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m Michael Hardy moved page Bartels-Stewart algorithm to Bartels–Stewart algorithm |
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1.Compute the [[Schur decomposition|real Schur decompositions]]
: <math>R = U^TAU,</math>
: <math>S = V^TB^TV.</math>
The matrices <math>R^T</math> and <math>S</math> are block-upper triangular matrices, with square blocks of size no greater than <math>2</math>.
2. Set <math>F = U^TCV.</math>
3. Solve the simplified system <math>RY - YS^T = F</math>, where <math>Y = U^TXV</math>. This can be done using forward substitution on the blocks. Specifically, if <math>s_{k-1, k} = 0</math>, then
: <math>(R - s_{kk}I)y_k = f_{k} + \sum_{j = k+1}^n s_{kj}y_j,</math>
where <math>y_k</math>is the <math>k</math>th column of <math>Y</math>. When <math>s_{k-1, k} \neq 0</math>, columns <math>[ y_{k-1} \mid y_{k}]</math> should be concatenated and solved for simultaneously.
4. Set <math>X = UYV^T.</math>
=== Computational cost ===
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