On Computing Eigenvectors of Symmetric Tridiagonal Matrices

Mastronardi, Nicola; Taeter, Harold; Dooren, Paul Van

doi:10.1007/978-3-030-04088-8_9

Cited by 3 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mastronardi [3,12] developed a procedure for computing an eigenvector of a symmetric tridiagonal matrix once its associate eigenvalue is known and gave the corresponding Matlab codes in [12].…”

Section: Comparing With Mastronardi's Methodsmentioning

confidence: 99%

“…However, Algorithm 7 has a significant advantage in time cost. In addition, Mastronardi seems unstable when computing the eigenvector (corresponding to the maximal eigenvalue) of Matrix Φ 1 and W 1 : the Matlab routine provided in [12] failed to converge. The instability also arises in computing some eigenvectors of random matrices.…”

Section: Comparing With Mastronardi's Methodsmentioning

confidence: 99%

“…Mastronardi and Van Dooreen discovered this instability when obtaining an ST eigenvector and solved the problem by a modified implicit QR decomposition method [12]. Their method can ensure an accurate calculation.…”

Section: Reorthogonalization 41 General Q Iterationmentioning

confidence: 99%

“…Mastronardi and Van Dooreen [12] proposed an ingenious method to determine the accurate eigenvector of a symmetric tridiagonal matrix once an approximation of the eigenvalue is known. In addition, they applied this method to calculate the weights of the Gaussian quadrature rules [3].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

2022

View full text Add to dashboard Cite

This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration and a significant modification when calculating severely clustered eigenvectors. The numerical results show that this method is competitive with other existing methods, especially when computing part eigenvectors or severely clustered ones.

show abstract

Section: Comparing With Mastronardi's Methodsmentioning

confidence: 99%

Section: Comparing With Mastronardi's Methodsmentioning

confidence: 99%

Section: Reorthogonalization 41 General Q Iterationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

2022

View full text Add to dashboard Cite

show abstract

“…In that paper, which mainly addresses the special case of symmetric weight functions, the authors indicate that the use of LAPACK subroutine DLASQ1 (which was called by the algorithm TNEigen-Values used in [28]) provides high relative accuracy in the computation of the nodes. As for the computation of the weights, the authors of [21] use an approach which is partly based on the eigenvector computation presented in [32]. Our approach to the computation of the weights, which is presented in detail in Section 4, is based on the computation of the right singular vectors of a bidiagonal matrix by means of the LAPACK routine DBDSQR.…”

mentioning

confidence: 99%

Accurate computations with totally positive matrices applied to the computation of Gaussian quadrature formulae

Marco

Martínez

Viaña

2022

ELA

View full text Add to dashboard Cite

For some families of classical orthogonal polynomials defined on appropriate intervals, it is shown that the corresponding Jacobi matrices are totally positive and their bidiagonal factorizations can be accurately computed. By exploiting these facts, an algorithm to compute with high relative accuracy the eigenvalues of those Jacobi matrices, and consequently the nodes of Gaussian quadrature formulae for those families of orthogonal polynomials, is presented. An algorithm is also presented for the computation of the eigenvectors of these Jacobi matrices, and hence the weights of Gaussian quadrature formulae. Although in this case high relative accuracy is not theoretically guaranteed, the numerical experiments with our algorithm provide very accurate results.

show abstract

Computing Gaussian quadrature rules with high relative accuracy

2022

View full text Add to dashboard Cite

The computation of n-point Gaussian quadrature rules for symmetric weight functions is considered in this paper. It is shown that the nodes and the weights of the Gaussian quadrature rule can be retrieved from the singular value decomposition of a bidiagonal matrix of size n/2. The proposed numerical method allows to compute the nodes with high relative accuracy and a computational complexity of $ \mathcal {O} (n^{2}). $ O ( n 2 ) . We also describe an algorithm for computing the weights of a generic Gaussian quadrature rule with high relative accuracy. Numerical examples show the effectiveness of the proposed approach.

show abstract

On Computing Eigenvectors of Symmetric Tridiagonal Matrices

Cited by 3 publications

References 14 publications

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

Accurate computations with totally positive matrices applied to the computation of Gaussian quadrature formulae

Computing Gaussian quadrature rules with high relative accuracy

Contact Info

Product

Resources

About