Computing tall skinny solutions of AX - XB = C
Mathematics and Computers in Simulation - MODELLING 2001 - Second IMACS conference on mathematical modelling and computational methods in mechanics, physics, biomechanics and geodynamics
Updating the QR decomposition of block tridiagonal and block Hessenberg matrices
Applied Numerical Mathematics
Bi-CGSTAB as an induced dimension reduction method
Applied Numerical Mathematics
Interpreting IDR as a Petrov-Galerkin Method
SIAM Journal on Scientific Computing
A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
A new family of global methods for linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
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Block BICG (BBICG) is an appealing method for solving $AX=B$ with $A\in{\Bbb R}^{n\times n}$ and $X, B \in {\Bbb R}^{n\times s}$. Because of its short-term recurrence form, memory allocation and computational cost do not depend on additional parameters. Unfortunately, loss of orthogonality prevents convergence in many cases.We present a new version of the algorithm that generates blocks of vectors that are vector-wise $A$-biorthogonal; moreover, a near-breakdown safeguard strategy inside the block stabilizes the computation of the coefficients. In order to smooth the possibly erratic behavior of the residual norm curve, the approximate solution is determined using a block QMR procedure. The new method considerably improves the robustness of the original algorithm, showing very good performance on dense or preconditioned matrices over both BBICG and the single right-hand side solver coupled two-term QMR method applied on each system.