Solution of the matrix equation AX + XB = C [F4]

  title={Solution of the matrix equation AX + XB = C [F4]},
  author={Richard H. Bartels and G. W. Stewart},
  journal={Commun. ACM},
Method and apparatus for welding together the internal diameters of two bellows members wherein the apparatus has a frame means carrying an indexible table having a fixture thereon, the fixture being adapted to receive and hold the bellows members with the internal diameters thereof in a position for being welded together. [] Key Method The table is indexed to cause the fixture to move from a loading station thereof to a welding station thereof adjacent welding means carried by the frame means. Automatic…
Computing tall skinny solutions of AX-XB=C
LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs
This work reorganizes the standard algorithm for this problem to use Level 3 BLAS operations, like matrix multiplication, in its inner loop, and develops and compares several condition estimation algorithms, which inexpensively but accurately estimate the sensitivity of the solution of this linear system.
Accurate solutions of M-matrix Sylvester equations
It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entry of the solution X, so the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations.
Algorithm 894: On a block Schur--Parlett algorithm for ϕ-functions based on the sep-inverse estimate
Modifications to the block Schur--Parlett algorithm are described together with the results of numerical experiments on the matrix values of ϕ-functions required in exponential integrators of FORTRAN 95.
Low-Rank Solution of
It is shown by numerical examples that the rational Krylov subspace generated by the CF-ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.
Perturbation theory and backward error forAX−XB=C
An “LAPACK-style” bound is shown to be efficiently computable and potentially much smaller than a sep-based bound, and a Fortran 77 code has been written that solves the Sylvester equation and computes this bound, making use of LAPACK routines.
A Hessenberg-Schur method for the problem AX + XB= C
A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form, and the resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B.
Parallel Algorithms for Certain Matrix Computations


The Direct Solution of the Discrete Poisson Equation on a Rectangle
where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue.
Matrix and other direct methods for the solution of systems of linear difference equations
  • W. G. Bickley, J. McNamee
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1960
The investigations described in this paper were initiated in an attempt to replace by direct methods the successive approximation methods such as those of Southwell and Thom for the solution of
TheQ R algorithm for real hessenberg matrices
The volume of work involved in a QR step is far less if the matrix is of Hessenberg form, and since there are several stable ways of reducing a general matrix to this form, the QR algorithm is invariably used after such a reduction.
The algebraic eigenvalue problem
Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of