A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
On growth in Gaussian elimination with complete pivoting
SIAM Journal on Matrix Analysis and Applications
LAPACK's user's guide
SIAM Journal on Matrix Analysis and Applications
Solution of the Sylvester matrix equation AXBT + CXDT = E
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
On computing condition numbers for the nonsymmetric eigenproblem
ACM Transactions on Mathematical Software (TOMS)
A Perturbation Analysis of the Generalized Sylvester Equation $(AR - LB, DR - LE) = (C, F)$
SIAM Journal on Matrix Analysis and Applications
ACM Transactions on Mathematical Software (TOMS)
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
LAPACK Working Note 75: LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation Between Regular Matrix Pairs
LAPACK Working Note 87: Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair and Condition Estimation: Theory, Algorithms and Software
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Parallel Triangular Sylvester-Type Matrix Equation Solvers for SMP Systems Using Recursive Blocking
PARA '00 Proceedings of the 5th International Workshop on Applied Parallel Computing, New Paradigms for HPC in Industry and Academia
Parallel Two-Sided Sylvester-Type Matrix Equation Solvers for SMP Systems Using Recursive Blocking
PARA '02 Proceedings of the 6th International Conference on Applied Parallel Computing Advanced Scientific Computing
PARA '02 Proceedings of the 6th International Conference on Applied Parallel Computing Advanced Scientific Computing
Block algorithms for reordering standard and generalized Schur forms
ACM Transactions on Mathematical Software (TOMS)
Parallel Algorithms for Triangular Periodic Sylvester-Type Matrix Equations
Euro-Par '08 Proceedings of the 14th international Euro-Par conference on Parallel Processing
Recursive blocked algorithms for solving periodic triangular Sylvester-type matrix equations
PARA'06 Proceedings of the 8th international conference on Applied parallel computing: state of the art in scientific computing
Automatic derivation of linear algebra algorithms with application to control theory
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
An efficient algorithm for solving general coupled matrix equations and its application
Mathematical and Computer Modelling: An International Journal
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Robust and fast software to solve the generalized Sylvester equation (AR - LB = C, DR - LE = F) for unknowns R and L is presented. This special linear system of equations, and its transpose, arises in computing error bounds for computed eigenvalues and eigenspaces of the generalized eigenvalue problem S-&lgr;T, in computing deflating subspaces of the same problem, and in computing certain decompositions of transfer matrices arising in control theory. Our contributions are twofold. First, we reorganize the standard algorithm for this problem to use Level 3 BLAS operations, like matrix multiplication, in its inner loop. This speeds up the algorithm by a factor of 9 on an IBM RS6000. Second, we develop and compare several condition estimation algorithms, which inexpensively but accurately estimate the sensitivity of the solution of this linear system.