1 edition of **On computing accurate singular values and eigenvalues of acyclic matrices** found in the catalog.

On computing accurate singular values and eigenvalues of acyclic matrices

J. W. Demmel

- 41 Want to read
- 7 Currently reading

Published
**1992**
by Naval Postgraduate School, Available from National Technical Information Service in Monterey, Calif, Springfield, Va
.

Written in English

- COMPUTATIONS,
- EIGENVALUES,
- MATRICES(MATHEMATICS)

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**

Other titles | NPS-MA-92-010. |

Statement | J.W. Demmel, W.B. Gragg |

Contributions | Gragg, William B., Naval Postgraduate School (U.S.). Dept. of Mathematics |

The Physical Object | |
---|---|

Pagination | 12 p. ; |

Number of Pages | 12 |

ID Numbers | |

Open Library | OL25480341M |

[3] J. DEMMEL ANDW. B. 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.

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An illustration of an open book. Books. An illustration of two cells of a film strip. Video An illustration of an audio speaker. 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 computing 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. 4. 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 .