Minimal Residual Method

The conjugate gradient method can be viewed as a special variant of the Lanczos method for positive definite symmetric systems. The minimal residual method (MINRES) and symmetric LQ method (SYMMLQ) methods are variants that can be applied to symmetric indefinite systems.

The vector sequences in the conjugate gradient method correspond to a factorization of a tridiagonal matrix similar to the coefficient matrix. Therefore, a breakdown of the algorithm can occur corresponding to a zero pivot if the matrix is indefinite. Furthermore, for indefinite matrices the minimization property of the conjugate gradient method is no longer well-defined. The MINRES methods is a variant of the conjugate gradient method that avoids the LU decomposition and does not suffer from breakdown. MINRES minimizes the residual in the 2-norm. The convergence behavior of the conjugate gradient and MINRES methods for indefinite systems was analyzed by Paige et al. (1995).

When A is not positive definite, but symmetric, we can still construct an orthogonal basis for the Krylov subspace by three-term recurrence relations. Eliminating the search directions in the equations of the conjugate gradient method gives a recurrence


which can be written in matrix form as


where T^__i is an (i+1)×i tridiagonal matrix.

In this case we have the problem that (·,·)_(A) no longer defines an inner product. However we can still try to minimize the residual in the 2-norm by obtaining

 x^((i)) in {r^((0)),Ar^((0)),...,A^(i-1)r^((0))},    x^((i))=R_iy^_

that minimizes


Now we exploit the fact that if


then R_(i+1)D_(i+1)^(-1) is an orthonormal transformation with respect to the current Krylov subspace


and this final expression can simply be seen as a minimum norm least squares problem.

The element in the (i+1,i) position of T^__i can be annihilated by a simple Givens rotation and the resulting upper bidiagonal system (the other subdiagonal elements having been removed in previous iteration steps) can simply be solved, which leads to the MINRES method (Paige and Saunders 1975).

See also

Biconjugate Gradient Method, Chebyshev Iteration, Conjugate Gradient Method on the Normal Equations Conjugate Gradient Method, Conjugate Gradient Squared Method, Flexible Generalized Minimal Residual Method, Generalized Minimal Residual Method, Linear System of Equations, Nonstationary Iterative Method, Preconditioner, Quasi-Minimal Residual Method Stationary Iterative Method, Symmetric LQ Method, Transpose-Free Quasi-Minimal Residual Method

This entry contributed by Noel Black and Shirley Moore, adapted from Barrett et al. (1994) (author's link)

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Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; and van der Vorst, H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. Philadelphia, PA: SIAM, 1994., C.; Parlett, B.; and van der Vorst, H. "Approximate Solutions and Eigenvalue Bounds from Krylov Subspaces." Numer. Lin. Alg. Appl. 29, 115-134, 1995.Paige, C. and Saunders, M. "Solution of Sparse Indefinite Systems of Linear Equations." SIAM J. Numer. Anal. 12, 617-629, 1975.

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Minimal Residual Method

Cite this as:

Black, Noel and Moore, Shirley. "Minimal Residual Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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