Ấn phẩm:
Iterative methods for singular linear equations and least-squares problems
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CG, MINRES, and SYMMLQ are Krylov subspace methods for solving large symmetric systems of linear equations. CG (the conjugate-gradient method) is reliable on positive-definite systems, while MINRES and SYMMLQ are designed for indefinite systems. When these methods are applied to an inconsistent system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length solution (often called the pseudoinverse solution). This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems.
MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). Our algorithm uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form), and so we call it MINRES-QLP. On singular or nonsingular systems, MINRES-QLP can give more accurate solutions than MINRES or SYMMLQ. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm and condition number.
For a singular matrix of arbitrary shape, we observe that null vectors can be obtained by solving least-squares problems involving the transpose of the matrix. For sparse rectangular matrices, this suggests an application of the iterative solver LSQR. In the square case, MINRES, MINRES-QLP, or LSQR are applicable. Results are given for solving homogeneous systems, computing the stationary probability vector for Markov Chain models, and finding null vectors for sparse systems arising in helioseismology.
Tác giả
Choi, Sou-Cheng (Terrya)
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Stanford University
Năm xuất bản
2006
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