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### NAME

ZGELSS - compute the minimum norm solution to a complex linear least squares problem

### SYNOPSIS

SUBROUTINE ZGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, INFO ) INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND DOUBLE PRECISION RWORK( * ), S( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )

### PURPOSE

ZGELSS computes the minimum norm solution to a complex linear least squares problem: Minimize 2-norm(| b - A*x |). using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. The effective rank of A is determined by treating as zero those singu‐ lar values which are less than RCOND times the largest singular value.

### ARGUMENTS

M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The number of columns of the matrix A. N >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. A (input/output) COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) On entry, the M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,M,N). S (output) DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A in decreasing order. The condition S(1)/S(min(m,n)). RCOND (input) DOUBLE PRECISION RCOND is used to determine the effective rank of A. Singular RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead. RANK (output) INTEGER The effective rank of A, i.e., the number of singular values RCOND*S(1). WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 1, and also: LWORK >= 2*min(M,N) + max(M,N,NRHS) For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) DOUBLE PRECISION array, dimension (5*min(M,N)) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiag‐ onal form did not converge to zero.