Previous: zgelss Up: ../lapack-z.html Next: zgeql2

NAME ZGELSX - compute the minimum-norm solution to a complex linear least squares problem SYNOPSIS SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, RWORK, INFO ) INTEGER INFO, LDA, LDB, M, N, NRHS, RANK DOUBLE PRECISION RCOND INTEGER JPVT( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) PURPOSE ZGELSX computes the minimum-norm solution to a complex linear least squares problem: minimize || A * X - B || using a complete orthogonal factorization 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 routine first computes a QR factorization with column pivoting: A * P = Q * [ R11 R12 ] [ 0 R22 ] with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/RCOND. The order of R11, RANK, is the effective rank of A. Then, R22 is considered to be negligible, and R12 is annihi- lated by unitary transformations from the right, arriving at the complete orthogonal factorization: A * P = Q * [ T11 0 ] * Z [ 0 0 ] The minimum-norm solution is then X = P * Z' [ inv(T11)*Q1'*B ] [ 0 ] where Q1 consists of the first RANK columns of Q. 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 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, A has been overwritten by details of its complete orthogonal factorization. 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, 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). JPVT (input/output) INTEGER array, dimension (N) On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial column, otherwise it is a free column. Before the QR factorization of A, all initial columns are permuted to the leading positions; only the remaining free columns are moved as a result of column pivoting during the factorization. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A. RCOND (input) DOUBLE PRECISION RCOND is used to determine the effective rank of A, which is defined as the order of the largest leading triangular submatrix R11 in the QR factorization with pivoting of A, whose estimated condition number < 1/RCOND. RANK (output) INTEGER The effective rank of A, i.e., the order of the sub- matrix R11. This is the same as the order of the submatrix T11 in the complete orthogonal factoriza- tion of A. WORK (workspace) COMPLEX*16 array, dimension (min(M,N) + max( N, 2*min(M,N)+NRHS )), RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value