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NAME SGELSS - compute the minimum norm solution to a real linear least squares problem SYNOPSIS SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO ) INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK REAL RCOND REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) PURPOSE SGELSS computes the minimum norm solution to a real 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 singular 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) REAL 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) REAL 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,MAX(M,N)). S (output) REAL array, dimension (min(M,N)) The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)). RCOND (input) REAL RCOND is used to determine the effective rank of A. Singular values S(i) <= 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 singu- lar values which are greater than RCOND*S(1). WORK (workspace/output) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 1, and also: LWORK >= 3*N+MAX(2*N,NRHS,M) if M >= N, LWORK >= 3*M+MAX(2*M,NRHS,N) if M < N. For good perfor- mance, LWORK should generally be larger. 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 bidiagonal form did not converge to zero.