By C. T. Kelley
I like to recommend this booklet as a leap begin to the sector. this can be a comprehenshive publication, that incorporates examples of the topics mentioned and script records in Matlab which are supplied on-line.
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Additional info for Iterative methods for linear and nonlinear equations
One can take a second pass through the modified Gram-Schmidt process and restore lost orthogonality , . 3. gmresb(x, b, A, e, kmax, p) 2. While p > e\\b\\2 and k < krnax do GMRES ITERATION 41 Even if modified Gram- Schmidt orthogonalization is used, one can still lose orthogonality in the columns of V. One can test for loss of orthogonality , , and reorthogonalize if needed or use a more stable means to create the matrix V . These more complex implementations are necessary if A is ill conditioned or many iterations will be taken.
In our description of the implementation, which is described and motivated fully in , we use fo = TO and follow the notation of . 3. bicgstab(x, 6, A, e, kmax) 2. Do While ||r|| Note that the iteration can break down in steps 2b and 2e. We provide an implementation of Algorithm bicgstab in the collection of MATLAB codes. The cost in storage and in floating-point operations per iteration remains bounded for the entire iteration. One must store seven vectors (x, 6, r,p, v,fo,£), letting s overwrite r when needed.
Hence the performance of CGNR and CGNE is determined by the distribution of singular values. 7. Examples for preconditioned conjugate iteration In the collection of MATLAB codes we provide a code for preconditioned conjugate gradient iteration. The inputs, described in the comment lines, are the initial iterate, #o, the right hand side vector 6, MATLAB functions for the matrix-vector product and (optionally) the preconditioner, and iteration parameters to specify the maximum number of iterations and the termination criterion.