On 2D Bisection Method for Double Eigenvalue Problems

✍ Scribed by Xingzhi Ji

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 339 KB
• Volume: 126
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.0122

✦ Synopsis

Ϫ( p 1 (x 1 ) yЈ 1 )Ј ϩ q 1 (x 1 ) y 1 ϭ (s 11 (x 1 ) ϩ Ȑs 12 (x 1 )) y 1 ,

The two-dimensional bisection method presented in (SIAM J. Matrix Anal. Appl. 13(4), 1085 (1992)) is efficient for solving a class

of double eigenvalue problems. This paper further extends the 2D bisection method to full matrix cases and analyses its stablity. As

in a single parameter case, the 2D bisection method is very stable for the tridiagonal matrix triples satisfying the symmetric-definite y 1 (b 1 ) cos ͱ 1 Ϫ ( p 1 yЈ 1 )(b 1 ) sin ͱ 1 ϭ 0, condition. Since the double eigenvalue problems arise from twoparameter boundary value problems, an estimate of the discretiza-

tion error in eigenpairs is also given. Some numerical examples are included. ᮊ 1996 Academic Press, Inc.

where 0 Յ Ͱ i Ͻ ȏ, 0 Ͻ ͱ i Յ ȏ, p i Ͼ 0 and pЈ i , q i , s ij are * The author thanks Professor P. A. Binding and Professor P. J. Browne BЉ(ͱ) ϩ ͕Ϫh ϩ ( ϩ 1)(1 Ϫ kЈ 2 sn 2 (ͱ, kЈ))͖B(ͱ) ϭ 0, for many stimulating discussions when he visited the University of Calgary.

B(ϪKЈ) ϭ BЈ(KЈ) ϭ 0, 91