SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, \$ SWORK, ITER, INFO ) * * -- LAPACK PROTOTYPE driver routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. * .. 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/output) 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 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 : 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 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 .. LOGICAL DOITREF PARAMETER ( DOITREF = .TRUE. ) * INTEGER ITERMAX PARAMETER ( ITERMAX = 30 ) * DOUBLE PRECISION BWDMAX PARAMETER ( BWDMAX = 1.0E+00 ) * DOUBLE PRECISION NEGONE, ONE PARAMETER ( NEGONE = -1.0D+0, ONE = 1.0D+0 ) * * .. Local Scalars .. INTEGER I, IITER, PTSA, PTSX DOUBLE PRECISION ANRM, 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 .. * 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 indices 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 normwise 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 ) \$ GO TO 10 END DO * * If we are here, the NRHS normwise 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 ) * DO I = 1, NRHS CALL DAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 ) END DO * * 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 normwise 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 ) \$ GO TO 20 END DO * * If we are here, the NRHS normwise 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 ) * IF( INFO.NE.0 ) \$ RETURN * CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX ) CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, X, LDX, \$ INFO ) * RETURN * * End of DSGESV. * END