Rayleigh–Ritz method: Difference between revisions

Revision as of 21:33, 23 March 2022

Revision as of 22:19, 23 March 2022

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=== Using the normal matrix ===

The definition of the singular value <math>\sigma</math> and the corresponding left and right singular vectors is <math>M v=\sigma u</math> and <math>M^* u=\sigma v</math>. Having found one set (left of right) of singular vectors by applying naively the Rayleigh–Ritz method to the [[Hermitian matrix|Hermitian]] '''normal matrix''' <math> A = M^* M \in \mathbb{C}^{N \times N}</math> of sizeor <math>N</math>-by-<math>N M M^* \in \mathbb{C}^{M \times M}</math>, allowswhichever directlyone applyingis thesmaller Rayleigh–Ritzsize, methodone takingcould advantage ofdetermine the factother thatset forof aleft givenof <math>N</math>-by-<math>m</math>right singular matrixvectors <math>simply Wby \individing \mathbb{C}^{Nby \timesthe m}singular </math>values. withHowever, [[orthonormal]]the columnsdivision theis eigenvalueunstable problemor fails for thesmall <math>m</math>-by-<math>m</math>or matrixzero singular values.

An alternative approach, e.g., defining the normal matrix as <math> A = M^* M \in \mathbb{C}^{N \times N}</math> of size <math>N</math>-by-<math>N</math>, takes advantage of the fact that for a given <math>N</math>-by-<math>m</math> matrix <math> W \in \mathbb{C}^{N \times m} </math> with [[orthonormal]] columns the eigenvalue problem of the Rayleigh–Ritz method for the <math>m</math>-by-<math>m</math> matrix

:<math> W^* A W = W^* M^* M W = (M W)^* M W</math>

can be interpreted as a singular value problem for the <math>N</math>-by-<math>m</math> matrix <math>M W</math>. This interpretation allows simple calculation of both left and right approximate singular vectors as follows.

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# Compute the matrices of the Ritz left <math>U = \mathbf {U}</math> and right <math>V_h = \mathbf {V}_h W^*</math> singular vectors

# Output approximations <math>U, \Sigma, V_h</math>, called the Ritz singular triplets, to selected singular values and the corresponding left and right singular vectors of the original matrix <math>M</math> representing an approximate [[Singular_value_decomposition#Truncated_SVD | Truncated singular value decomposition (SVD)]]

==== Example ====