# Systems Optimization Laboratory

## SYMMLQ: Sparse Symmetric Equations

• AUTHORS: C. C. Paige, M. A. Saunders.
• CONTRIBUTORS: Jeffery Kline, Dominique Orban.
• CONTENTS: Implementation of a conjugate-gradient type method for solving sparse linear equations:  Solve \begin{align*} Ax = b \ \text{ or } \ (A - sI)x = b. \end{align*} The matrix $$A - sI$$ must be symmetric and nonsingular, but it may be definite or indefinite. The scalar $$s$$ is a shifting parameter -- it may be any number. The method is based on Lanczos tridiagonalization. You may provide a preconditioner, but it must be positive definite.

If A - sI is symmetric but singular, use MINRES.
If A is unsymmetric, use LSQR.
• REFERENCES:
C. C. Paige and M. A. Saunders (1975). Solution of sparse indefinite systems of linear equations, SINUM 12, 617--629.