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Iterative Methods
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
by a similarity transformation (based for example on Householder reflectors; see Sec. 2.8). Only after this is completed do we begin performing the QR method. The matrix M can be made to be orthogonal, and since orthogonal matrices have nice properties that is what we do. Note that if A is symmetric then the upper Hessenberg matrix is actually tridiagonal. Sometimes Givens rotations are used instead of Householder reflectors, and sometimes we reduce the matrix to bidiagonal form (where only the main diagonal and the subdiagonal or only the main diagonal and the superdiagonal are nonzero) instead. In the common case that A is symmetric we can realize some additional savings.
Regularization Techniques for MR Image Reconstruction
Published in Joseph Suresh Paul, Raji Susan Mathew, Regularized Image Reconstruction in Parallel MRI with MATLAB®, 2019
Joseph Suresh Paul, Raji Susan Mathew
The Arnoldi process is an orthogonal projection method onto Km applied to general non-Hermitian matrices. This method was first introduced as a procedure for reducing a dense matrix into the Hessenberg form that involves a unitary transformation. As pointed out by Arnoldi [35], the eigenvalues of the Hessenberg matrix, obtained with fewer steps than N, can provide accurate approximations to the eigenvalues of the original matrix. An advantage of this approach is that it leads to an efficient method for approximating the eigenvalues of large sparse matrices and hence can be used for obtaining the solution of large sparse linear systems of equations. Arnoldi’s procedure uses the stabilized Gram–Schmidt process to generate a sequence of orthonormal vectors, v1,v2,v3,…, called the Arnoldi vectors, such that for every N, the vectors v1,…,vN span the Krylov sub-space Km.
Programs for Analysis of Constant Linear Systems
Published in Ernest S. Armstrong, ORACLS, 2020
Subroutine EIGEN computes all the eigenvalues and selected eigenvectors of a real n × n matrix A stored as a variable-dimensioned two-dimensional array. The input matrix is first balanced by exact similarity transformations such that the norms of corresponding rows and columns are nearly equal [4-1]. The balanced matrix is reduced to upper Hessenberg form by stabilized elementary similarity transformations [4-2]. All of the eigenvalues of the Hessenberg matrix are found by the double shift QR algorithm [4-3]. The desired eigenvectors of the Hessenberg matrix are then found by the inverse iteration method [4-4].
An overset generalised minimal residual method for the multi-solver paradigm
Published in International Journal of Computational Fluid Dynamics, 2020
Dylan Jude, Jayanarayanan Sitaraman, Vinod Lakshminarayan, James Baeder
As recommended by Saad (2003), the Hessenberg matrix from Algorithm 2 is made upper-triangular with Givens rotations in each Krylov iteration, resulting in an easily solvable minimisation problem and approximation of the residual in each linear solver iteration. For time-accurate simulations, the residual can therefore be tracked for each temporal step, non-linear sub-iteration and linear solver iteration (Krylov iteration).