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Iterative methods for singular linea...

Choi, Sou-Cheng (Terrya).

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Iterative methods for singular linear equations and least-squares problems.

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Iterative methods for singular linear equations and least-squares problems./
作者:	Choi, Sou-Cheng (Terrya).
面頁冊數:	101 p.
附註:	Advisers: Michael A. Saunders; Gene H. Golub.
Contained By:	Dissertation Abstracts International67-11B.
標題:	Engineering, General. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3242533
ISBN:	9780542983108

Iterative methods for singular linear equations and least-squares problems.
Choi, Sou-Cheng (Terrya).

Iterative methods for singular linear equations and least-squares problems. - 101 p.

Advisers: Michael A. Saunders; Gene H. Golub.

Thesis (Ph.D.)--Stanford University, 2007.

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.

ISBN: 9780542983108Subjects--Topical Terms:

1020744
Engineering, General.

Iterative methods for singular linear equations and least-squares problems.
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100 1 $a Choi, Sou-Cheng (Terrya). $3 1288285
245 1 0 $a Iterative methods for singular linear equations and least-squares problems.
300 $a 101 p.
500 $a Advisers: Michael A. Saunders; Gene H. Golub.
500 $a Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6436.
502 $a Thesis (Ph.D.)--Stanford University, 2007.
520 $a 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.
520 $a 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.
520 $a 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.
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