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.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

          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.EQ.0 and ITER.GE.0) or the double precision


          factorization (if INFO.EQ.0 and ITER.LT.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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016