Generating conjugate directions for arbitrary matrices by matrix equations I

Cs.J. Hegedüs
1991 Computers and Mathematics with Applications  
Recursive methods for generating conjugate directions with respect to an arbitrary matrix are investigated. There are three basic techniques to achieve this aim: (i) minimizing a quadratic form, (ii) generation by projections, and (iii) use of matrix equations. These techniques are equivalent to each other, however, the third one is stressed in this paper because of its versatility. Among matrix equation forms Hestenes -Stiefel type recursions and L~nczos type recursions are mentioned, where
more » ... recursion matrices are bidiagonal matrices in the simple case. With respect to the choice of recursion matrices, direct and reverse methods are introduced. The recursion matrices may have lower and upper triangular forms in the direct case and they may be lower and upper Hessenberg matrices in the reverse case. The recurslon matrices chosen here are as simple as possible, actually they have no more nonzero elements than that of a bidiagonal matrix. Consequently, the storage of four vectors suffices to perform the recursions in all cases. It is shown that restructuring the bidiagonal matrices makes it posssible to avoid zero divisors for the Hestenes -Stiefel type schemes.
doi:10.1016/0898-1221(91)90232-s fatcat:nsqvkm5it5httdqtad2aforbfu