Data type in double but solving problem using single precision routines for general (i.e., unsymmetric, in some cases rectangular) matrix

• dsgesv : Solves a general system of linear equations AX=B using iterativement refinement.

dsgesv

USAGE:
  ipiv, x, iter, info, a = NumRu::Lapack.dsgesv( a, b)
    or
  NumRu::Lapack.dsgesv  # print help


FORTRAN MANUAL
      SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, SWORK, ITER, INFO)

*  Purpose
*  =======
*
*  DSGESV computes the solution to a real system of linear equations
*     A * X = B,
*  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*
*  DSGESV first attempts to factorize the matrix in SINGLE PRECISION 
*  and use this factorization within an iterative refinement procedure to
*  produce a solution with DOUBLE PRECISION normwise backward error
*  quality (see below). If the approach fails the method switches to a
*  DOUBLE PRECISION factorization and solve.
*
*  The iterative refinement is not going to be a winning strategy if
*  the ratio SINGLE PRECISION performance over DOUBLE PRECISION performance
*  is too small. A reasonable strategy should take the number of right-hand
*  sides and the size of the matrix into account. This might be done with a 
*  call to ILAENV in the future. Up to now, we always try iterative refinement.
*
*  The iterative refinement process is stopped if
*      ITER > ITERMAX
*  or for all the RHS we have:
*      RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX 
*  where
*      o ITER is the number of the current iteration in the iterative
*        refinement process
*      o RNRM is the infinity-norm of the residual
*      o XNRM is the infinity-norm of the solution
*      o ANRM is the infinity-operator-norm of the matrix A
*      o EPS is the machine epsilon returned by DLAMCH('Epsilon')
*  The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.
*

*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input or input/ouptut) DOUBLE PRECISION array,
*          dimension (LDA,N)
*          On entry, the N-by-N coefficient matrix A.
*          On exit, if iterative refinement has been successfully used
*          (INFO.EQ.0 and ITER.GE.0, see description below), then A is
*          unchanged, if double precision factorization has been used
*          (INFO.EQ.0 and ITER.LT.0, see description below), then the
*          array A contains the factors L and U from the factorization
*          A = P*L*U; the unit diagonal elements of L are not stored.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          The pivot indices that define the permutation matrix P;
*          row i of the matrix was interchanged with row IPIV(i).
*          Corresponds either to the single precision factorization 
*          (if INFO.EQ.0 and ITER.GE.0) or the double precision 
*          factorization (if INFO.EQ.0 and ITER.LT.0).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          The N-by-NRHS matrix of right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
*          If INFO = 0, the N-by-NRHS solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N*NRHS)
*          This array is used to hold the residual vectors.
*
*  SWORK   (workspace) REAL array, dimension (N*(N+NRHS))
*          This array is used to use the single precision matrix and the 
*          right-hand sides or solutions in single precision.
*
*  ITER    (output) INTEGER
*          < 0: iterative refinement has failed, double precision
*               factorization has been performed
*               -1 : taking into account machine parameters, N, NRHS, it
*                    is a priori not worth working in SINGLE PRECISION
*               -2 : overflow of an entry when moving from double to
*                    SINGLE PRECISION
*               -3 : failure of SGETRF
*               -31: stop the iterative refinement after the 30th
*                    iterations
*          > 0: iterative refinement has been sucessfully used.
*               Returns the number of iterations
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, U(i,i) computed in DOUBLE PRECISION is
*                exactly zero.  The factorization has been completed,
*                but the factor U is exactly singular, so the solution
*                could not be computed.
*
*  =========
*