.TH "SRC/zgelsy.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/zgelsy.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBzgelsy\fP (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, rwork, info)" .br .RI "\fB ZGELSY solves overdetermined or underdetermined systems for GE matrices\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine zgelsy (integer m, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer, dimension( * ) jpvt, double precision rcond, integer rank, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)" .PP \fB ZGELSY solves overdetermined or underdetermined systems for GE matrices\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> ZGELSY 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 annihilated !> 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**H [ inv(T11)*Q1**H*B ] !> [ 0 ] !> where Q1 consists of the first RANK columns of Q\&. !> !> This routine is basically identical to the original xGELSX except !> three differences: !> o The permutation of matrix B (the right hand side) is faster and !> more simple\&. !> o The call to the subroutine xGEQPF has been substituted by the !> the call to the subroutine xGEQP3\&. This subroutine is a Blas-3 !> version of the QR factorization with column pivoting\&. !> o Matrix B (the right hand side) is updated with Blas-3\&. !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf !> M is INTEGER !> The number of rows of the matrix A\&. M >= 0\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The number of columns of the matrix A\&. N >= 0\&. !> .fi .PP .br \fINRHS\fP .PP .nf !> NRHS is INTEGER !> The number of right hand sides, i\&.e\&., the number of !> columns of matrices B and X\&. NRHS >= 0\&. !> .fi .PP .br \fIA\fP .PP .nf !> A is 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\&. !> .fi .PP .br \fILDA\fP .PP .nf !> LDA is INTEGER !> The leading dimension of the array A\&. LDA >= max(1,M)\&. !> .fi .PP .br \fIB\fP .PP .nf !> B is 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 = 0 or N = 0, B is not referenced\&. !> .fi .PP .br \fILDB\fP .PP .nf !> LDB is INTEGER !> The leading dimension of the array B\&. LDB >= max(1,M,N)\&. !> .fi .PP .br \fIJPVT\fP .PP .nf !> JPVT is INTEGER array, dimension (N) !> On entry, if JPVT(i) \&.ne\&. 0, the i-th column of A is permuted !> to the front of AP, otherwise column i is a free column\&. !> On exit, if JPVT(i) = k, then the i-th column of A*P !> was the k-th column of A\&. !> .fi .PP .br \fIRCOND\fP .PP .nf !> RCOND is 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\&. !> .fi .PP .br \fIRANK\fP .PP .nf !> RANK is INTEGER !> The effective rank of A, i\&.e\&., the order of the submatrix !> R11\&. This is the same as the order of the submatrix T11 !> in the complete orthogonal factorization of A\&. !> If NRHS = 0, RANK = 0 on output\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. !> .fi .PP .br \fILWORK\fP .PP .nf !> LWORK is INTEGER !> The dimension of the array WORK\&. !> The unblocked strategy requires that: !> LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) !> where MN = min(M,N)\&. !> The block algorithm requires that: !> LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) !> where NB is an upper bound on the blocksize returned !> by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, !> and ZUNMRZ\&. !> !> 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\&. !> .fi .PP .br \fIRWORK\fP .PP .nf !> RWORK is DOUBLE PRECISION array, dimension (2*N) !> .fi .PP .br \fIINFO\fP .PP .nf !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA .br E\&. Quintana-Orti, Depto\&. de Informatica, Universidad Jaime I, Spain .br G\&. Quintana-Orti, Depto\&. de Informatica, Universidad Jaime I, Spain .br .RE .PP .PP Definition at line \fB210\fP of file \fBzgelsy\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.