Parallel homotopy algorithm for symmetric large sparse eigenproblems

• Published in: Journal of computational and applied mathematics Vol. 60; no. 1; pp. 77 - 100
• Main Authors: Huang, Liang Jiao, Li, Tien-Yien
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 20-06-1995; Elsevier
• Subjects: Eigenproblems; Exact sciences and technology; Homotopy algorithm; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Parallelism; Sciences and techniques of general use; Homotopy algorithm; Parallelism; Eigenproblems; Matrix algebra; Symmetric matrix; Homotopy; Starting; Sparse matrix; Eigenvalues and eigenfunctions; Continuation method; Parallel algorithms
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/0377-0427(94)00085-F

In this paper, the homotopy continuation method is applied to solve the eigenproblem
Ax =
λx,
λ ∈ R,
x ∈ R
n
⧹ {0} for a symmetric large sparse matrix
A. A one-parameter family of matrices
A(
t) =
tA + (1 −
t)
D is introduced and the eigenproblem
A(
t)
x(
t) =
λ(
t)
x(
t) is considered for
t ∈ [0,1]. We discuss the problem of choosing an optimal starting matrix
A(0) =
D and consider the regularity and bifurcation problem of
λ(
t) and
x(
t). A homotopy continuation algorithm is constructed and implemented on both parallel and vector machines for several types of matrices. The numerical experiments show that our method is efficient and highly parallel.