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A Hessenberg-Schur method for the problem AX + XB= C

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3 Author(s)
Golub, G. ; Stanford University, Stanford, CA, USA ; Nash, S. ; Van Loan, C.

One of the most effective methods for solving the matrix equation AX+XB=C is the Bartels-Stewart algorithm. Key to this technique is the orthogonal reduction of A and B to triangular form using the QR algorithm for eigenvalues. A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form. The resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B . The stability of the new method is demonstrated through a roundoff error analysis and supported by numerical tests. Finally, it is shown how the techniques described can be applied and generalized to other matrix equation problems.

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Automatic Control, IEEE Transactions on  (Volume:24 ,  Issue: 6 )