CGELSY(3)      LAPACK routine of NEC Numeric Library Collection      CGELSY(3)



NAME
       CGELSY

SYNOPSIS
       SUBROUTINE CGELSY (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
           LWORK, RWORK, INFO)



PURPOSE
            CGELSY 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.




ARGUMENTS
           M         (input)
                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           NRHS      (input)
                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of
                     columns of matrices B and X. NRHS >= 0.

           A         (input/output)
                     A is COMPLEX 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)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           B         (input/output)
                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the M-by-NRHS right hand side matrix B.
                     On exit, the N-by-NRHS solution matrix X.

           LDB       (input)
                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,M,N).

           JPVT      (input/output)
                     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.

           RCOND     (input)
                     RCOND is REAL
                     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)
                     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.

           WORK      (output)
                     WORK is COMPLEX array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     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 CGEQP3, CTZRZF, CTZRQF, CUNMQR,
                     and CUNMRZ.

                     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     (output)
                     RWORK is REAL array, dimension (2*N)

           INFO      (output)
                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value



LAPACK routine                  31 October 2017                      CGELSY(3)