path: root/SRC/cgesvx.f

      SUBROUTINE CGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
     $                   EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
     $                   WORK, RWORK, INFO )
*
*  -- LAPACK driver routine (version 3.2) --
*     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
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      REAL               BERR( * ), C( * ), FERR( * ), R( * ),
     $                   RWORK( * )
      COMPLEX            A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  Purpose
*  =======
*
*  CGESVX uses the LU factorization to compute the solution to a complex
*  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  (Conjugate 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) COMPLEX 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) COMPLEX 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 CGETRF.  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 CGETRF; 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) REAL 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) REAL 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) COMPLEX 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) COMPLEX 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) REAL
*          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) REAL 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) REAL 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) COMPLEX array, dimension (2*N)
*
*  RWORK   (workspace/output) REAL array, dimension (2*N)
*          On exit, RWORK(1) contains the reciprocal pivot growth
*          factor norm(A)/norm(U). The "max absolute element" norm is
*          used. If RWORK(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
*          RWORK(1) contains the reciprocal pivot growth factor for the
*          leading INFO columns of A.
*
*  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 ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
      CHARACTER          NORM
      INTEGER            I, INFEQU, J
      REAL               AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
     $                   ROWCND, RPVGRW, SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               CLANGE, CLANTR, SLAMCH
      EXTERNAL           LSAME, CLANGE, CLANTR, SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           CGECON, CGEEQU, CGERFS, CGETRF, CGETRS, CLACPY,
     $                   CLAQGE, 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 = SLAMCH( '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( 'CGESVX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
         CALL CGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
            CALL CLAQGE( 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 CLACPY( 'Full', N, N, A, LDA, AF, LDAF )
         CALL CGETRF( 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 = CLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
     $               RWORK )
            IF( RPVGRW.EQ.ZERO ) THEN
               RPVGRW = ONE
            ELSE
               RPVGRW = CLANGE( 'M', N, INFO, A, LDA, RWORK ) /
     $                  RPVGRW
            END IF
            RWORK( 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 = CLANGE( NORM, N, N, A, LDA, RWORK )
      RPVGRW = CLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK )
      IF( RPVGRW.EQ.ZERO ) THEN
         RPVGRW = ONE
      ELSE
         RPVGRW = CLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW
      END IF
*
*     Compute the reciprocal of the condition number of A.
*
      CALL CGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
*
*     Compute the solution matrix X.
*
      CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL CGETRS( 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 CGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
     $             LDX, FERR, BERR, WORK, RWORK, 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
*
*     Set INFO = N+1 if the matrix is singular to working precision.
*
      IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
     $   INFO = N + 1
*
      RWORK( 1 ) = RPVGRW
      RETURN
*
*     End of CGESVX
*
      END