On computing accurate singular values and eigenvalues of acyclic matrices by J. W. Demmel

Cover of: On computing accurate singular values and eigenvalues of acyclic matrices | J. W. Demmel

Published by Naval Postgraduate School, Available from National Technical Information Service in Monterey, Calif, Springfield, Va .

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  • COMPUTATIONS,
  • EIGENVALUES,
  • MATRICES(MATHEMATICS)

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It is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values, independent of the values of the matrix entries. In this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic. We also show how to compute the singular values of such a matrix to high relative accuracy. The same algorithm can compute eigenvalues of symmetric acyclic matrices with tiny component-wise relative backward error. This class includes tridragonal matfices, arrow matrices, and exponentially many others.

Edition Notes

Book details

Other titlesNPS-MA-92-010.
StatementJ.W. Demmel, W.B. Gragg
ContributionsGragg, William B., Naval Postgraduate School (U.S.). Dept. of Mathematics
The Physical Object
Pagination12 p. ;
Number of Pages12
ID Numbers
Open LibraryOL25480341M

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On computing accurate singular values and eigenvalues of acyclic matrices On computing accurate singular values and eigenvalues of acyclic matrices by Demmel, J.

W.;Gragg, William B. Publication date Pages:   On Computing Accurate Singular Values and Eigenvalues of Matrices With Acyclic Graphs James W. Demmel* Computer Science Division and Department of Mathematics University of California Berkeley, California and William Graggt Department of Mathematics Naval Postgraduate School Monterey, California Submitted by Richard A.

Brualdi Cited by: BibTeX @MISC{Demmel92oncomputing, author = {James W. Demmel and William Cragg}, title = {On Computing Accurate Singular Values and Eigenvalues of Matrices with Acyclic.

ric acyclic matrices with tiny componentwise relative backward error, and applies this to compute the singular values of biacyclic matrices to high relative accuracy. Section 5 gives some examples of matrices with acyclic sparsity patterns. Section 6. On computing accurate singular values and eigenvalues of matrices with acyclic graphs.

In this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic. We also show how to compute the singular values of such a matrix to high relative accuracy. The same algorithm can compute eigenvalues of Cited by:   Computing the singular values of a bidiagonal matrix is the final phase of the standard algorithm for the singular value decomposition of a general matrix.

On On computing accurate singular values and eigenvalues of acyclic matrices book accurate singular values and eigenvalues of matrices with acyclic graphs. Linear Algebra and its ApplicationsAccurate Singular Values and Differential QD.

For diagonally dominant M-matrices and some subclasses of nonsingular totally nonnegative matrices, accurate methods for computing their singular values, eigenvalues or inverses have been obtained. In this paper, we consider how to accurately solve the nonsymmetric eigenvalue problem for a class of generalized sign regular matrices including extremely ill-conditioned quasi-Cauchy and quasi-Vandermonde matrices.

The problem of performing accurate computations with structured matrices is very much a representation problem. We first develop a relatively.

inite matrices [13] and matrices that admit accurate rank-revealing factorizations [8], which will form the basis of our works. For such matrices, the Jacobi method can be used to efficiently compute, respectively, the smaller eigenvalue and singular values to high relative accuracy. Some other examples are bidiagonal and acyclic matrices [4, 7.

These classes include many graded matrices and all symmetric positive-definite matrices that can be consistently ordered (i.e., all symmetric positive-definite tridiagonal matrices).

In particular, singular values and eigenvalues are determined to high relative precision, independent of their magnitudes, and there are algorithms to compute them. Accurate Computation of Singular Values and Eigenvalues of Symmetric Matrices Article (PDF Available) July with 51 Reads How we measure 'reads'.

Lecture Computing Eigenvalues and Singular Values Course Home The book LAPACK has 10 authors, and you can download any of these codes, like the eigenvalue code. So that's where Matlab, naturally--that's the Bible for code in linear algebra.

Then the singular value of the matrix--SV for singular values. Ye, Q. ( a), ‘ Computing singular values of diagonally dominant matrices to high relative accuracy ’, Math.

Comp., to appear. Ye, Q. ( b), ‘ Relative perturbation bounds for eigenvalues of symmetric positive definite diagonally dominant matrices ’, SIAM J.

Matrix Anal. Appl., to appear. A real square matrix is said to be totally nonpositive if all of its minors are nonpositive. In this paper, strong relative perturbation bounds are developed for eigenvalues and singular values of.

In this paper, we consider the product eigenvalue problem for a wide class of structured matrices containing the well-known Vandermonde and Cauchy matrices. A periodic qd-type reduction method is developed for computing eigenvalues of products of these rectangular matrices so that no subtraction of like-signed numbers occurs.

Consequently, all the eigenvalues. Finite elements, relative accuracy, singular values. 1 INTRODUCTION AND SUMMARY To find the singular values ai of a matrix with a high relative accuracy means to get the relative errors 6ai/ai small even then when ai differ greatly in their magnitude: I~1 ~ c~ (1) for the corresponding singular values ai, ai + ~ai, respectively.

ON COMPUTING ACCURATE SINGULAR VALUES AND EIGENVALUES OF ACYCLIC MATRICES 12 PERSONAL AUTHOR(S) James W. Demmel and William Gragg 13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year, Month. Day) 15 PAGE COUNT Technical Report FROM 10/1/91TO3/31/92 August 7, 10 16 SUPPLEMENTARY.

Eigenvalues and Singular Values This chapter is about eigenvalues and singular values of matrices. Computational algorithms and sensitivity to perturbations are both discussed. Eigenvalue and Singular Value Decompositions An eigenvalue and eigenvector of a square matrix A are a scalar λ and a nonzero vector x so that Ax = λx.

In this video you will learn how to calculate the singular values of a matrix by finding the eigenvalues of A transpose A. We will also do. Singular values of the SVD decomposition of the matrix A is the square root of the eigenvalues of the matrix (A multiplied by A transpose) or(A transpose multplied by A), the two ar identical with positive eigenvalues.

where is the characteristic polynomial of A. We have some properties of the eigenvalues of a matrix. Theorem. Let A be a square matrix of order n. If is an eigenvalue of A, then: 1. is an eigenvalue of A m, for 2. If A is invertible, then is an eigenvalue of A 3.

A is not invertible if and only if is an eigenvalue of A. If is any number, then is an eigenvalue of. In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any × matrix via an extension of the polar decomposition.

Specifically, the singular value decomposition of an × real or complex matrix is a factorization of the form ∗, where is an × real or complex unitary matrix. In the standard procedure for computing the singular values, one first reduces the input matrix to an upper bidiagonal matrix by orthogonal transformations and then computes the singular values.

Singular Value Decomposition Computing the SVD The SVD can be computed by performing an eigenvalue computation for the normal matrix A?A (a positive-semide nite matrix). This squares the condition number for small singular values and is not numerically-stable.

Instead, modern algorithms use an algorithm based on computing. The computed accuracy of smaller eigenvalues of a matrix has been discussed extensively in the context of the dense matrix eigenvalue problems in the last two decades. Starting with a work by Demmel and Kahan [23] on computing singular values of bidiagonal matrices, there is a large body of literature on when and how.

[3] J. DEMMEL ANDW. GRAGG, On computing accurate singular values and eigenvalues of matrices with acyclic graphs, Linear Algebra Appl. (), – [4] J.

D EMMEL AND W. K AHAN, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. [V,D,W] = eig(A,B) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'*B.

The generalized eigenvalue problem is to determine the solution to the equation Av = λBv, where A and B are n-by-n matrices, v is a column vector of length n, and λ is a scalar.

The values of λ that satisfy the equation are the generalized eigenvalues. orthogonal matrix, V vv1, n is an nn or- thogonal matrix, and S diag, 1 n is an mn diagonal matrix. The singular values of A are assumed to be nonnegative and sorted to satisfy 12 n 0.

() The columns of U and V are called left singular vectors and right singular vectors, respectively. These. p(A)x=p()x, so eigenvalues of polynomial in matrix are values of polynomial evaluated at eigenvalues of original matrix Michael T.

Heath Scientific Computing 21 / 87 Eigenvalue Problems Computing Eigenvalues and Eigenvectors Problem Transformations Other Methods Similarity Transformation issimilar toAif there is nonsingular matrixT such that.

@article{osti_, title = {Computing small singular values of bidiagonal matrices with guaranteed high relative accuracy: LAPACK working note number 3}, author = {Demmel, J and Kahan, W}, abstractNote = {Computing the singular values of a bidiagonal matrix is the final phase of the standard algorithm for the singular value decomposition of a general matrix.

qmult Pre-multiply matrix by random orthogonal matrix. randcolu Random matrix with normalized cols and specified singular values. randcorr Random correlation matrix with specified eigenvalues. randhess Random, orthogonal upper Hessenberg matrix. randjorth Random J-orthogonal (hyperbolic, pseudo-orthogonal) matrix.

rando Random matrix with. (from Lay’s book) Find a singular value decomposition of A= 4 11 14 8 7 2: Step 1. We rst need to nd the eigenvalues of ATA.

We compute that ATA= 0 @ 80 40 40 1 A: We know that at least one of the eigenvalues is 0, because this matrix can have rank at most 2. In fact, we can compute that the eigenvalues are p 1 = acterization of eigenvalues as extreme values of Rayleigh quotients.

Theorem [Perron-Frobenius, Symmetric Case] Let Gbe a connected weighted graph, let A be its adjacency matrix, and let 1 2 n be its eigenvalues. Then a. 1 n, and b. 1 > 2, c. The eigenvalue 1 has a strictly positive eigenvector.

vs 2-D y. The classes that represent matrices, and basic operations, such as matrix multiplications and transpose are a part of convenience, we summarize the differences between and y here. is matrix class that has a more convenient interface than y for matrix.

Matrix factorization type of the singular value decomposition (SVD) of a matrix A. This is the return type of svd(_), the corresponding matrix factorization function. If F::SVD is the factorization object, U, S, V and Vt can be obtained via F.U, F.S, F.V andsuch that A = U * Diagonal(S) * Vt.

The singular values in S are sorted in. A non-square matrix A does not have eigenvalues. In their place, one uses the square roots of the eigenvalues of the associated square Gram matrix K = ATA, which are called singular values of the original matrix. The numerical computation of eigenvalues and eigenvectors is a challenging issue, and must be be deferred until later.

In volume 2, Raf Vandebril, Marc Van Barel, and Nicola Mastronardi discuss the theory of structured eigenvalue and singular value computations for semiseparable matrices. These matrices have hidden properties that allow the development of efficient methods and algorithms to accurately compute the matrix : Raf Vandebril, Marc van Van Barel, Nicola Mastronardi.

very true. can take it like this: any matrix can be diagonalized by using appropriate elementary matrices and we know the eigen values of diagonal matrices are the diagonal elements and so if any of the eigen value is zero then determinant value of matrix is zero and so it is Singular.

Singular Values of Special Matrices In this section, we present some matrices where the singular values (or some of the singular values) are known, and facts about the singular values of certain structured matrices.

Facts: The following results can be obtained by straightforward computations if no specific reference is given. Let D. Computing Eigenvalues and Eigenvectors Characteristic Polynomial Relevant Properties of Matrices Conditioning Characteristic Polynomial Equation Ax = x is equivalent to (A I)x = 0 which has nonzero solution x if, and only if, its matrix is singular Eigenvalues of A are roots iof characteristic polynomial det(A I) = 0 in of degree n.

I was a bit misleading originally because I worked with the notion of PSD matrices rather than covariance matrices. The answer before the editing was fine. It addressed exactly why your eigen-decomposition did not fit your singular value decomposition. With the editting I show why your singular value decomposition did not fit the eigen.Chapter Computing Eigenvalues and Singular Values to High Relative Accuracy.

Zlatko Drmač. University of Zagreb. To compute the eigenvalues and singular values to high relative accuracy means to have a guaranteed number of accurate digits in all computed approximate values.Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

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