ZGELSD computes the minimum-norm solution to a real linear least
 squares problem:
     minimize 2-norm(| b - A*x |)
 using the singular value decomposition (SVD) 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 problem is solved in three steps:
 (1) Reduce the coefficient matrix A to bidiagonal form with
     Householder transformations, reducing the original problem
     into a 'bidiagonal least squares problem' (BLS)
 (2) Solve the BLS using a divide and conquer approach.
 (3) Apply back all the Householder transformations to solve
     the original least squares problem.
 The effective rank of A is determined by treating as zero those
 singular values which are less than RCOND times the largest singular
 value.

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, A has been destroyed.

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 M-by-NRHS right hand side matrix B.
          On exit, B is overwritten by 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 the modulus of elements n+1:m in that column.

LDB

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

S

          S is DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A in decreasing order.
          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

          RCOND is DOUBLE PRECISION
          RCOND is used to determine the effective rank of A.
          Singular values S(i) <= RCOND*S(1) are treated as zero.
          If RCOND < 0, machine precision is used instead.

RANK

          RANK is INTEGER
          The effective rank of A, i.e., the number of singular values
          which are greater than RCOND*S(1).

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 must be at least 1.
          The exact minimum amount of workspace needed depends on M,
          N and NRHS. As long as LWORK is at least
              2*N + N*NRHS
          if M is greater than or equal to N or
              2*M + M*NRHS
          if M is less than N, the code will execute correctly.
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the array WORK and the
          minimum sizes of the arrays RWORK and IWORK, and returns
          these values as the first entries of the WORK, RWORK and
          IWORK arrays, and no error message related to LWORK is issued
          by XERBLA.

RWORK

          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
          LRWORK >=
             10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
          if M is greater than or equal to N or
             10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
          if M is less than N, the code will execute correctly.
          SMLSIZ is returned by ILAENV and is equal to the maximum
          size of the subproblems at the bottom of the computation
          tree (usually about 25), and
             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
          On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
          where MINMN = MIN( M,N ).
          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0:  the algorithm for computing the SVD failed to converge;
                if INFO = i, i off-diagonal elements of an intermediate
                bidiagonal form did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA