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May, 2006IWASEP VI1 Accurate symmetric rank revealing and eigen decompositions of structured matrices Froilán M. Dopico Universidad Carlos III de Madrid (Spain) joint work with Plamen Koev and Juan M. Molera

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May, 2006IWASEP VI2 Rank Revealing Decompositions (RRD)

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May, 2006IWASEP VI3 RRDs Computed with high relative accuracy Computed factors satisfy: is the machine precision.

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May, 2006IWASEP VI4 Main application of Accurate RRDs Given an accurate RRD there are O(n 3 ) algorithms to compute SVDs to high relative accuracy (Demmel et al. (1999)). where

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May, 2006IWASEP VI5 How to compute accurate RRDs Rank Revealing property: In practice from Gaussian Elimination with Complete Pivoting (GECP). Guaranteed RRDs by Pan (2000), Miranian and Gu (2003). Accuracy: Only for matrices with special structures. Demmel: Cauchy, Scaled-Cauchy, Vandermonde. Demmel and Veselic: Well Scaled Positive Definite Matrices. Demmel and Koev: M-Matrices, Polynomial Vandermonde. Demmel and Gragg: Acyclic Matrices (include bidiagonal)… ACCURACY IN A FINITE COMPUTATION (GECP) GUARANTEES ACCURACY IN SVD

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May, 2006IWASEP VI6 Our goal: Compute Accurate Symmetric RRDs For symmetric matrices compute: We consider four classes of symmetric matrices: Symmetric Cauchy and Scaled-Cauchy Symmetric Vandermonde Symmetric Totally Nonnegative Symmetric Graded Matrices well conditioned diagonal, and

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May, 2006IWASEP VI7 For symmetric indefinite matrices GECP may not preserve the symmetry. Given an accurate symmetric RRD, eigenvalues and vectors are computed with high relative accuracy by the O(n 3 ) symmetric J orthogonal algorithm (Veselic- Slapnicar) : The same accuracy can be obtained given an accurate nonsymmetric RRD of a symmetric matrix using the O(n 3 ) SSVD algorithm (FMD, Molera and Moro). Remarks on computing symmetric RRDs

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May, 2006IWASEP VI8 Algorithm 1.Compute accurate Schur Complements (Gohberg, Kailath, Olshevsky) and (Demmel) Symmetric RRDs of Scaled Cauchy Matrices 2.Use Diagonal Pivoting Method with the Bunch-Parlett complete pivoting strategy. 3.Orthogonal diagonalization of the 2 x 2 pivots. CAREFUL ERROR ANALYSIS IS NEEDED, BUT IT IS GENERAL IF ACCURATE SCHUR COMPLEMENTS ARE AVAILABLE. This approach has been used by M. J. Peláez and J. Moro to compute accurate SYMMETRIC RRDs of DSTU and TSC matrices

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May, 2006IWASEP VI9 RRDs of Symmetric Vandermonde Matrices (I) Letbe a real number Remarks 1.NONSYMMETRIC approach by Demmel does NOT respect the symmetry: SCALED CAUCHY = VANDERMONDE x DFT 2. Schur complements of Vandermonde matrices are NOT Vandermonde. 3.Symmetric Pivoting DESTROY the symmetric Vandermonde structure. 4. We avoid pivoting and use EXACT formulas for A = LDL T and its converse.

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May, 2006IWASEP VI10 RRDs of Symmetric Vandermonde Matrices (II) The LDL T can be accurately computed in 2 n 2 flops CONDITION NUMBER OF L?

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May, 2006IWASEP VI11 RRDs of Symmetric Vandermonde Matrices (III) Large Small LargeSmall

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May, 2006IWASEP VI12 RRDs of Symmetric Vandermonde Matrices (IV) ? At present only NONSYMMETRIC approach

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May, 2006IWASEP VI13 RRDs of Totally Nonnegative Matrices (I) Matrices with all minors nonnegative are called totally nonnegative (TN). We consider only nonsingular matrices. TN can be parametrized as products of nonnegative bidiagonal matrices. BIDIAGONAL FACTORIZATION NEED NOT BE AN RRD OF A

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May, 2006IWASEP VI14 Very Good Numerical Virtues of BD(A) P. Koev has shown that given the bidiagonal factorization of a TN matrix, BD(A), it is possible to compute accurately and efficiently O(n 3 ): 1.Eigenvalues of nonsymmetric TN. 2.Singular values and vectors of TNs. 3.Inverse. 4.BDs of products of TNs. 5.LDU decompositions. 6.BDs of Schur complements. 7.BDs of submatrices and Minors. 8.Apply Givens rotations: BD(A) BD(GA). 9.QR factorization……………

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May, 2006IWASEP VI15 RRDs of Totally Nonnegative Matrices (II) Our Goal: Compute accurate RRDs given BD(A) 1.In O(n 3 ) time 2.Using a finite process, 3.and, respecting the symmetry. This is nontrivial because PIVOTING STRATEGIES DESTROY TN STRUCTURE. Two algorithms: non-symmetric and symmetric. Why?

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May, 2006IWASEP VI16 Accurate RRDs of nonsymmetric TNs 1.BD(A) BD(B) by applying Givens rotations, and B bidiagonal (Golub-Kahan): A = Q B H T 2.B = P 1 LDU P 2 using accurate GECP for acyclic matrices (Demmel et al.) A = (Q P 1 L) D (U P 2 H T ) COST: 14 n 3

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May, 2006IWASEP VI17 Accurate RRDs of Symmetric TNs (I) A is POSITIVE DEFINITE COMPLETE PIVOTING IS DIAGONAL PIVOTING, AND IT PRESERVES THE SYMMETRY

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May, 2006IWASEP VI18 1. BD(A) BD(T), by applying Givens rotations, and T tridiagonal A = Q T Q T Accurate RRDs of Symmetric TNs (II) 2.Compute a symmetric RRD of T T = P L D L T P T a)GECP (DIAGONAL SYMMETRIC PIVOTING). b)ELEMENTS OF D AND L COMPUTED AS SIGNED QUOTIENTS OF MINORS OF T. c)MINORS OF T COST O(n) FLOPS. d)SUBTRACTION FREE APPROACH 3.Multiply to get: A = Q TQ T = (Q P L) D (Q P L) T COST: 21 n 3

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May, 2006IWASEP VI19 RRDs of graded matrices

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May, 2006IWASEP VI20 compute an accurate RRD of A? What is known for RRDs of NONSYMMETRIC graded matrices? Under which conditions does GECP applied on A Notice thatimplies

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May, 2006IWASEP VI21 What is known for RRDs of NONSYMMETRIC graded matrices? COMPUTED WITH GECP JUST FOR THEORY Demmel et al. (1999)

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May, 2006IWASEP VI22 compute an accurate SYMMETRIC RRD of A? Accurate RRDs of SYMMETRIC graded matrices Under which conditions does the DIAGONAL PIVOTING METHOD with the BUNCH-PARLETT COMPLETE PIVOTING strategy If is small

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May, 2006IWASEP VI23 Numerical Example: Symmetric Vandermonde for a=1/2

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May, 2006IWASEP VI24 CONCLUSIONS If accurate RRDs for a nonsymmetric class of matrices can be computed, accurate symmetric RRDs can (almost always) be computed for the symmetric counterparts. Non-Trivial algorithms may be required. Different classes of matrices require very different approaches: there is not an UNIVERSAL approach. If accurate Schur complements are available DIAGONAL PIVOTING METHOD WITH BUNCH AND PARLETT STRATEGY will do the job. Nonsymmetric RRDs plus SSVD algorithm computes accurate eigenvalues and eigenventors.

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