ZGELST solves overdetermined or underdetermined real linear systems
 involving an M-by-N matrix A, or its conjugate-transpose, using a QR
 or LQ factorization of A with compact WY representation of Q.
 It is assumed that A has full rank.
 The following options are provided:
 1. If TRANS = 'N' and m >= n:  find the least squares solution of
    an overdetermined system, i.e., solve the least squares problem
                 minimize || B - A*X ||.
 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
    an underdetermined system A * X = B.
 3. If TRANS = 'C' and m >= n:  find the minimum norm solution of
    an underdetermined system A**T * X = B.
 4. If TRANS = 'C' and m < n:  find the least squares solution of
    an overdetermined system, i.e., solve the least squares problem
                 minimize || B - A**T * X ||.
 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.

TRANS

          TRANS is CHARACTER*1
          = 'N': the linear system involves A;
          = 'C': the linear system involves A**H.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

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

NRHS

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

A

          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
            if M >= N, A is overwritten by details of its QR
                       factorization as returned by ZGEQRT;
            if M <  N, A is overwritten by details of its LQ
                       factorization as returned by ZGELQT.

LDA

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

B

          B is COMPLEX*16 array, dimension (LDB,NRHS)
          On entry, the matrix B of right hand side vectors, stored
          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
          if TRANS = 'C'.
          On exit, if INFO = 0, B is overwritten by the solution
          vectors, stored columnwise:
          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
          squares solution vectors; the residual sum of squares for the
          solution in each column is given by the sum of squares of
          modulus of elements N+1 to M in that column;
          if TRANS = 'N' and m < n, rows 1 to N of B contain the
          minimum norm solution vectors;
          if TRANS = 'C' and m >= n, rows 1 to M of B contain the
          minimum norm solution vectors;
          if TRANS = 'C' and m < n, rows 1 to M of B contain the
          least squares solution vectors; the residual sum of squares
          for the solution in each column is given by the sum of
          squares of the modulus of elements M+1 to N in that column.

LDB

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

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          LWORK >= max( 1, MN + max( MN, NRHS ) ).
          For optimal performance,
          LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
          where MN = min(M,N) and NB is the optimum block size.
          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.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO =  i, the i-th diagonal element of the
                triangular factor of A is zero, so that A does not have
                full rank; the least squares solution could not be
                computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  November 2022,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley