SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
+ SWORK, ITER, INFO)
*
* -- LAPACK PROTOTYPE driver routine (version 3.1.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* February 2007
*
* ..
* .. WARNING: PROTOTYPE ..
* This is an LAPACK PROTOTYPE routine which means that the
* interface of this routine is likely to be changed in the future
* based on community feedback.
*
* ..
* .. Scalar Arguments ..
INTEGER INFO,ITER,LDA,LDB,LDX,N,NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV(*)
REAL SWORK(*)
DOUBLE PRECISION A(LDA,*),B(LDB,*),WORK(N,*),X(LDX,*)
* ..
*
* 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.
*
* =========
*
* .. Parameters ..
DOUBLE PRECISION NEGONE,ONE
PARAMETER (NEGONE=-1.0D+0,ONE=1.0D+0)
*
* .. Local Scalars ..
LOGICAL DOITREF
INTEGER I,IITER,ITERMAX,OK,PTSA,PTSX
DOUBLE PRECISION ANRM,BWDMAX,CTE,EPS,RNRM,XNRM
*
* .. External Subroutines ..
EXTERNAL DAXPY,DGEMM,DLACPY,DLAG2S,SLAG2D,
+ SGETRF,SGETRS,XERBLA
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH,DLANGE
EXTERNAL IDAMAX,DLAMCH,DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS,DBLE,MAX,SQRT
* ..
* .. Executable Statements ..
*
ITERMAX = 30
BWDMAX = 1.0E+00
DOITREF = .TRUE.
*
OK = 0
INFO = 0
ITER = 0
*
* Test the input parameters.
*
IF (N.LT.0) THEN
INFO = -1
ELSE IF (NRHS.LT.0) THEN
INFO = -2
ELSE IF (LDA.LT.MAX(1,N)) THEN
INFO = -4
ELSE IF (LDB.LT.MAX(1,N)) THEN
INFO = -7
ELSE IF (LDX.LT.MAX(1,N)) THEN
INFO = -9
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DSGESV',-INFO)
RETURN
END IF
*
* Quick return if (N.EQ.0).
*
IF (N.EQ.0) RETURN
*
* Skip single precision iterative refinement if a priori slower
* than double precision factorization.
*
IF (.NOT.DOITREF) THEN
ITER = -1
GO TO 40
END IF
*
* Compute some constants.
*
ANRM = DLANGE('I',N,N,A,LDA,WORK)
EPS = DLAMCH('Epsilon')
CTE = ANRM*EPS*SQRT(DBLE(N))*BWDMAX
*
* Set the pointers PTSA, PTSX for referencing SA and SX in SWORK.
*
PTSA = 1
PTSX = PTSA + N*N
*
* Convert B from double precision to single precision and store the
* result in SX.
*
CALL DLAG2S(N,NRHS,B,LDB,SWORK(PTSX),N,INFO)
*
IF (INFO.NE.0) THEN
ITER = -2
GO TO 40
END IF
*
* Convert A from double precision to single precision and store the
* result in SA.
*
CALL DLAG2S(N,N,A,LDA,SWORK(PTSA),N,INFO)
*
IF (INFO.NE.0) THEN
ITER = -2
GO TO 40
END IF
*
* Compute the LU factorization of SA.
*
CALL SGETRF(N,N,SWORK(PTSA),N,IPIV,INFO)
*
IF (INFO.NE.0) THEN
ITER = -3
GO TO 40
END IF
*
* Solve the system SA*SX = SB.
*
CALL SGETRS('No transpose',N,NRHS,SWORK(PTSA),N,IPIV,
+ SWORK(PTSX),N,INFO)
*
* Convert SX back to double precision
*
CALL SLAG2D(N,NRHS,SWORK(PTSX),N,X,LDX,INFO)
*
* Compute R = B - AX (R is WORK).
*
CALL DLACPY('All',N,NRHS,B,LDB,WORK,N)
*
CALL DGEMM('No Transpose','No Transpose',N,NRHS,N,NEGONE,A,LDA,X,
+ LDX,ONE,WORK,N)
*
* Check whether the NRHS normwised backward errors satisfy the
* stopping criterion. If yes, set ITER=0 and return.
*
DO I = 1,NRHS
XNRM = ABS(X(IDAMAX(N,X(1,I),1),I))
RNRM = ABS(WORK(IDAMAX(N,WORK(1,I),1),I))
IF (RNRM.GT.XNRM*CTE) GOTO 10
END DO
*
* If we are here, the NRHS normwised backward errors satisfy the
* stopping criterion. We are good to exit.
*
ITER = 0
RETURN
*
10 CONTINUE
*
DO 30 IITER = 1,ITERMAX
*
* Convert R (in WORK) from double precision to single precision
* and store the result in SX.
*
CALL DLAG2S(N,NRHS,WORK,N,SWORK(PTSX),N,INFO)
*
IF (INFO.NE.0) THEN
ITER = -2
GO TO 40
END IF
*
* Solve the system SA*SX = SR.
*
CALL SGETRS('No transpose',N,NRHS,SWORK(PTSA),N,IPIV,
+ SWORK(PTSX),N,INFO)
*
* Convert SX back to double precision and update the current
* iterate.
*
CALL SLAG2D(N,NRHS,SWORK(PTSX),N,WORK,N,INFO)
*
CALL DAXPY(N*NRHS,ONE,WORK,1,X,1)
*
* Compute R = B - AX (R is WORK).
*
CALL DLACPY('All',N,NRHS,B,LDB,WORK,N)
*
CALL DGEMM('No Transpose','No Transpose',N,NRHS,N,NEGONE,A,
+ LDA,X,LDX,ONE,WORK,N)
*
* Check whether the NRHS normwised backward errors satisfy the
* stopping criterion. If yes, set ITER=IITER>0 and return.
*
DO I = 1,NRHS
XNRM = ABS(X(IDAMAX(N,X(1,I),1),I))
RNRM = ABS(WORK(IDAMAX(N,WORK(1,I),1),I))
IF (RNRM.GT.XNRM*CTE) GOTO 20
END DO
*
* If we are here, the NRHS normwised backward errors satisfy the
* stopping criterion, we are good to exit.
*
ITER = IITER
*
RETURN
*
20 CONTINUE
*
30 CONTINUE
*
* If we are at this place of the code, this is because we have
* performed ITER=ITERMAX iterations and never satisified the stopping
* criterion, set up the ITER flag accordingly and follow up on double
* precision routine.
*
ITER = -ITERMAX - 1
*
40 CONTINUE
*
* Single-precision iterative refinement failed to converge to a
* satisfactory solution, so we resort to double precision.
*
CALL DGETRF(N,N,A,LDA,IPIV,INFO)
*
CALL DLACPY('All',N,NRHS,B,LDB,X,LDX)
*
IF (INFO.EQ.0) THEN
CALL DGETRS('No transpose',N,NRHS,A,LDA,IPIV,X,LDX,INFO)
END IF
*
RETURN
*
* End of DSGESV.
*
END