AB-GMRES iterative solver with a RIF preconditioner
x=splspc_rifgmresab(A, b) x=splspc_rifgmresab(A, b, trif) x=splspc_rifgmresab(A, b, trif, restart) x=splspc_rifgmresab(A, b, trif, restart, maxiter) x=splspc_rifgmresab(A, b, trif, restart, maxiter, tol) [x, nbiter]= splsc_rifgmresab(...) [x, nbiter, res]= splsc_rifgmresab(...) [x, nbiter, res, result]= splsc_rifgmresab(...) [x, nbiter, res, result, pretime]= splsc_rifgmresab(...) [x, nbiter, res, result, pretime, itertime]= splsc_rifgmresab(...)
sparse matrix (size m x n) : the matrix A in the least square problem min(b-Ax)
matrix of double (size m x 1) : the vector b in the least square problem min(b-Ax)
matrix of double (size 1 x 1) : a dropping tolerance parameter for the eliminations of the smaller elements during the algorithm (default = 1.e-1)
matrix of integer (size 1 x 1) : the number of iterations before restarting the algorithm (default = min(m,n))
matrix of integer (size 1 x 1) : the maximum number of iterations in the algorithm (default = min(m,n))
matrix of integer (size 1 x 1) : the threshold of tolerance for stopping the algorithm (default = 1.e-6)
matrix of double (size n x 1) : solution vector x in the least square problem min(b-Ax)
matrix of integer (size 1 x 1) : total number of necessary iterations to find the solution vector
matrix of double (size 1 x 1) : norm of the computed residual r=||(A^T)(b-Ax)||/||(A^T)b||
matrix of integer (size 1 x 1) : 0 if the solution is reached, 1 otherwise
matrix of double (size 1 x 1) : computation time for the preconditioner
matrix of double (size 1 x 1) : computation time for the iterative method
This function solves the sparse linear least squares problem min |Ax-b|, with the AB-GMRES method with RIF preconditionning.
Any optional argument equal to the empty matrix [] is replaced by its default value.
The GMRES method consist in solving the least square problem min(b-Ax), for A square matrix. But when A is not square, GMRES can't be directly used and need to be slightly modified.
The AB-GMRES method consist in solving the least square problem min(b-ABx), ie finding x such as the norm of b-ABx is minimal. This method is mainly useful for underdetermined problems, ie m < n for A of size m x n, because the matrix obtained by doing the product of B and A is of size m x m.
The Robust Incomplete Factorization (RIF) is a method applied before an iterative method (such as GMRES or CGLS). Depending on the larger dimension of the matrix A, this methods seeks to calculate an approximation of A^(T)A or AA^(T). Indeed, it gives an upper triangular matrix Z such as A^(T)A (or AA^(T)) is almost equal to (Z^-T)D(Z^-1) with D diagonal. Thus, the preconditioner is computed in this way : if m > n, we have C={Z(D^-1)(Z^T)} and the preconditioner is B=C(A^T). Alternatively, if m < n, we have C={Z(D^-1)(Z^T)} and the preconditioner is B=(A^T)C. The accuracy and speed of the method is strongly dependant of the choice of the dropping tolerance parameter. If m < n, A^T is automatically used rather than A to calculate the preconditioner and then B^T is taken as a preconditioner in the iterative method.
Restart can be used to save memory but is not adviced in this case. The initial vector x0 for the iterative process has been set up to a vector of zeros.
path = fullfile(splspc_getpath(),"tests","matrices"); filename=fullfile(path,"illc1033.mtx"); A = mmread(filename); A = A'; e = ones(size(A,2),1); b = A * e; // With default settings x=splspc_rifgmresab(A,b); norm(A*x-b)/norm(b) // Configure the options trif=1.e-1; restart=1000; max_iter=3000; tol=1.e-6; [x,nbiter,resid,result,pretime,itertime]=splspc_rifgmresab(A,b,trif,restart,max_iter,tol); // Configure only max_iter x=splspc_rifgmresab(A,b,[],[],max_iter);
1994 - 2008 - Sparselib++ v1.7 - R. Pozo, K. Remington, and A. Lumsdaine
2010 - National Institute of Informatics - K. Hayami, T. Ito and J.-F. Yin
2011 - DIGITEO - Michael Baudin
2011 - National Institute of Informatics - Benoit Goepfert