An Arnoldi Gram-Schmidt process and Hessenberg matrices for Orthonormal Polynomials
|
|
- Isabella Shaw
- 6 years ago
- Views:
Transcription
1 [ 1 ] University of Cyprus An Arnoldi Gram-Schmidt process and Hessenberg matrices for Orthonormal Polynomials Nikos Stylianopoulos, University of Cyprus New Perspectives in Univariate and Multivariate OPs Banff International Research Station October 2010 Construction Asymptotics Matrices
2 [ 2 ] University of Cyprus Bergman polynomials {p n } on an archipelago G G1 GN Γ1 G2 ΓN Γ2 Γ j, j = 1,..., N, a system of disjoint and mutually exterior Jordan curves in C, G j := int(γ j ), Γ := N j=1γ j, G := N j=1g j. f, g := f (z)g(z)da(z), f L2 (G) := f, f 1/2 G The Bergman polynomials {p n } n=0 of G are the orthonormal polynomials w.r.t. the area measure on G: p m, p n = p m (z)p n (z)da(z) = δ m,n, G with p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... Construction Asymptotics Matrices
3 [ 3 ] University of Cyprus Construction of p n s Algorithm: Conventional Gram-Schmidt (GS) Apply the Gram-Schmidt process to the monomials 1, z, z 2, z 3,... Main ingredient: the moments µ m,k := z m, z k = z m z k da(z), m, k = 0, 1,.... G The above algorithm has been been suggested by pioneers of Numerical Conformal Mapping (like P. Davis and D. Gaier) as the standard procedure for constructing Bergman polynomials. It was subsequently used by researchers in this area (e.g. Burbea, Kokkinos, Papamichael, Sideridis and Warby). It has been even employed in the numerical conformal FORTRAN package BKMPACK of Warby. Construction Asymptotics Matrices Conventional GS Arnoldi GS
4 [ 4 ] University of Cyprus Instability Indicator The GS method is notorious for its instability. For measuring it, when orthonormalizing a system S n := {u 0, u 1,..., u n } of functions, the following instability indicator has been proposed by J.M. Taylor, (Proc. R.S. Edin., 1978): u n 2 L I n := 2 (G) min u span(sn 1 ) u n u 2, n N. L 2 (G) Note that, when S n is an orthonormal system, then I n = 1. When S n is linearly dependent then I n =. Also, if G n := [ u m, u k ] n m,k=1, denotes the Gram matrix associated with S n then, κ 2 (G n ) I n, where κ 2 (G n ) is the spectral condition number of G n. Construction Asymptotics Matrices Conventional GS Arnoldi GS
5 [ 5 ] University of Cyprus Instability of the Conventional GS In the single-component case N = 1, consider the monomials u j := z j, j = 0, 1,..., n. Then, for the conventional GS we have the following result: Theorem (N. Papamichael and M. Warby, Numer. Math., 1986.) Assume that the curve Γ is piecewise-analytic without cusps and let L := z L (Γ)/cap(Γ) ( 1), where cap(γ) denotes the capacity of Γ. Then, c 1 (Γ) L 2n I n c 2 (Γ) L 2n. Note that L = 1, iff G D and that I n is sensitive to the relative position of G w.r.t. the origin. When G is the 8 2 rectangle centered at the origin, then L = 3/ In this case, I and the method breaks down in MATLAB, for n = 25. Construction Asymptotics Matrices Conventional GS Arnoldi GS
6 [ 6 ] University of Cyprus The Arnoldi algorithm in Numerical Liner Algebra Let A C m,m, b C m and consider the Krylov subspace K k := span{b, Ab, A 2 b..., A k 1 b}. The Arnoldi algorithm produces an orthonormal basis {v 1, v 2,..., v k } of K k as follows: W. Arnoldi (Quart. Appl. Math., 1951) At the n-th step, apply GS to orthonormalize the vector Av n 1 (instead of A n 1 b) against the (already computed) orthonormal vectors {v 1, v 2,..., v n 1 }. Construction Asymptotics Matrices Conventional GS Arnoldi GS
7 [ 7 ] University of Cyprus The Arnoldi algorithm for OP s Let µ be a (non-trivial) finite Borel measure with compact support Σ := supp(µ) on C and consider the series of orthonormal polynomials p n (z, µ) := λ n (µ)z n +, λ n (µ) > 0, n = 0, 1, 2,..., generated by the inner product f, g µ = f (z)g(z)dµ(z). Arnoldi GS for Orthonormal Polynomials At the n-th step, apply GS to orthonormalize the polynomial zp n 1 (instead of z n ) against the (already computed) orthonormal polynomials {p 0, p 1,..., p n 1 }. Used in B. Gragg & L. Reichel, Linear Algebra Appl. (1987), for the construction of Szegö polynomials. Construction Asymptotics Matrices Conventional GS Arnoldi GS
8 [ 8 ] University of Cyprus Stability of the Arnoldi GS In the case of the Arnoldi GS, the instability indicator is given by: zp n 1 2 L I n := 2 (G) min p Pn 1 zp n 1 p 2, n N. L 2 (G) Theorem It holds, 1 I n z L (Σ) λ 2 n 1 (µ) λ 2 n(µ), n N. Typically: When dµ dz (Szegö polynomials), or dµ da (Bergman polynomials), then c 1 (Γ) λ n 1(µ) λ n (µ) c 2 (Γ), n N. When dµ w(x)dx on [a, b] R, this ratio tends to a constant. Construction Asymptotics Matrices Conventional GS Arnoldi GS
9 [ 9 ] University of Cyprus Zeros of Bergman polys: Three Disks Zeros of the Bergman polynomials p n, n = 140, 150 and 160. Theory in: Gustafsson, Putinar, Saff & St, Adv. Math., Construction Asymptotics Matrices Conventional GS Arnoldi GS
10 [ 10 ] University of Cyprus Single-component case N = 1 Ω Φ Δ G ζ ϕζ D 0 Γ Ω := C \ G Φ(z) = γz + γ 0 + γ 1 z + γ 2 +. z2 cap(γ) = 1/γ The Bergman polynomials of G: p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... Construction Asymptotics Matrices N = 1 Archipelago
11 [ 11 ] University of Cyprus Strong asymptotics when Γ is analytic Ω Φ Δ G D 0 ρ 1 Lρ Γ T. Carleman, Ark. Mat. Astr. Fys. (1922) If ρ < 1 is the smallest index for which Φ is conformal in ext(l ρ ), then n + 1 γ 2(n+1) π λ 2 = 1 α n, where 0 α n c 1 (Γ) ρ 2n, n where p n (z) = n + 1 π Φn (z)φ (z){1 + A n (z)}, n N, A n (z) c 2 (Γ) n ρ n, z Ω. Construction Asymptotics Matrices N = 1 Archipelago
12 [ 12 ] University of Cyprus Strong asymptotics when Γ is smooth We say that Γ C(p, α), for some p N and 0 < α < 1, if Γ is given by z = g(s), where s is the arclength, with g (p) Lipα. Then both Φ and Ψ := Φ 1 are p times continuously differentiable in Ω \ { } and \ { } respectively, with Φ (p) and Ψ (p) Lipα. P.K. Suetin, Proc. Steklov Inst. Math. AMS (1974) Assume that Γ C(p + 1, α), with p + α > 1/2. Then n + 1 γ 2(n+1) 1 π λ 2 = 1 α n, where 0 α n c 1 (Γ) n n, 2(p+α) where p n (z) = n + 1 π Φn (z)φ (z){1 + A n (z)}, n N, A n (z) c 2 (Γ) log n, z Ω. np+α Construction Asymptotics Matrices N = 1 Archipelago
13 [ 13 ] University of Cyprus Strong asymptotics for Γ non-smooth Theorem (St, C. R. Acad. Sci. Paris, 2010) Assume that Γ is piecewise analytic without cusps. Then, n + 1 γ 2(n+1) π λ 2 = 1 α n, where 0 α n c(γ) 1 n n, n N. and for any z Ω, where p n (z) = A n (z) n + 1 π Φn (z)φ (z) {1 + A n (z)}, c 1 (Γ) dist(z, Γ) Φ (z) 1 n + c 2 (Γ) 1 n, n N. Construction Asymptotics Matrices N = 1 Archipelago
14 [ 14 ] University of Cyprus Ratio asymptotics for λ n Corollary (St, C. R. Acad. Sci. Paris, 2010) n + 1 n λ n 1 λ n = cap(γ) + ξ n, where ξ n c(γ) 1 n, n N. The above relation provides the means for computing approximations to the capacity of Γ, by using only the leading coefficients of the Bergman polynomials. In addition it yiels: Corollary c 1 (Γ) I n c 2 (Γ), n N. Hence, under the assumptions of the previous theorem, the Arnoldi GS for Bergman polynomials, in the single component case, is stable. Construction Asymptotics Matrices N = 1 Archipelago
15 [ 15 ] University of Cyprus Leading coefficients in an archipelago G1 GN Γ1 G2 ΓN Γ2 Theorem (Gustafsson, Putinar, Saff & St, Adv. Math., 2009) Assume that every Γ j is analytic, j = 1, 2,..., N. Then, c 1 (Γ) Corollary n + 1 π 1 cap(γ) n+1 λ n c 2 (Γ) n + 1 c 3 (Γ) I n c 4 (Γ), n N. π 1, n N. cap(γ) n+1 Hence, the Arnoldi GS, for Bergman polynomials on an archipelago, is stable. Construction Asymptotics Matrices N = 1 Archipelago
16 [ 16 ] University of Cyprus The inverse conformal map Ψ Ω Φ Δ G Γ ζ ϕζ D 0 Recall that Φ(z) = γz + γ 0 + γ 1 z + γ 2 z 2 +, and let Ψ := Φ 1 : {w : w > 1} Ω, denote the inverse conformal map. Then, where Ψ(w) = bw + b 0 + b 1 w + b 2 +, w > 1, w 2 b = cap(γ) = 1/γ. Construction Asymptotics Matrices Faber Hessenberg Recurrence
17 [ 17 ] University of Cyprus Faber polynomials of G The Faber polynomial F n (z) (n N) of G, is the polynomial part of the Laurent series expansion of Φ n (z) at : ( ) 1 F n (z) = Φ n (z) + O, z. z The Faber polynomial of the 2nd kind G n (z), is the polynomial part of the expansion of the Laurent series expansion of Φ n (z)φ (z) at : ( ) 1 G n (z) = Φ n (z)φ (z) + O, z. z Note: G n (z) = F n+1 (z) n + 1. Construction Asymptotics Matrices Faber Hessenberg Recurrence
18 [ 18 ] University of Cyprus The Faber matrix G The Faber polynomials of the 2nd kind satisfy the recurrence relation, zg n (z) = bg n+1 (z) + n b k G n k (z), n = 0, 1,..., k=0 and induce the upper Hessenberg Toeplitz matrix: b 0 b 1 b 2 b 3 b 4 b 5 b 6 b b 0 b 1 b 2 b 3 b 4 b 5 0 b b 0 b 1 b 2 b 3 b b b 0 b 1 b 2 b 3 G = b b 0 b 1 b b b 0 b b b Construction Asymptotics Matrices Faber Hessenberg Recurrence
19 [ 19 ] University of Cyprus The upper Hessenberg matrix M The Bergman polynomials satisfy the recurrence relation, n+1 zp n (z) = a k,n p k (z), n = 0, 1,..., k=0 where a k,n are Fourier coefficients: a k,n = zp n, p k, and induce the (infinite) upper Hessenberg matrix: a 00 a 01 a 02 a 03 a 04 a 05 a 06 a 10 a 11 a 12 a 13 a 14 a 15 a 16 0 a 21 a 22 a 23 a 24 a 25 a a 32 a 33 a 34 a 35 a 36 M = a 43 a 44 a 45 a a 54 a 55 a a 65 a Construction Asymptotics Matrices Faber Hessenberg Recurrence
20 [ 20 ] University of Cyprus Eigenvalues = Zeros Note The eigenvalues of the finite n n Faber matrix are the zeros of the Faber polynomial G n (z). Note The eigenvalues of the finite n n Hessenberg matrix are the zeros of the Bergman polynomial p n (z). Construction Asymptotics Matrices Faber Hessenberg Recurrence
21 [ 21 ] University of Cyprus Main subdiagonal Consider the main subdiagonal of the Hessenberg matrix: a n+1,n = zp n, p n+1 = λ n z n+1 +, p n+1 = λ n z n+1, p n+1 = Since cap(γ) = b, it follows from the ratio asymptotics for λ n, that: λ n λ n+1. Lemma n + 2 n + 1 a n+1,n = b + O ( ) 1, n N. n That is, the main subdiagonal of the Hessenberg matrix tends to the main subdiagonal of the Faber matrix. Construction Asymptotics Matrices Faber Hessenberg Recurrence
22 [ 22 ] University of Cyprus M G More generally, using the theory on strong asymptotics for non-smooth curves we have: Theorem (Saff & St) Assume that Γ is piecewise analytic w/o cusps, then for any fixed k N {0}, ( ) n n + k + 1 a n,n+k = b k + O n, n. That is, the k-th diagonal of the Hessenberg matrix tends to the k-th diagonal of the Faber matrix. Construction Asymptotics Matrices Faber Hessenberg Recurrence
23 [ 23 ] University of Cyprus Ratio asymptotics for p n (z) Theorem (St, C. R. Acad. Sci. Paris, 2010) Assume that Γ is piecewise analytic without cusps. Then, for any z Ω, and sufficiently large n N, n + 1 p n+1 (z) = Φ(z){1 + B n (z)}, n + 2 p n (z) where B n (z) c 1 (Γ) dist(z, Γ) Φ (z) 1 n + c 2 (Γ) 1 n. Construction Asymptotics Matrices Faber Hessenberg Recurrence
24 [ 24 ] University of Cyprus Only ellipses carry finite-term recurrences for {p n } Definition We say that the polynomials {p n } n=0 satisfy a (M + 1)-term recurrence relation, if for any n M 1, zp n (z) = a n+1,n p n+1 (z) + a n,n p n (z) a n M+1,n p n M+1 (z). Theorem (St, C. R. Acad. Sci. Paris, 2010) Assume that: Γ = G is piecewise analytic without cusps; the Bergman polynomials {p n } n=0 satisfy a (M + 1)-term recurrence relation, with some M 2. Then M = 2 and Γ is an ellipse. The above theorem refines results of Putinar & St (CAOT, 2007) and Khavinson & St (Springer, 2010). Construction Asymptotics Matrices Faber Hessenberg Recurrence
25 [ 25 ] University of Cyprus Banded Hessenberg matrices are tridiagonal Corollary If the Hessenberg matrix is banded with constant bandwidth 3, then is tridiagonal. This result should put an end to the long search in Numerical Linear Algebra, for practical polynomial iteration methods, based on short-term recurrence relations of orthogonal polynomials,. Construction Asymptotics Matrices Faber Hessenberg Recurrence
Bergman shift operators for Jordan domains
[ 1 ] University of Cyprus Bergman shift operators for Jordan domains Nikos Stylianopoulos University of Cyprus Num An 2012 Fifth Conference in Numerical Analysis University of Ioannina September 2012
More informationEstimates for Bergman polynomials in domains with corners
[ 1 ] University of Cyprus Estimates for Bergman polynomials in domains with corners Nikos Stylianopoulos University of Cyprus The Real World is Complex 2015 in honor of Christian Berg Copenhagen August
More informationProblems Session. Nikos Stylianopoulos University of Cyprus
[ 1 ] University of Cyprus Problems Session Nikos Stylianopoulos University of Cyprus Hausdorff Geometry Of Polynomials And Polynomial Sequences Institut Mittag-Leffler Djursholm, Sweden May-June 2018
More informationASYMPTOTICS FOR HESSENBERG MATRICES FOR THE BERGMAN SHIFT OPERATOR ON JORDAN REGIONS
ASYMPTOTICS FOR HESSENBERG MATRICES FOR THE BERGMAN SHIFT OPERATOR ON JORDAN REGIONS EDWARD B. SAFF AND NIKOS STYLIANOPOULOS Abstract. Let G be a bounded Jordan domain in the complex plane. The Bergman
More informationZeros of Polynomials: Beware of Predictions from Plots
[ 1 / 27 ] University of Cyprus Zeros of Polynomials: Beware of Predictions from Plots Nikos Stylianopoulos a report of joint work with Ed Saff Vanderbilt University May 2006 Five Plots Fundamental Results
More informationAsymptotics for Polynomial Zeros: Beware of Predictions from Plots
Computational Methods and Function Theory Volume 00 (0000), No. 0,? CMFT-MS XXYYYZZ Asymptotics for Polynomial Zeros: Beware of Predictions from Plots E. B. Saff and N. S. Stylianopoulos Dedicated to Walter
More informationNatural boundary and Zero distribution of random polynomials in smooth domains arxiv: v1 [math.pr] 2 Oct 2017
Natural boundary and Zero distribution of random polynomials in smooth domains arxiv:1710.00937v1 [math.pr] 2 Oct 2017 Igor Pritsker and Koushik Ramachandran Abstract We consider the zero distribution
More informationON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH
ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH V. FABER, J. LIESEN, AND P. TICHÝ Abstract. Numerous algorithms in numerical linear algebra are based on the reduction of a given matrix
More informationLarge Deviations, Linear Statistics, and Scaling Limits for Mahler Ensemble of Complex Random Polynomials
Large Deviations, Linear Statistics, and Scaling Limits for Mahler Ensemble of Complex Random Polynomials Maxim L. Yattselev joint work with Christopher D. Sinclair International Conference on Approximation
More informationMatrix Functions and their Approximation by. Polynomial methods
[ 1 / 48 ] University of Cyprus Matrix Functions and their Approximation by Polynomial Methods Stefan Güttel stefan@guettel.com Nicosia, 7th April 2006 Matrix functions Polynomial methods [ 2 / 48 ] University
More informationRatio Asymptotics for General Orthogonal Polynomials
Ratio Asymptotics for General Orthogonal Polynomials Brian Simanek 1 (Caltech, USA) Arizona Spring School of Analysis and Mathematical Physics Tucson, AZ March 13, 2012 1 This material is based upon work
More information4.8 Arnoldi Iteration, Krylov Subspaces and GMRES
48 Arnoldi Iteration, Krylov Subspaces and GMRES We start with the problem of using a similarity transformation to convert an n n matrix A to upper Hessenberg form H, ie, A = QHQ, (30) with an appropriate
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationNumerische Mathematik
Numer. Math. (1997) 76: 489 513 Numerische Mathematik c Springer-Verlag 1997 Electronic Edition Approximation of conformal mappings of annular regions N. Papamichael 1, I.E. Pritsker 2,, E.B. Saff 3,,
More informationMatrix functions and their approximation. Krylov subspaces
[ 1 / 31 ] University of Cyprus Matrix functions and their approximation using Krylov subspaces Matrixfunktionen und ihre Approximation in Krylov-Unterräumen Stefan Güttel stefan@guettel.com Nicosia, 24th
More informationCONSTRUCTIVE APPROXIMATION 2001 Springer-Verlag New York Inc.
Constr. Approx. (2001) 17: 267 274 DOI: 10.1007/s003650010021 CONSTRUCTIVE APPROXIMATION 2001 Springer-Verlag New York Inc. Faber Polynomials Corresponding to Rational Exterior Mapping Functions J. Liesen
More informationMATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More informationOrthogonal polynomials with respect to generalized Jacobi measures. Tivadar Danka
Orthogonal polynomials with respect to generalized Jacobi measures Tivadar Danka A thesis submitted for the degree of Doctor of Philosophy Supervisor: Vilmos Totik Doctoral School in Mathematics and Computer
More informationComputation of Rational Szegő-Lobatto Quadrature Formulas
Computation of Rational Szegő-Lobatto Quadrature Formulas Joint work with: A. Bultheel (Belgium) E. Hendriksen (The Netherlands) O. Njastad (Norway) P. GONZÁLEZ-VERA Departament of Mathematical Analysis.
More informationOrthogonal polynomials
Orthogonal polynomials Gérard MEURANT October, 2008 1 Definition 2 Moments 3 Existence 4 Three-term recurrences 5 Jacobi matrices 6 Christoffel-Darboux relation 7 Examples of orthogonal polynomials 8 Variable-signed
More informationSZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART
Georgian Mathematical Journal Volume 4 (27), Number 4, 673 68 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationOn rational approximation of algebraic functions. Julius Borcea. Rikard Bøgvad & Boris Shapiro
On rational approximation of algebraic functions http://arxiv.org/abs/math.ca/0409353 Julius Borcea joint work with Rikard Bøgvad & Boris Shapiro 1. Padé approximation: short overview 2. A scheme of rational
More informationThe Lanczos and conjugate gradient algorithms
The Lanczos and conjugate gradient algorithms Gérard MEURANT October, 2008 1 The Lanczos algorithm 2 The Lanczos algorithm in finite precision 3 The nonsymmetric Lanczos algorithm 4 The Golub Kahan bidiagonalization
More informationPseudospectra and Nonnormal Dynamical Systems
Pseudospectra and Nonnormal Dynamical Systems Mark Embree and Russell Carden Computational and Applied Mathematics Rice University Houston, Texas ELGERSBURG MARCH 1 Overview of the Course These lectures
More informationCourse Notes: Week 1
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues
More information1. Introduction. Let µ(t) be a distribution function with infinitely many points of increase in the interval [ π, π] and let
SENSITIVITY ANALYSIS OR SZEGŐ POLYNOMIALS SUN-MI KIM AND LOTHAR REICHEL Abstract Szegő polynomials are orthogonal with respect to an inner product on the unit circle Numerical methods for weighted least-squares
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationM.A. Botchev. September 5, 2014
Rome-Moscow school of Matrix Methods and Applied Linear Algebra 2014 A short introduction to Krylov subspaces for linear systems, matrix functions and inexact Newton methods. Plan and exercises. M.A. Botchev
More informationSolutions: Problem Set 3 Math 201B, Winter 2007
Solutions: Problem Set 3 Math 201B, Winter 2007 Problem 1. Prove that an infinite-dimensional Hilbert space is a separable metric space if and only if it has a countable orthonormal basis. Solution. If
More informationLast Time. Social Network Graphs Betweenness. Graph Laplacian. Girvan-Newman Algorithm. Spectral Bisection
Eigenvalue Problems Last Time Social Network Graphs Betweenness Girvan-Newman Algorithm Graph Laplacian Spectral Bisection λ 2, w 2 Today Small deviation into eigenvalue problems Formulation Standard eigenvalue
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
More informationKatholieke Universiteit Leuven Department of Computer Science
Convergence of the isometric Arnoldi process S. Helsen A.B.J. Kuijlaars M. Van Barel Report TW 373, November 2003 Katholieke Universiteit Leuven Department of Computer Science Celestijnenlaan 200A B-3001
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.4: Krylov Subspace Methods 2 / 51 Krylov Subspace Methods In this chapter we give
More informationEXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE
EXTREMAL DOMAINS FOR SELF-COMMUTATORS IN THE BERGMAN SPACE MATTHEW FLEEMAN AND DMITRY KHAVINSON Abstract. In [10], the authors have shown that Putnam's inequality for the norm of self-commutators can be
More informationA New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space
A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space Raúl Curto IWOTA 2016 Special Session on Multivariable Operator Theory St. Louis, MO, USA; July 19, 2016 (with
More informationComputation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated.
Math 504, Homework 5 Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated 1 Find the eigenvalues and the associated eigenspaces
More informationApproximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method
Approximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method Antti Koskela KTH Royal Institute of Technology, Lindstedtvägen 25, 10044 Stockholm,
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationKey words. cyclic subspaces, Krylov subspaces, orthogonal bases, orthogonalization, short recurrences, normal matrices.
THE FABER-MANTEUFFEL THEOREM FOR LINEAR OPERATORS V FABER, J LIESEN, AND P TICHÝ Abstract A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of
More informationOn the reduction of matrix polynomials to Hessenberg form
Electronic Journal of Linear Algebra Volume 3 Volume 3: (26) Article 24 26 On the reduction of matrix polynomials to Hessenberg form Thomas R. Cameron Washington State University, tcameron@math.wsu.edu
More informationORTHOGONAL POLYNOMIALS WITH EXPONENTIALLY DECAYING RECURSION COEFFICIENTS
ORTHOGONAL POLYNOMIALS WITH EXPONENTIALLY DECAYING RECURSION COEFFICIENTS BARRY SIMON* Dedicated to S. Molchanov on his 65th birthday Abstract. We review recent results on necessary and sufficient conditions
More informationarxiv: v1 [math.cv] 30 Jun 2008
EQUIDISTRIBUTION OF FEKETE POINTS ON COMPLEX MANIFOLDS arxiv:0807.0035v1 [math.cv] 30 Jun 2008 ROBERT BERMAN, SÉBASTIEN BOUCKSOM Abstract. We prove the several variable version of a classical equidistribution
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 19: More on Arnoldi Iteration; Lanczos Iteration Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 17 Outline 1
More informationAlgorithms that use the Arnoldi Basis
AMSC 600 /CMSC 760 Advanced Linear Numerical Analysis Fall 2007 Arnoldi Methods Dianne P. O Leary c 2006, 2007 Algorithms that use the Arnoldi Basis Reference: Chapter 6 of Saad The Arnoldi Basis How to
More informationArnoldi Methods in SLEPc
Scalable Library for Eigenvalue Problem Computations SLEPc Technical Report STR-4 Available at http://slepc.upv.es Arnoldi Methods in SLEPc V. Hernández J. E. Román A. Tomás V. Vidal Last update: October,
More informationCharacterization of half-radial matrices
Characterization of half-radial matrices Iveta Hnětynková, Petr Tichý Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8, Czech Republic Abstract Numerical radius r(a) is the
More informationInverses of regular Hessenberg matrices
Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2010 27 30 June 2010. Inverses of regular Hessenberg matrices J. Abderramán
More information1 Extrapolation: A Hint of Things to Come
Notes for 2017-03-24 1 Extrapolation: A Hint of Things to Come Stationary iterations are simple. Methods like Jacobi or Gauss-Seidel are easy to program, and it s (relatively) easy to analyze their convergence.
More informationCOMPOSITION OPERATORS INDUCED BY SYMBOLS DEFINED ON A POLYDISK
COMPOSITION OPERATORS INDUCED BY SYMBOLS DEFINED ON A POLYDISK MICHAEL STESSIN AND KEHE ZHU* ABSTRACT. Suppose ϕ is a holomorphic mapping from the polydisk D m into the polydisk D n, or from the polydisk
More informationOPUC, CMV MATRICES AND PERTURBATIONS OF MEASURES SUPPORTED ON THE UNIT CIRCLE
OPUC, CMV MATRICES AND PERTURBATIONS OF MEASURES SUPPORTED ON THE UNIT CIRCLE FRANCISCO MARCELLÁN AND NIKTA SHAYANFAR Abstract. Let us consider a Hermitian linear functional defined on the linear space
More informationKrylov Subspaces. Lab 1. The Arnoldi Iteration
Lab 1 Krylov Subspaces Lab Objective: Discuss simple Krylov Subspace Methods for finding eigenvalues and show some interesting applications. One of the biggest difficulties in computational linear algebra
More informationCommutants of Finite Blaschke Product. Multiplication Operators on Hilbert Spaces of Analytic Functions
Commutants of Finite Blaschke Product Multiplication Operators on Hilbert Spaces of Analytic Functions Carl C. Cowen IUPUI (Indiana University Purdue University Indianapolis) Universidad de Zaragoza, 5
More informationAlgebra Exam Topics. Updated August 2017
Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have
More informationLARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems
LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization General Tools for Solving Large Eigen-Problems
More informationLab 1: Iterative Methods for Solving Linear Systems
Lab 1: Iterative Methods for Solving Linear Systems January 22, 2017 Introduction Many real world applications require the solution to very large and sparse linear systems where direct methods such as
More informationError Estimation and Evaluation of Matrix Functions
Error Estimation and Evaluation of Matrix Functions Bernd Beckermann Carl Jagels Miroslav Pranić Lothar Reichel UC3M, Nov. 16, 2010 Outline: Approximation of functions of matrices: f(a)v with A large and
More informationDerivatives of Faber polynomials and Markov inequalities
Derivatives of Faber polynomials and Markov inequalities Igor E. Pritsker * Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, OK 74078-1058, U.S.A. E-mail: igor@math.okstate.edu
More informationSpectral Theory of Orthogonal Polynomials
Spectral Theory of Orthogonal Polynomials Barry Simon IBM Professor of Mathematics and Theoretical Physics California Institute of Technology Pasadena, CA, U.S.A. Lecture 1: Introduction and Overview Spectral
More informationIntroduction to Arnoldi method
Introduction to Arnoldi method SF2524 - Matrix Computations for Large-scale Systems KTH Royal Institute of Technology (Elias Jarlebring) 2014-11-07 KTH Royal Institute of Technology (Elias Jarlebring)Introduction
More informationLARGE SPARSE EIGENVALUE PROBLEMS
LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization 14-1 General Tools for Solving Large Eigen-Problems
More informationHADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS. PATRICE LASSÈRE & NGUYEN THANH VAN
HADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS. PATRICE LASSÈRE & NGUYEN THANH VAN Résumé. Principally, we extend the Hadamard-Fabry gap theorem for power series to Faber-Erokhin
More informationA complex geometric proof of Tian-Yau-Zelditch expansion
A complex geometric proof of Tian-Yau-Zelditch expansion Zhiqin Lu Department of Mathematics, UC Irvine, Irvine CA 92697 October 21, 2010 Zhiqin Lu, Dept. Math, UCI A complex geometric proof of TYZ expansion
More informationCompact symetric bilinear forms
Compact symetric bilinear forms Mihai Mathematics Department UC Santa Barbara IWOTA 2006 IWOTA 2006 Compact forms [1] joint work with: J. Danciger (Stanford) S. Garcia (Pomona
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationELE/MCE 503 Linear Algebra Facts Fall 2018
ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2
More informationKrylov subspace projection methods
I.1.(a) Krylov subspace projection methods Orthogonal projection technique : framework Let A be an n n complex matrix and K be an m-dimensional subspace of C n. An orthogonal projection technique seeks
More informationModel reduction of large-scale dynamical systems
Model reduction of large-scale dynamical systems Lecture III: Krylov approximation and rational interpolation Thanos Antoulas Rice University and Jacobs University email: aca@rice.edu URL: www.ece.rice.edu/
More informationThe restarted QR-algorithm for eigenvalue computation of structured matrices
Journal of Computational and Applied Mathematics 149 (2002) 415 422 www.elsevier.com/locate/cam The restarted QR-algorithm for eigenvalue computation of structured matrices Daniela Calvetti a; 1, Sun-Mi
More information1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal
. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P
More informationKrylov Subspaces. The order-n Krylov subspace of A generated by x is
Lab 1 Krylov Subspaces Lab Objective: matrices. Use Krylov subspaces to find eigenvalues of extremely large One of the biggest difficulties in computational linear algebra is the amount of memory needed
More informationSimple iteration procedure
Simple iteration procedure Solve Known approximate solution Preconditionning: Jacobi Gauss-Seidel Lower triangle residue use of pre-conditionner correction residue use of pre-conditionner Convergence Spectral
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationAugmented GMRES-type methods
Augmented GMRES-type methods James Baglama 1 and Lothar Reichel 2, 1 Department of Mathematics, University of Rhode Island, Kingston, RI 02881. E-mail: jbaglama@math.uri.edu. Home page: http://hypatia.math.uri.edu/
More informationOn multivariable Fejér inequalities
On multivariable Fejér inequalities Linda J. Patton Mathematics Department, Cal Poly, San Luis Obispo, CA 93407 Mihai Putinar Mathematics Department, University of California, Santa Barbara, 93106 September
More informationFEM and sparse linear system solving
FEM & sparse linear system solving, Lecture 9, Nov 19, 2017 1/36 Lecture 9, Nov 17, 2017: Krylov space methods http://people.inf.ethz.ch/arbenz/fem17 Peter Arbenz Computer Science Department, ETH Zürich
More informationSpurious Poles in Padé Approximation of Algebraic Functions
Spurious Poles in Padé Approximation of Algebraic Functions Maxim L. Yattselev University of Oregon Department of Mathematical Sciences Colloquium Indiana University - Purdue University Fort Wayne Transcendental
More informationADJOINT OPERATOR OF BERGMAN PROJECTION AND BESOV SPACE B 1
AJOINT OPERATOR OF BERGMAN PROJECTION AN BESOV SPACE B 1 AVI KALAJ and JORJIJE VUJAINOVIĆ The main result of this paper is related to finding two-sided bounds of norm for the adjoint operator P of the
More informationOn the influence of eigenvalues on Bi-CG residual norms
On the influence of eigenvalues on Bi-CG residual norms Jurjen Duintjer Tebbens Institute of Computer Science Academy of Sciences of the Czech Republic duintjertebbens@cs.cas.cz Gérard Meurant 30, rue
More informationMatrix Algebra: Summary
May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................
More informationThe topology of random lemniscates
The topology of random lemniscates Erik Lundberg, Florida Atlantic University joint work (Proc. London Math. Soc., 2016) with Antonio Lerario and joint work (in preparation) with Koushik Ramachandran elundber@fau.edu
More informationNumerische Mathematik
Numer. Math. 2002) 91: 503 542 Digital Object Identifier DOI) 10.1007/s002110100224 Numerische Mathematik L 2 -Approximations of power and logarithmic functions with applications to numerical conformal
More informationTRUNCATED TOEPLITZ OPERATORS ON FINITE DIMENSIONAL SPACES
TRUNCATED TOEPLITZ OPERATORS ON FINITE DIMENSIONAL SPACES JOSEPH A. CIMA, WILLIAM T. ROSS, AND WARREN R. WOGEN Abstract. In this paper, we study the matrix representations of compressions of Toeplitz operators
More informationNumerical Methods in Matrix Computations
Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices
More informationLecture 17 Methods for System of Linear Equations: Part 2. Songting Luo. Department of Mathematics Iowa State University
Lecture 17 Methods for System of Linear Equations: Part 2 Songting Luo Department of Mathematics Iowa State University MATH 481 Numerical Methods for Differential Equations Songting Luo ( Department of
More informationLinear Algebra problems
Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X
More informationON LINEAR COMBINATIONS OF
Física Teórica, Julio Abad, 1 7 (2008) ON LINEAR COMBINATIONS OF ORTHOGONAL POLYNOMIALS Manuel Alfaro, Ana Peña and María Luisa Rezola Departamento de Matemáticas and IUMA, Facultad de Ciencias, Universidad
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2018/19 Part 4: Iterative Methods PD
More informationLarge-scale eigenvalue problems
ELE 538B: Mathematics of High-Dimensional Data Large-scale eigenvalue problems Yuxin Chen Princeton University, Fall 208 Outline Power method Lanczos algorithm Eigenvalue problems 4-2 Eigendecomposition
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationIs Every Matrix Similar to a Toeplitz Matrix?
Is Every Matrix Similar to a Toeplitz Matrix? D. Steven Mackey Niloufer Mackey Srdjan Petrovic 21 February 1999 Submitted to Linear Algebra & its Applications Abstract We show that every n n complex nonderogatory
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationBoundary behaviour of optimal polynomial approximants
Boundary behaviour of optimal polynomial approximants University of South Florida Laval University, in honour of Tom Ransford, May 2018 This talk is based on some recent and upcoming papers with various
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationThis ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0
Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x
More informationLinear Independence. Stephen Boyd. EE103 Stanford University. October 9, 2017
Linear Independence Stephen Boyd EE103 Stanford University October 9, 2017 Outline Linear independence Basis Orthonormal vectors Gram-Schmidt algorithm Linear independence 2 Linear dependence set of n-vectors
More informationLinear Algebra using Dirac Notation: Pt. 2
Linear Algebra using Dirac Notation: Pt. 2 PHYS 476Q - Southern Illinois University February 6, 2018 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018
More information