A transpose-free “Lanczos/Orthodir” algorithm for linear systems

✍ Scribed by Claude Brezinski

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 360 KB
• Volume: 324
• Category: Article
• ISSN: 0764-4442
• DOI: 10.1016/s0764-4442(99)80374-7

✦ Synopsis

The method of Lanczos for solving systems of linear equations is implemented via recurrence relationships between formal orthogonal polynomials.

In this Note, a new procedure for computing the coefficients of these recurrence relations is proposed. In contrast with all other procedures. it does not make use of the transpose of the matrix of the systetn. Un algorithne cc Lm~czoslOrthodir J) saris trnmposition pour les systhes lin&ires Version franqaise abdgke On considkre le systPme d'Cquations IinCaires de dimension p, AI. = 6. La mtthode de Lanczos [9] consiste & construire la suite d'itCrCs ;I'~. E Rl' definis par (XL. -:K(~) E Ii-k(il, ro) = span(,l.o. A,/.(,, . . , P'7.()): '1.1. = (h -h,,. ) i Kn. ( .4T. y) = spanj y, -45,. . . izTi ' y).

oti :I:() es1 un vecteur arbitraire et y un vectcur presquc arbitraire. Ces deux conditions dkfinissent complktement k vecteur XL.. On montre que '/'k = 2, -zk~:l; = p,.(L4)r,,, oti l'k est un poly&me de de.& au plus k tel quc PA (0) = 1, qui satisfait les conditions d'orthogonalitk