A novel class of block methods based on the blockAA^T-Lanczos bi-orthogonalization process for matrix equations

A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

“…Due to simple calculations and less information required, block Krylov subspace methods are always designed to solve system (1) efficiently [3,4]. Recently, Tadano et al presented the block conjugate orthogonal conjugate gradient (BCOCG) method [5], which can exploit the symmetry of A naturally.…”

“…Due to simple calculations and less information required, block Krylov subspace methods are always designed to solve system (1) efficiently [3,4]. Recently, Tadano et al presented the block conjugate orthogonal conjugate gradient (BCOCG) method [5], which can exploit the symmetry of A naturally. The BCOCG is also deemed a natural generalization of the conjugate orthogonal conjugate gradient (COCG) method [6][7][8] for solving systems (1).…”

“…Recently, Tadano et al presented the block conjugate orthogonal conjugate gradient (BCOCG) method [5], which can exploit the symmetry of A naturally. The BCOCG is also deemed a natural generalization of the conjugate orthogonal conjugate gradient (COCG) method [6][7][8] for solving systems (1). Besides COCG-type methods, the COCR method described in [2,7,8] can also exploit the symmetry of A when p = 1 in (1).…”

“…The BCOCG is also deemed a natural generalization of the conjugate orthogonal conjugate gradient (COCG) method [6][7][8] for solving systems (1). Besides COCG-type methods, the COCR method described in [2,7,8] can also exploit the symmetry of A when p = 1 in (1). In [1], Gu et al introduced a block version of the COCR method (BCOCR) by employing the residual orthonormalization technique.…”

Improvement of the accuracy of the approximate solution of the Block BiCR method

“…As one of Block Krylov subspace methods, the Block Bi-Conjugate Residual (BiCR) method [6] has been proposed by Zhang et al This method is a natural extension of the BiCR method [7] for linear systems with single right-hand side proposed by Sogabe et al The Block BiCR method often shows smooth convergence behavior compared with the Block BiCG method. However, the accuracy of the approximate solution generated by the Block BiCR method may deteriorate due to an error matrix that arises from the matrix multiplication with respect to the coefficient matrix A.…”