simultaneous
Hermite{Pade(cid:19) approximation of a system of several functions. The paper compares multiple orthog-
onal polynomials with the classical families of orthogonal polynomials, such as Hermite, Laguerre,
Jacobi, and Bessel polynomials, using characterization theorems.
Bultheel, Gonza(cid:19)lez-Vera, Hendriksen, and Nja(cid:23)stad consider orthogonal rational functions with pre-
scribed poles, and discuss quadrature rules for their exact integration. These quadrature rules may
be viewed as extensions of quadrature rules for Szego} polynomials. The latter rules are exact for
rational functions with poles at the origin and at in(cid:12)nity.
Many of the papers of this volume are concerned with quadrature or cubature rules related to
orthogonal polynomials. The analysis of multivariable orthogonal polynomials forms the foundation
of many cubature formulas. The contribution by Cools, Mysovskikh, and Schmid discusses the con-
nection between cubature formulas and orthogonal polynomials. The paper reviews the development
initiated by Radon’s seminal contribution from 1948 and discusses open questions. The work by Xu
deals with multivariate orthogonal polynomials and cubature formulas for several regions in Rd. Xu
shows that orthogonal structures and cubature formulas for these regions are closely related.
The paper by Milovanovic(cid:19) deals with the properties of quadrature rules with multiple nodes. These
rules generalize the Gauss{Tura(cid:19)n rules. Moment-preserving approximation by defective splines is
considered as an application.
Computational issues related to Gauss quadrature rules are the topic of the contributions by Ehrich
and Laurie. The latter paper discusses numerical methods for the computation of the nodes and
weights of Gauss-type quadrature rules, when moments, modi(cid:12)ed moments, or the recursion coef-
(cid:12)cients of the orthogonal polynomials associated with a nonnegative measure are known. Ehrich
is concerned with how to estimate the error of quadrature rules of Gauss type. This question is
important, e.g., for the design of adaptive quadrature routines based on rules of Gauss type.
The contribution by Mori and Sugihara reviews the double exponential transformation in numerical
integration and in a variety of Sinc methods. This transformation enables e(cid:14)cient evaluation of the
integrals of analytic functions with endpoint singularities.
Many algorithms for the solution of large-scale problems in science and engineering are based
on orthogonal polynomials and Gauss-type quadrature rules. Calvetti, Morigi, Reichel, and Sgallari
describe an application of Gauss quadrature to the computation of bounds or estimates of the Eu-
clidean norm of the error in iterates (approximate solutions) generated by an iterative method for the
solution of large linear systems of equations with a symmetric matrix. The matrix may be positive
de(cid:12)nite or inde(cid:12)nite.
The computation of zeros of polynomials is a classical problem in numerical analysis. The contri-
bution by Ammar, Calvetti, Gragg, and Reichel describes algorithms based on Szego} polynomials.
Preface xi
In particular, knowledge of the location of zeros of Szego} polynomials is important for the analysis
and implementation of (cid:12)lters for time series.
Walter Gautschia
Francisco Marcella(cid:19)nb
Lothar Reichelc
aDepartment of Computer Sciences,
Purdue University,
West Lafayette, IN 47907-1398, USA
bDepartamento de Matema(cid:19)ticas,
Universidad Carlos III de Madrid,
Avenida Universidad 30,
E-28911 Legane(cid:19)s, Madrid, Spain
cDepartment of Mathematics and Computer Science,
Kent State University,
Kent OH 44242-0001, USA
Journal of Computational and Applied Mathematics 127 (2001) 1{16
www.elsevier.nl/locate/cam
Polynomial zero(cid:12)nders based on Szego} polynomials
G.S. Ammara;(cid:3), D. Calvettib;1, W.B. Graggc, L. Reicheld;2
aDepartment of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA
bDepartment of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA
cDepartment of Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA
dDepartment of Mathematics and Computer Science, Kent State University, Kent, OH 44242, USA
Received 28 December 1999; received in revised form 27 March 2000
Abstract
The computation of zeros of polynomials is a classical computational problem. This paper presents two new zero(cid:12)nders
that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member
of a family of Szego} polynomials. Numerical experiments indicate that these methods generally give higher accuracy than
computing the eigenvalues of the companion matrix associated with the polynomial. (cid:13)c 2001 Elsevier Science B.V. All
rights reserved.
Keywords: Szego}{Hessenbergmatrix;Companionmatrix;Eigenvalueproblem;Continuationmethod;Parallelcomputation
1. Introduction
The computation of the zeros of a polynomial
(z)=zn+(cid:11) zn−1+(cid:1)(cid:1)(cid:1)+(cid:11) z+(cid:11) ; (cid:11) 2 C; (1)
n n−1 1 0 j
is a fundamental problem in scienti(cid:12)c computation that arises in many diverse applications. The
conditioning of this problem has been investigated by Gautschi [8,9]. Several classical methods for
determining zeros of polynomials are described by Henrici [17, Chapter 6] and Stoer and Bulirsch
[26, Chapter 5]. A recent extensive bibliography of zero(cid:12)nders is provided by McNamee [21].
(cid:3)Corresponding author.
E-mail addresses: [email protected] (G.S. Ammar), [email protected] (D. Calvetti), [email protected]
(W.B. Gragg), [email protected] (L. Reichel).
1Supported in part by NSF Grant DMS 9806702.
2Supported in part by NSF Grant DMS 9806413.
0377-0427/01/$-see front matter (cid:13)c 2001 Elsevier Science B.V. All rights reserved.
PII: S0377-0427(00)00491-X
2 G.S. Ammar et al./Journal of Computational and Applied Mathematics 127 (2001) 1{16
Among the most popular numerical methods for computing zeros of polynomials are the Jenkins{
Traub algorithm [18], and the computation of the zeros as eigenvalues of the companion matrix
2 3
0 (cid:1)(cid:1)(cid:1) 0 −(cid:11)
0
6 7
61 0 (cid:1)(cid:1)(cid:1) 0 −(cid:11) 7
6 1 7
6 7
6 1 0 (cid:1)(cid:1)(cid:1) 0 −(cid:11) 7
6 2 7
6 . . 7
C =6 .. .. 7 2 Cn(cid:2)n (2)
n 6 7
6 7
66 ... ... 77
6 7
6 7
4 1 0 −(cid:11)n−25
0 1 −(cid:11)
n−1
associated with the polynomial (1) by the QR algorithm after balancing; see Edelman and Murakami
[7] and Moler [22]. Recently, Goedecker [10] compared these methods and found the latter approach
to be competitive with several available implementations of the Jenkins{Traub algorithm with regard
to both accuracy and execution time for polynomials of small to moderate degree.
This paper describes two new methods for computing zeros of polynomials. The methods are
based on the observation that, after a change of variable, any polynomial can be considered a
member of a family of Szego} polynomials. The new zero(cid:12)nders use the recursion relation for the
Szego} polynomials, which are de(cid:12)ned as follows. Let ! be a nondecreasing distribution function
with in(cid:12)nitely many points of increase on the unit circle in the complex plane and de(cid:12)ne the inner
product
Z
1 (cid:25) p
(f;g):= f(z)g(z)d!(t); z:=exp(it); i:= −1; (3)
2(cid:25)
−(cid:25)
for polynomials f and g, where the bar denotes complex conjugation. We assume for notational
convenience that d!(t) is scaled so that (1;1) = 1. Introduce orthonormal polynomials with re-
spect to this inner product, (cid:30) ;(cid:30) ;(cid:30) ;:::; where (cid:30) is of degree j with positive leading coe(cid:14)cient.
0 1 2 j
These polynomials are known as Szego} polynomials and many of their properties are discussed by
Grenander and Szego} [16]. In particular, they satisfy the recursion relation
(cid:30) (z)=(cid:30)(cid:3)(z)=1;
0 0
(cid:27) (cid:30) (z)=z(cid:30) (z)+(cid:13) (cid:30)(cid:3)(z); j=0;1;2:;:::;n−1;
j+1 j+1 j j+1 j
(cid:27) (cid:30)(cid:3) (z)=(cid:13)(cid:22) z(cid:30) (z)+(cid:30)(cid:3)(z); (4)
j+1 j+1 j+1 j j
where the recursion coe(cid:14)cients (cid:13) and the auxiliary coe(cid:14)cients (cid:27) are de(cid:12)ned by
j+1 j+1
(z(cid:30) ;1)
(cid:13) =− j ;
j+1 (cid:14)
j
(cid:27) =(cid:27) (1−j(cid:13) j2); j=0;1;2;:::;
j+1 j j+1
(cid:14) =(cid:14) (cid:27) ; (cid:14) =(cid:27) =1: (5)
j+1 j j+1 0 0
G.S. Ammar et al./Journal of Computational and Applied Mathematics 127 (2001) 1{16 3
It follows from (4) that the auxiliary polynomials (cid:30)(cid:3) satisfy
j
(cid:30)(cid:3)(z):=zj(cid:30)(cid:22) (1=z): (6)
j j
The zeros of the Szego} polynomials are strictly inside the unit circle and all recursion coe(cid:14)cients
(cid:13) are of magnitude smaller than one; see, e.g., [1,16]. The leading coe(cid:14)cient of (cid:30) is 1=(cid:14) .
j j j
The (cid:12)rst step in the new zero(cid:12)nders of this paper is to determine recursion coe(cid:14)cients f(cid:13) gn ,
j j=1
such that the Szego} polynomial (cid:30) satis(cid:12)es
n
(cid:14) (cid:30) ((cid:16))=(cid:17)n (z); (7)
n n 1 n
where
(cid:16)=(cid:17) z+(cid:17) ; (8)
1 2
and the constants (cid:17) and (cid:17) are chosen so that the zeros z of are mapped to zeros (cid:16) of (cid:30) inside
1 2 j n j n
the unit circle. We refer to this change of variable as a rescaling of the monic polynomial (z). Its
n
construction is discussed in Section 2. Thus, the problem of determining the zeros of is reduced
n
to the problem of computing the zeros of a Szego} polynomial of degree n. Section 3 considers two
methods for this purpose, based on a matrix formulation of the recursion relation (4). This gives
an n(cid:2)n upper Hessenberg matrix whose eigenvalues are the zeros of (cid:30) . We refer to this matrix,
n
which is described in [11], as the Szego}{Hessenberg matrix associated with (cid:30) . Having computed
n
the eigenvalues (cid:16) of this matrix, we use the relation (8) to compute the zeros z of .
j j n
A third method for computing the zeros of (z) is to use the power-basis coe(cid:14)cients of the
n
monic Szego} polynomial (cid:8) ((cid:16)):=(cid:14) (cid:30) ((cid:16)) of (7) to form the companion matrix associated with (cid:8) ,
n n n n
compute its eigenvalues, and transform these back to the z-variable using (8). In other words, to
use the companion matrix of the rescaled monic polynomial (cid:8) instead of that of . This method
n n
is included in the numerical results we report in Section 4.
Section 4 compares the use of the QR algorithm with balancing for computing the eigenvalues of
the Szego}{Hessenberg, the companion matrix (2) of , and the companion matrix of the rescaled
n
polynomial (cid:8) . We note in passing that these are all upper Hessenberg matrices. Balancing is
n
commonly used for improving the accuracy of the computed eigenvalues; see [7] for a discussion on
balancing of the companion matrix. In our experiments we found that when the parameters (cid:17) and
1
(cid:17) for the rescaling are chosen so that all zeros of (cid:30) are inside the unit circle and one zero is close
2 n
to the unit circle, the computed eigenvalues of the Szego}{Hessenberg matrix and of the companion
matrix of the rescaled polynomial (7) generally provide more accurate zeros of than those of
n
the companion matrix of . This rescaling is achieved by application of the Schur{Cohn test as
n
described in Section 3. Numerous computed examples, some of which are reported in Section 4,
indicate that computing eigenvalues of the Szego}{Hessenberg matrix after balancing often gives the
zeros of with higher accuracy than computing eigenvalues of the companion matrix of the scaled
n
polynomial (7) after balancing. Both methods, in general, give higher accuracy in the computed
zeros than computing the zeros of as eigenvalues of the balanced companion matrix.
n
The other zero(cid:12)nder for Szego} polynomials discussed in Section 3 is the continuation method
previously introduced in [2]. For many polynomials , this method yields higher accuracy than the
n
computation of the eigenvalues of the associated companion or Szego}{Hessenberg matrices. Section
4 presents numerical examples and Section 5 contains concluding remarks.