zporfs.f

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      SUBROUTINE zporfs( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
     $                   ldx, ferr, berr, work, rwork, info )
*
      include 'plasmaf.h'
*
*  -- LAPACK routine (version 3.2) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
*
*     .. Scalar Arguments ..
      CHARACTER          uplo
      INTEGER            info, lda, ldaf, ldb, ldx, n, nrhs
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   berr( * ), ferr( * ), rwork( * )
      COMPLEX*16         a( lda, * ), af( ldaf, * ), b( ldb, * ),
     $                   work( * ), x( ldx, * )
*     ..
*
*  Purpose
*  =======
*
*  ZPORFS improves the computed solution to a system of linear
*  equations when the coefficient matrix is Hermitian positive definite,
*  and provides error bounds and backward error estimates for the
*  solution.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          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) COMPLEX*16 array, dimension (LDA,N)
*          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N
*          upper triangular part of A contains the upper triangular part
*          of the matrix A, and the strictly lower triangular part of A
*          is not referenced.  If UPLO = 'L', the leading N-by-N lower
*          triangular part of A contains the lower triangular part of
*          the matrix A, and the strictly upper triangular part of A is
*          not referenced.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  AF      (input) COMPLEX*16 array, dimension (LDAF,N)
*          The triangular factor U or L from the Cholesky factorization
*          A = U**H*U or A = L*L**H, as computed by ZPOTRF.
*
*  LDAF    (input) INTEGER
*          The leading dimension of the array AF.  LDAF >= max(1,N).
*
*  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
*          The right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
*          On entry, the solution matrix X, as computed by ZPOTRS.
*          On exit, the improved solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  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) COMPLEX*16 array, dimension (2*N)
*
*  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  Internal Parameters
*  ===================
*
*  ITMAX is the maximum number of steps of iterative refinement.
*
*  ====================================================================
*
*     .. Parameters ..
      INTEGER            itmax
      parameter( itmax = 5 )
      DOUBLE PRECISION   zero
      parameter( zero = 0.0d+0 )
      COMPLEX*16         one
      parameter( one = ( 1.0d+0, 0.0d+0 ) )
      DOUBLE PRECISION   two
      parameter( two = 2.0d+0 )
      DOUBLE PRECISION   three
      parameter( three = 3.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            upper
      INTEGER            count, i, j, k, kase, nz, plasma_uplo
      DOUBLE PRECISION   eps, lstres, s, safe1, safe2, safmin, xk
      COMPLEX*16         zdum
*     ..
*     .. Local Arrays ..
      INTEGER            isave( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zaxpy, zcopy, zhemv, zlacn2, zpotrs
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      DOUBLE PRECISION   dlamch
      EXTERNAL           lsame, dlamch
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   cabs1
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( nrhs.LT.0 ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -5
      ELSE IF( ldaf.LT.max( 1, n ) ) THEN
         info = -7
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -9
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
         info = -11
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZPORFS', -info )
         return
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
         DO 10 j = 1, nrhs
            ferr( j ) = zero
            berr( j ) = zero
   10    continue
         return
      END IF
*
      IF ( lsame( uplo, 'U' ) ) THEN
          plasma_uplo = plasmaupper
      ELSE
          plasma_uplo = plasmalower
      ENDIF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      nz = n + 1
      eps = dlamch( 'Epsilon' )
      safmin = dlamch( 'Safe minimum' )
      safe1 = nz*safmin
      safe2 = safe1 / eps
*
*     Do for each right hand side
*
      DO 140 j = 1, nrhs
*
         count = 1
         lstres = three
   20    continue
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X
*
         CALL zcopy( n, b( 1, j ), 1, work, 1 )
         CALL zhemv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         DO 30 i = 1, n
            rwork( i ) = cabs1( b( i, j ) )
   30    continue
*
*        Compute abs(A)*abs(X) + abs(B).
*
         IF( upper ) THEN
            DO 50 k = 1, n
               s = zero
               xk = cabs1( x( k, j ) )
               DO 40 i = 1, k - 1
                  rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
                  s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
   40          continue
               rwork( k ) = rwork( k ) + abs( dble( a( k, k ) ) )*xk + s
   50       continue
         ELSE
            DO 70 k = 1, n
               s = zero
               xk = cabs1( x( k, j ) )
               rwork( k ) = rwork( k ) + abs( dble( a( k, k ) ) )*xk
               DO 60 i = k + 1, n
                  rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
                  s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
   60          continue
               rwork( k ) = rwork( k ) + s
   70       continue
         END IF
         s = zero
         DO 80 i = 1, n
            IF( rwork( i ).GT.safe2 ) THEN
               s = max( s, cabs1( work( i ) ) / rwork( i ) )
            ELSE
               s = max( s, ( cabs1( work( i ) )+safe1 ) /
     $             ( rwork( i )+safe1 ) )
            END IF
   80    continue
         berr( j ) = s
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
     $       count.LE.itmax ) THEN
*
*           Update solution and try again.
*
            CALL plasma_zpotrs( plasma_uplo, n, 1, af, ldaf, 
     $                         work, n, info )
            CALL zaxpy( n, one, work, 1, x( 1, j ), 1 )
            lstres = berr( j )
            count = count + 1
            go to 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
*        Use ZLACN2 to estimate the infinity-norm of the matrix
*           inv(A) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
*
         DO 90 i = 1, n
            IF( rwork( i ).GT.safe2 ) THEN
               rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
            ELSE
               rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
     $                      safe1
            END IF
   90    continue
*
         kase = 0
  100    continue
         CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
         IF( kase.NE.0 ) THEN
            IF( kase.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(A').
*
               CALL plasma_zpotrs( plasma_uplo, n, 1, af, ldaf, 
     $                             work, n, info )
               DO 110 i = 1, n
                  work( i ) = rwork( i )*work( i )
  110          continue
            ELSE IF( kase.EQ.2 ) THEN
*
*              Multiply by inv(A)*diag(W).
*
               DO 120 i = 1, n
                  work( i ) = rwork( i )*work( i )
  120          continue
               CALL plasma_zpotrs( plasma_uplo, n, 1, af, ldaf, 
     $                              work, n, info )
            END IF
            go to 100
         END IF
*
*        Normalize error.
*
         lstres = zero
         DO 130 i = 1, n
            lstres = max( lstres, cabs1( x( i, j ) ) )
  130    continue
         IF( lstres.NE.zero )
     $      ferr( j ) = ferr( j ) / lstres
*
  140 continue
*
      return
*
*     End of ZPORFS
*
      END