.TH "zgelsy.f" 3 "Tue Nov 14 2017" "Version 3.8.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
zgelsy.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBzgelsy\fP (M, \fBN\fP, \fBNRHS\fP, A, \fBLDA\fP, B, \fBLDB\fP, 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.
.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.
.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
\fBDate:\fP
.RS 4
December 2016 
.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 212 of file zgelsy\&.f\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.
