dgesvx.f

          SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
         $                   EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
         $                   WORK, IWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          EQUED, FACT, TRANS
          INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
          DOUBLE PRECISION   RCOND
    *     ..
    *     .. Array Arguments ..
          INTEGER            IPIV( * ), IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
         $                   BERR( * ), C( * ), FERR( * ), R( * ),
         $                   WORK( * ), X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DGESVX uses the LU factorization to compute 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.
    *
    *  Error bounds on the solution and a condition estimate are also
    *  provided.
    *
    *  Description
    *  ===========
    *
    *  The following steps are performed:
    *
    *  1. If FACT = 'E', real scaling factors are computed to equilibrate
    *     the system:
    *        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
    *        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
    *        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
    *     Whether or not the system will be equilibrated depends on the
    *     scaling of the matrix A, but if equilibration is used, A is
    *     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
    *     or diag(C)*B (if TRANS = 'T' or 'C').
    *
    *  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
    *     matrix A (after equilibration if FACT = 'E') as
    *        A = P * L * U,
    *     where P is a permutation matrix, L is a unit lower triangular
    *     matrix, and U is upper triangular.
    *
    *  3. If some U(i,i)=0, so that U is exactly singular, then the routine
    *     returns with INFO = i. Otherwise, the factored form of A is used
    *     to estimate the condition number of the matrix A.  If the
    *     reciprocal of the condition number is less than machine precision,
    *     INFO = N+1 is returned as a warning, but the routine still goes on
    *     to solve for X and compute error bounds as described below.
    *
    *  4. The system of equations is solved for X using the factored form
    *     of A.
    *
    *  5. Iterative refinement is applied to improve the computed solution
    *     matrix and calculate error bounds and backward error estimates
    *     for it.
    *
    *  6. If equilibration was used, the matrix X is premultiplied by
    *     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
    *     that it solves the original system before equilibration.
    *
    *  Arguments
    *  =========
    *
    *  FACT    (input) CHARACTER*1
    *          Specifies whether or not the factored form of the matrix A is
    *          supplied on entry, and if not, whether the matrix A should be
    *          equilibrated before it is factored.
    *          = 'F':  On entry, AF and IPIV contain the factored form of A.
    *                  If EQUED is not 'N', the matrix A has been
    *                  equilibrated with scaling factors given by R and C.
    *                  A, AF, and IPIV are not modified.
    *          = 'N':  The matrix A will be copied to AF and factored.
    *          = 'E':  The matrix A will be equilibrated if necessary, then
    *                  copied to AF and factored.
    *
    *  TRANS   (input) CHARACTER*1
    *          Specifies the form of the system of equations:
    *          = 'N':  A * X = B     (No transpose)
    *          = 'T':  A**T * X = B  (Transpose)
    *          = 'C':  A**H * X = B  (Transpose)
    *
    *  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 matrices B and X.  NRHS >= 0.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
    *          not 'N', then A must have been equilibrated by the scaling
    *          factors in R and/or C.  A is not modified if FACT = 'F' or
    *          'N', or if FACT = 'E' and EQUED = 'N' on exit.
    *
    *          On exit, if EQUED .ne. 'N', A is scaled as follows:
    *          EQUED = 'R':  A := diag(R) * A
    *          EQUED = 'C':  A := A * diag(C)
    *          EQUED = 'B':  A := diag(R) * A * diag(C).
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
    *          If FACT = 'F', then AF is an input argument and on entry
    *          contains the factors L and U from the factorization
    *          A = P*L*U as computed by DGETRF.  If EQUED .ne. 'N', then
    *          AF is the factored form of the equilibrated matrix A.
    *
    *          If FACT = 'N', then AF is an output argument and on exit
    *          returns the factors L and U from the factorization A = P*L*U
    *          of the original matrix A.
    *
    *          If FACT = 'E', then AF is an output argument and on exit
    *          returns the factors L and U from the factorization A = P*L*U
    *          of the equilibrated matrix A (see the description of A for
    *          the form of the equilibrated matrix).
    *
    *  LDAF    (input) INTEGER
    *          The leading dimension of the array AF.  LDAF >= max(1,N).
    *
    *  IPIV    (input or output) INTEGER array, dimension (N)
    *          If FACT = 'F', then IPIV is an input argument and on entry
    *          contains the pivot indices from the factorization A = P*L*U
    *          as computed by DGETRF; row i of the matrix was interchanged
    *          with row IPIV(i).
    *
    *          If FACT = 'N', then IPIV is an output argument and on exit
    *          contains the pivot indices from the factorization A = P*L*U
    *          of the original matrix A.
    *
    *          If FACT = 'E', then IPIV is an output argument and on exit
    *          contains the pivot indices from the factorization A = P*L*U
    *          of the equilibrated matrix A.
    *
    *  EQUED   (input or output) CHARACTER*1
    *          Specifies the form of equilibration that was done.
    *          = 'N':  No equilibration (always true if FACT = 'N').
    *          = 'R':  Row equilibration, i.e., A has been premultiplied by
    *                  diag(R).
    *          = 'C':  Column equilibration, i.e., A has been postmultiplied
    *                  by diag(C).
    *          = 'B':  Both row and column equilibration, i.e., A has been
    *                  replaced by diag(R) * A * diag(C).
    *          EQUED is an input argument if FACT = 'F'; otherwise, it is an
    *          output argument.
    *
    *  R       (input or output) DOUBLE PRECISION array, dimension (N)
    *          The row scale factors for A.  If EQUED = 'R' or 'B', A is
    *          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
    *          is not accessed.  R is an input argument if FACT = 'F';
    *          otherwise, R is an output argument.  If FACT = 'F' and
    *          EQUED = 'R' or 'B', each element of R must be positive.
    *
    *  C       (input or output) DOUBLE PRECISION array, dimension (N)
    *          The column scale factors for A.  If EQUED = 'C' or 'B', A is
    *          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
    *          is not accessed.  C is an input argument if FACT = 'F';
    *          otherwise, C is an output argument.  If FACT = 'F' and
    *          EQUED = 'C' or 'B', each element of C must be positive.
    *
    *  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
    *          On entry, the N-by-NRHS right hand side matrix B.
    *          On exit,
    *          if EQUED = 'N', B is not modified;
    *          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
    *          diag(R)*B;
    *          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
    *          overwritten by diag(C)*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 or INFO = N+1, the N-by-NRHS solution matrix X
    *          to the original system of equations.  Note that A and B are
    *          modified on exit if EQUED .ne. 'N', and the solution to the
    *          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
    *          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
    *          and EQUED = 'R' or 'B'.
    *
    *  LDX     (input) INTEGER
    *          The leading dimension of the array X.  LDX >= max(1,N).
    *
    *  RCOND   (output) DOUBLE PRECISION
    *          The estimate of the reciprocal condition number of the matrix
    *          A after equilibration (if done).  If RCOND is less than the
    *          machine precision (in particular, if RCOND = 0), the matrix
    *          is singular to working precision.  This condition is
    *          indicated by a return code of INFO > 0.
    *
    *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
    *          The estimated forward error bound for each solution vector
    *          X(j) (the j-th column of the solution matrix X).
    *          If XTRUE is the true solution corresponding to X(j), FERR(j)
    *          is an estimated upper bound for the magnitude of the largest
    *          element in (X(j) - XTRUE) divided by the magnitude of the
    *          largest element in X(j).  The estimate is as reliable as
    *          the estimate for RCOND, and is almost always a slight
    *          overestimate of the true error.
    *
    *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
    *          The componentwise relative backward error of each solution
    *          vector X(j) (i.e., the smallest relative change in
    *          any element of A or B that makes X(j) an exact solution).
    *
    *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (4*N)
    *          On exit, WORK(1) contains the reciprocal pivot growth
    *          factor norm(A)/norm(U). The "max absolute element" norm is
    *          used. If WORK(1) is much less than 1, then the stability
    *          of the LU factorization of the (equilibrated) matrix A
    *          could be poor. This also means that the solution X, condition
    *          estimator RCOND, and forward error bound FERR could be
    *          unreliable. If factorization fails with 0<INFO<=N, then
    *          WORK(1) contains the reciprocal pivot growth factor for the
    *          leading INFO columns of A.
    *
    *  IWORK   (workspace) INTEGER array, dimension (N)
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *          > 0:  if INFO = i, and i is
    *                <= N:  U(i,i) is exactly zero.  The factorization has
    *                       been completed, but the factor U is exactly
    *                       singular, so the solution and error bounds
    *                       could not be computed. RCOND = 0 is returned.
    *                = N+1: U is nonsingular, but RCOND is less than machine
    *                       precision, meaning that the matrix is singular
    *                       to working precision.  Nevertheless, the
    *                       solution and error bounds are computed because
    *                       there are a number of situations where the
    *                       computed solution can be more accurate than the
    *                       value of RCOND would suggest.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
          CHARACTER          NORM
          INTEGER            I, INFEQU, J
          DOUBLE PRECISION   AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
         $                   ROWCND, RPVGRW, SMLNUM
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, DLANGE, DLANTR
          EXTERNAL           LSAME, DLAMCH, DLANGE, DLANTR
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DGECON, DGEEQU, DGERFS, DGETRF, DGETRS, DLACPY,
         $                   DLAQGE, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
          INFO = 0
          NOFACT = LSAME( FACT, 'N' )
          EQUIL = LSAME( FACT, 'E' )
          NOTRAN = LSAME( TRANS, 'N' )
          IF( NOFACT .OR. EQUIL ) THEN
             EQUED = 'N'
             ROWEQU = .FALSE.
             COLEQU = .FALSE.
          ELSE
             ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
             COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
             SMLNUM = DLAMCH( 'Safe minimum' )
             BIGNUM = ONE / SMLNUM
          END IF
    *
    *     Test the input parameters.
    *
          IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
         $     THEN
             INFO = -1
          ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
         $         LSAME( TRANS, 'C' ) ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( NRHS.LT.0 ) THEN
             INFO = -4
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -6
          ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
             INFO = -8
          ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
         $         ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
             INFO = -10
          ELSE
             IF( ROWEQU ) THEN
                RCMIN = BIGNUM
                RCMAX = ZERO
                DO 10 J = 1, N
                   RCMIN = MIN( RCMIN, R( J ) )
                   RCMAX = MAX( RCMAX, R( J ) )
       10       CONTINUE
                IF( RCMIN.LE.ZERO ) THEN
                   INFO = -11
                ELSE IF( N.GT.0 ) THEN
                   ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
                ELSE
                   ROWCND = ONE
                END IF
             END IF
             IF( COLEQU .AND. INFO.EQ.0 ) THEN
                RCMIN = BIGNUM
                RCMAX = ZERO
                DO 20 J = 1, N
                   RCMIN = MIN( RCMIN, C( J ) )
                   RCMAX = MAX( RCMAX, C( J ) )
       20       CONTINUE
                IF( RCMIN.LE.ZERO ) THEN
                   INFO = -12
                ELSE IF( N.GT.0 ) THEN
                   COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
                ELSE
                   COLCND = ONE
                END IF
             END IF
             IF( INFO.EQ.0 ) THEN
                IF( LDB.LT.MAX( 1, N ) ) THEN
                   INFO = -14
                ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
                   INFO = -16
                END IF
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DGESVX', -INFO )
             RETURN
          END IF
    *
          IF( EQUIL ) THEN
    *
    *        Compute row and column scalings to equilibrate the matrix A.
    *
             CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
             IF( INFEQU.EQ.0 ) THEN
    *
    *           Equilibrate the matrix.
    *
                CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
         $                   EQUED )
                ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
                COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
             END IF
          END IF
    *
    *     Scale the right hand side.
    *
          IF( NOTRAN ) THEN
             IF( ROWEQU ) THEN
                DO 40 J = 1, NRHS
                   DO 30 I = 1, N
                      B( I, J ) = R( I )*B( I, J )
       30          CONTINUE
       40       CONTINUE
             END IF
          ELSE IF( COLEQU ) THEN
             DO 60 J = 1, NRHS
                DO 50 I = 1, N
                   B( I, J ) = C( I )*B( I, J )
       50       CONTINUE
       60    CONTINUE
          END IF
    *
          IF( NOFACT .OR. EQUIL ) THEN
    *
    *        Compute the LU factorization of A.
    *
             CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
             CALL DGETRF( N, N, AF, LDAF, IPIV, INFO )
    *
    *        Return if INFO is non-zero.
    *
             IF( INFO.GT.0 ) THEN
    *
    *           Compute the reciprocal pivot growth factor of the
    *           leading rank-deficient INFO columns of A.
    *
                RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
         $               WORK )
                IF( RPVGRW.EQ.ZERO ) THEN
                   RPVGRW = ONE
                ELSE
                   RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
                END IF
                WORK( 1 ) = RPVGRW
                RCOND = ZERO
                RETURN
             END IF
          END IF
    *
    *     Compute the norm of the matrix A and the
    *     reciprocal pivot growth factor RPVGRW.
    *
          IF( NOTRAN ) THEN
             NORM = '1'
          ELSE
             NORM = 'I'
          END IF
          ANORM = DLANGE( NORM, N, N, A, LDA, WORK )
          RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
          IF( RPVGRW.EQ.ZERO ) THEN
             RPVGRW = ONE
          ELSE
             RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
          END IF
    *
    *     Compute the reciprocal of the condition number of A.
    *
          CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
    *
    *     Compute the solution matrix X.
    *
          CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
          CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
    *
    *     Use iterative refinement to improve the computed solution and
    *     compute error bounds and backward error estimates for it.
    *
          CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
         $             LDX, FERR, BERR, WORK, IWORK, INFO )
    *
    *     Transform the solution matrix X to a solution of the original
    *     system.
    *
          IF( NOTRAN ) THEN
             IF( COLEQU ) THEN
                DO 80 J = 1, NRHS
                   DO 70 I = 1, N
                      X( I, J ) = C( I )*X( I, J )
       70          CONTINUE
       80       CONTINUE
                DO 90 J = 1, NRHS
                   FERR( J ) = FERR( J ) / COLCND
       90       CONTINUE
             END IF
          ELSE IF( ROWEQU ) THEN
             DO 110 J = 1, NRHS
                DO 100 I = 1, N
                   X( I, J ) = R( I )*X( I, J )
      100       CONTINUE
      110    CONTINUE
             DO 120 J = 1, NRHS
                FERR( J ) = FERR( J ) / ROWCND
      120    CONTINUE
          END IF
    *
          WORK( 1 ) = RPVGRW
    *
    *     Set INFO = N+1 if the matrix is singular to working precision.
    *
          IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
         $   INFO = N + 1
          RETURN
    *
    *     End of DGESVX
    *
          END