An Algorithm for Symmetric Tridiagonal Eigenproblems: Divide and Conquer with Homotopy Continuation

Li, Kuiyuan; Li, Tien-Yien

doi:10.1137/0914046

Cited by 20 publications

(8 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main difference between the two methods lies in the merging process of the solutions of two smaller subproblems. Li and Li [12] present another homotopy divide-and-conquer algorithm. In contrast to our algorithm, they split the original matrix A just once and compute only approximations of the eigenvalues of D. We, however, need full accuracy in order to exploit possible deflation.…”

Section: Applying the Divide-and-conquer Principlementioning

confidence: 99%

“…A drawback of the method, as shown in other work [15,13,14,12], is the difficulty of implementing it in a reliable and efficient way. A new algorithm based on the homotopy method maintains the high accuracy of the algorithm from Li and Li [12], while considerably improving the performance for matrices with clustered eigenvalues. This is achieved by deflation of certain very close eigenpairs and the use of the divide-and-conquer principle.…”

Section: Introduction Various Stable Algorithms Are Known To Solve Tmentioning

confidence: 99%

“…Rank-one modification as starting matrix. While Li and Li [12] use a rank-two modification to split A, we apply a rank-one modification with some nice mathematical properties. A third way to split A would be to use a rank-one extension; see [16].…”

mentioning

confidence: 99%

See 2 more Smart Citations

A Robust, Parallel Homotopy Algorithm for the Symmetric Tridiagonal Eigenproblem

Oettli¹

1998

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

The homotopy method has been used in the past for finding zeros of nonlinear functions. Early algorithms based on the homotopy approach for solving the symmetric eigenproblem suffer either from insufficient orthogonality between computed eigenvectors or from low efficiency. A new algorithm presented in this article overcomes these problems by using deflation of close eigenpairs and maintains a high degree of parallelism by applying the divide-and-conquer principle. A numerical comparison of a sequential implementation of the new algorithm shows that it is competitive with other well-established algorithms both in speed and accuracy.Algorithms based on the homotopy method are excellent candidates for multicomputers because of their natural parallelism. This article examines how the algorithm can be implemented on distributed-memory multicomputers. The experimental performance results obtained on an Intel Paragon multicomputer are very satisfying.

show abstract

Section: Applying the Divide-and-conquer Principlementioning

confidence: 99%

Section: Introduction Various Stable Algorithms Are Known To Solve Tmentioning

confidence: 99%

See 1 more Smart Citation

A Robust, Parallel Homotopy Algorithm for the Symmetric Tridiagonal Eigenproblem

Oettli¹

1998

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…When A 1 is a real symmetric tridiagonal matrix with nonzero off-diagonal elements, a very successful homotopy method is known (see Li and Li [16] and Li, Zhang, and Sun [21]). The following phenomena, while absent in the symmetric tridiagonal case, are present for the general case:…”

mentioning

confidence: 99%

Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

Lui¹,

Keller²,

Kwok³

1997

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract.A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A 1 is presented. From the eigenpairs of some real matrix A 0 , the eigenpairs of A(t) ≡ (1 − t)A 0 + tA 1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A 1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix:• bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented.

show abstract

“…There are many very efficient algorithms for (2), for instant, the QR algorithm [8], the D&C algorithm [3], the bisection algorithm [5] and the homotopy algorithm [6]. Nhen A and B are both tridiagonal the above technique is unattractive because L-IAL-T is, in general, a full matrix.…”

mentioning

confidence: 99%

A fully parallel method for tridiagonal eigenvalue problem

1994

International Journal of Mathematics and Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

An Algorithm for Symmetric Tridiagonal Eigenproblems: Divide and Conquer with Homotopy Continuation

Cited by 20 publications

References 13 publications

A Robust, Parallel Homotopy Algorithm for the Symmetric Tridiagonal Eigenproblem

A Robust, Parallel Homotopy Algorithm for the Symmetric Tridiagonal Eigenproblem

Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

A fully parallel method for tridiagonal eigenvalue problem

Contact Info

Product

Resources

About