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Symmetric LQ 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 and symmetric LQ method (SYMMLQ) 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 and SYMMLQ methods are variants of the CG method that avoid the LU decomposition and do not suffer from breakdown. SYMMLQ solves the projected system, but does not minimize anything (it keeps the residual orthogonal to all previous ones).

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

Ar^((i))=r^((i+1))t_(i+1,i)+r^((i))t_(i,i)+r^((i-1))t_(i-1,i),
(1)

which can be written in matrix form as

AR_i=R_(i+1)T^__i,
(2)

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^_
(3)

that minimizes

|Ax^((i))-b|_2=|AR_iy^_-b|_2
(4)
=|R_(i+1)T^__iy-b|_2.
(5)

Now we exploit the fact that if

D_(i+1)=diag(|r^((0))|_2,|r^((1))|_2,...,|r^((i))|_2),
(6)

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

|Ax^((i))-b|_2=|D_(i+1)T^__iy-|r^((0))|_2e^((1))|_2,
(7)

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

One approach is to solve the system T_iy=|r^((0))|_2e^((1)), as in the conjugate gradient method (T_i is the upper i×i part of T^__i). However, other than in the conjugate gradient method, we cannot rely on the existence of a Cholesky decomposition (since A is not positive definite). An alternative is then to decompose T_i by an LQ decomposition. This leads to simple recurrences and the resulting method is known as SYMMLQ (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, Minimal Residual Method, Nonstationary Iterative Method, Preconditioner, Quasi-Minimal Residual Method Stationary Iterative 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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References

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. http://www.netlib.org/linalg/html_templates/Templates.html.Paige, 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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Symmetric LQ Method

Cite this as:

Black, Noel and Moore, Shirley. "Symmetric LQ Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SymmetricLQMethod.html

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