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## cgelsd()

subroutine cgelsd | ( | integer | m,
|  | integer | n,
|  | integer | nrhs,
|  | complex, dimension( lda, * ) | a,
|  | integer | lda,
|  | complex, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | real, dimension( * ) | s,
|  | real | rcond,
|  | integer | rank,
|  | complex, dimension( * ) | work,
|  | integer | lwork,
|  | real, dimension( * ) | rwork,
|  | integer, dimension( * ) | iwork,
|  | integer | info )

CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Download CGELSD + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> CGELSD 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  (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.
      !>
      !>

Parameters

  [in]

  M

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

  [in]

  N

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

  [in]

  NRHS

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

  [in,out]

  A

      !>          A is COMPLEX array, dimension (LDA,N)
      !>          On entry, the M-by-N matrix A.
      !>          On exit, A has been destroyed.
      !>

  [in]

  LDA

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

  [in,out]

  B

      !>          B is COMPLEX 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.
      !>

  [in]

  LDB

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

  [out]

  S

      !>          S is REAL 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)).
      !>

  [in]

  RCOND

      !>          RCOND is REAL
      !>          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.
      !>

  [out]

  RANK

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

  [out]

  WORK

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

  [in]

  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.
      !>

  [out]

  RWORK

      !>          RWORK is REAL 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.
      !>

  [out]

  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.
      !>

  [out]

  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.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

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