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IMSL_SVDCOMP

# IMSL_SVDCOMP

The IMSL_SVDCOMP function computes the singular value decomposition (SVD), A = USVT, of a real or complex rectangular matrix A. An estimate of the rank of A also can be computed.

The IMSL_SVDCOMP function computes the singular value decomposition of a real or complex matrix A. It reduces the matrix A to a bidiagonal matrix B by pre- and post-multiplying Householder transformations, then, it computes singular value decomposition of B using the implicit-shifted QR algorithm. An estimate of the rank of the matrix A is obtained by finding the smallest integer k such that:

sk,k â¤ TOL_RANK or sk,k â¤ TOL_RANK * ||A||infinity

Since si + 1, i + 1 â¤ s i,i , it follows that all the s i,i satisfy the same inequality for i = k, ..., min(m, n) â 2. The rank is set to the value k. If A = USVT, its generalized inverse is A+ = VS+UT. Here, S+ = diag (sâ1 0,0,..., sâ1 i,i, 0, ..., 0). Only singular values that are not negligible are reciprocated. If the keyword INVERSE is specified, the function first computes the singular value decomposition of the matrix A, then computes the generalized inverse. The IMSL_SVDCOMP function fails if the QR algorithm does not converge after 30 iterations.

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