# Solving linear systems by orthogonal tridiagonalization (GMINRES and/or GLSQR)

A general matrix A can be reduced to tridiagonal form by orthogonal

transformations on the left and right: UTAV = T. We can arrange that the

rst columns of U and V are proportional to given vectors b and c. An iterative

form of this process was given by Saunders, Simon, and Yip (SINUM 1988) and

used to solve square systems Ax = b and ATy = c simultaneously. (One of the

resulting solvers becomes MINRES when A is symmetric and b = c.)

The approach was rediscovered by Reichel and Ye (NLAA 2008) with emphasis

on rectangular A and least-squares problems Ax ~ b. The resulting solver was

regarded as a generalization of LSQR (although it doesn't become LSQR in

any special case). Careful choice of c was shown to improve convergence.

In his last year of life, Gene Golub became interested in \GLSQR" for

estimating cTx = bTy without computing x or y. Golub, Stoll, and Wathen

(ETNA 2008) revealed that the orthogonal tridiagonalization is equivalent to a

certain block Lanczos process. This reminds us of Golub, Luk, and Overton

(TOMS 1981): a block Lanczos approach to computing singular vectors.

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