Divide-and-conquer eigenvalue algorithm: Difference between revisions

Revision as of 20:02, 20 April 2005

Revision as of 03:30, 19 June 2005

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The only difference between <math>T_{1}</math> and <math>\hat{T}_{1}</math> is that the lower right entry <math>t_{nn}</math> in <math>\hat{T}_{1}</math> has been replaced with <math>t_{nn} - \beta</math> and similarly, in <math>\hat{t}_{2}</math> the top left entry <math>t_{n+1,n+1}</math> has been replaced with <math>t_{n+1,n+1} - \beta</math>.

The remainederremainder of the divide step is to solve for the eigenvalues (and if desired the eigenvectors) of <math>\hat{T}_{1}</math> and <math>\hat{T}_{2}</math>, that is to find the [[diagonalization]]s <math>\hat{T}_{1} = Q_{1} D_{1} Q_{1}^{T}</math> and <math>\hat{T}_{2} = Q_{2} D_{2} Q_{2}^{T}</math>. This can be accomplished with recursive calls to the divide-and-conquer algorithm, although practical implemntationsimplementations often switch to the QR algorithm for small enough submatrices.

==Conquer==