dsgesv.f3

!>
!> 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.
!> 

N

!>          N is INTEGER
!>          The number of linear equations, i.e., the order of the
!>          matrix A.  N >= 0.
!> 

NRHS

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

A

!>          A is 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 = 0 and ITER >= 0, see description below), then A is
!>          unchanged, if double precision factorization has been used
!>          (INFO = 0 and ITER < 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

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

IPIV

!>          IPIV is 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 = 0 and ITER >= 0) or the double precision
!>          factorization (if INFO = 0 and ITER < 0).
!> 

B

!>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
!>          The N-by-NRHS right hand side matrix B.
!> 

LDB

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

X

!>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
!>          If INFO = 0, the N-by-NRHS solution matrix X.
!> 

LDX

!>          LDX is INTEGER
!>          The leading dimension of the array X.  LDX >= max(1,N).
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (N,NRHS)
!>          This array is used to hold the residual vectors.
!> 

SWORK

!>          SWORK is 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

!>          ITER is INTEGER
!>          < 0: iterative refinement has failed, double precision
!>               factorization has been performed
!>               -1 : the routine fell back to full precision for
!>                    implementation- or machine-specific reasons
!>               -2 : narrowing the precision induced an overflow,
!>                    the routine fell back to full precision
!>               -3 : failure of SGETRF
!>               -31: stop the iterative refinement after the 30th
!>                    iterations
!>          > 0: iterative refinement has been successfully used.
!>               Returns the number of iterations
!> 

INFO

!>          INFO is 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.
!> 

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