dtbrfs.f

          SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
         $                   LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
    *
    *     .. Scalar Arguments ..
          CHARACTER          DIAG, TRANS, UPLO
          INTEGER            INFO, KD, LDAB, LDB, LDX, N, NRHS
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), BERR( * ),
         $                   FERR( * ), WORK( * ), X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DTBRFS provides error bounds and backward error estimates for the
    *  solution to a system of linear equations with a triangular band
    *  coefficient matrix.
    *
    *  The solution matrix X must be computed by DTBTRS or some other
    *  means before entering this routine.  DTBRFS does not do iterative
    *  refinement because doing so cannot improve the backward error.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  A is upper triangular;
    *          = 'L':  A is lower triangular.
    *
    *  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 = Transpose)
    *
    *  DIAG    (input) CHARACTER*1
    *          = 'N':  A is non-unit triangular;
    *          = 'U':  A is unit triangular.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KD      (input) INTEGER
    *          The number of superdiagonals or subdiagonals of the
    *          triangular band matrix A.  KD >= 0.
    *
    *  NRHS    (input) INTEGER
    *          The number of right hand sides, i.e., the number of columns
    *          of the matrices B and X.  NRHS >= 0.
    *
    *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
    *          The upper or lower triangular band matrix A, stored in the
    *          first kd+1 rows of the array. The j-th column of A is stored
    *          in the j-th column of the array AB as follows:
    *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    *          If DIAG = 'U', the diagonal elements of A are not referenced
    *          and are assumed to be 1.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KD+1.
    *
    *  B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
    *          The 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) DOUBLE PRECISION array, dimension (3*N)
    *
    *  IWORK   (workspace) INTEGER array, dimension (N)
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO
          PARAMETER          ( ZERO = 0.0D+0 )
          DOUBLE PRECISION   ONE
          PARAMETER          ( ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOTRAN, NOUNIT, UPPER
          CHARACTER          TRANST
          INTEGER            I, J, K, KASE, NZ
          DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
    *     ..
    *     .. Local Arrays ..
          INTEGER            ISAVE( 3 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DCOPY, DLACN2, DTBMV, DTBSV, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH
          EXTERNAL           LSAME, DLAMCH
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          NOTRAN = LSAME( TRANS, 'N' )
          NOUNIT = LSAME( DIAG, 'N' )
    *
          IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -1
          ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
         $         LSAME( TRANS, 'C' ) ) THEN
             INFO = -2
          ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
             INFO = -3
          ELSE IF( N.LT.0 ) THEN
             INFO = -4
          ELSE IF( KD.LT.0 ) THEN
             INFO = -5
          ELSE IF( NRHS.LT.0 ) THEN
             INFO = -6
          ELSE IF( LDAB.LT.KD+1 ) THEN
             INFO = -8
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -10
          ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
             INFO = -12
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DTBRFS', -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( NOTRAN ) THEN
             TRANST = 'T'
          ELSE
             TRANST = 'N'
          END IF
    *
    *     NZ = maximum number of nonzero elements in each row of A, plus 1
    *
          NZ = KD + 2
          EPS = DLAMCH( 'Epsilon' )
          SAFMIN = DLAMCH( 'Safe minimum' )
          SAFE1 = NZ*SAFMIN
          SAFE2 = SAFE1 / EPS
    *
    *     Do for each right hand side
    *
          DO 250 J = 1, NRHS
    *
    *        Compute residual R = B - op(A) * X,
    *        where op(A) = A or A', depending on TRANS.
    *
             CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
             CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ),
         $               1 )
             CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
    *
    *        Compute componentwise relative backward error from formula
    *
    *        max(i) ( abs(R(i)) / ( abs(op(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 20 I = 1, N
                WORK( I ) = ABS( B( I, J ) )
       20    CONTINUE
    *
             IF( NOTRAN ) THEN
    *
    *           Compute abs(A)*abs(X) + abs(B).
    *
                IF( UPPER ) THEN
                   IF( NOUNIT ) THEN
                      DO 40 K = 1, N
                         XK = ABS( X( K, J ) )
                         DO 30 I = MAX( 1, K-KD ), K
                            WORK( I ) = WORK( I ) +
         $                              ABS( AB( KD+1+I-K, K ) )*XK
       30                CONTINUE
       40             CONTINUE
                   ELSE
                      DO 60 K = 1, N
                         XK = ABS( X( K, J ) )
                         DO 50 I = MAX( 1, K-KD ), K - 1
                            WORK( I ) = WORK( I ) +
         $                              ABS( AB( KD+1+I-K, K ) )*XK
       50                CONTINUE
                         WORK( K ) = WORK( K ) + XK
       60             CONTINUE
                   END IF
                ELSE
                   IF( NOUNIT ) THEN
                      DO 80 K = 1, N
                         XK = ABS( X( K, J ) )
                         DO 70 I = K, MIN( N, K+KD )
                            WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
       70                CONTINUE
       80             CONTINUE
                   ELSE
                      DO 100 K = 1, N
                         XK = ABS( X( K, J ) )
                         DO 90 I = K + 1, MIN( N, K+KD )
                            WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
       90                CONTINUE
                         WORK( K ) = WORK( K ) + XK
      100             CONTINUE
                   END IF
                END IF
             ELSE
    *
    *           Compute abs(A')*abs(X) + abs(B).
    *
                IF( UPPER ) THEN
                   IF( NOUNIT ) THEN
                      DO 120 K = 1, N
                         S = ZERO
                         DO 110 I = MAX( 1, K-KD ), K
                            S = S + ABS( AB( KD+1+I-K, K ) )*
         $                      ABS( X( I, J ) )
      110                CONTINUE
                         WORK( K ) = WORK( K ) + S
      120             CONTINUE
                   ELSE
                      DO 140 K = 1, N
                         S = ABS( X( K, J ) )
                         DO 130 I = MAX( 1, K-KD ), K - 1
                            S = S + ABS( AB( KD+1+I-K, K ) )*
         $                      ABS( X( I, J ) )
      130                CONTINUE
                         WORK( K ) = WORK( K ) + S
      140             CONTINUE
                   END IF
                ELSE
                   IF( NOUNIT ) THEN
                      DO 160 K = 1, N
                         S = ZERO
                         DO 150 I = K, MIN( N, K+KD )
                            S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
      150                CONTINUE
                         WORK( K ) = WORK( K ) + S
      160             CONTINUE
                   ELSE
                      DO 180 K = 1, N
                         S = ABS( X( K, J ) )
                         DO 170 I = K + 1, MIN( N, K+KD )
                            S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
      170                CONTINUE
                         WORK( K ) = WORK( K ) + S
      180             CONTINUE
                   END IF
                END IF
             END IF
             S = ZERO
             DO 190 I = 1, N
                IF( WORK( I ).GT.SAFE2 ) THEN
                   S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
                ELSE
                   S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
         $             ( WORK( I )+SAFE1 ) )
                END IF
      190    CONTINUE
             BERR( J ) = S
    *
    *        Bound error from formula
    *
    *        norm(X - XTRUE) / norm(X) .le. FERR =
    *        norm( abs(inv(op(A)))*
    *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
    *
    *        where
    *          norm(Z) is the magnitude of the largest component of Z
    *          inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B))
    *        is incremented by SAFE1 if the i-th component of
    *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
    *
    *        Use DLACN2 to estimate the infinity-norm of the matrix
    *           inv(op(A)) * diag(W),
    *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
    *
             DO 200 I = 1, N
                IF( WORK( I ).GT.SAFE2 ) THEN
                   WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
                ELSE
                   WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
                END IF
      200    CONTINUE
    *
             KASE = 0
      210    CONTINUE
             CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
         $                KASE, ISAVE )
             IF( KASE.NE.0 ) THEN
                IF( KASE.EQ.1 ) THEN
    *
    *              Multiply by diag(W)*inv(op(A)').
    *
                   CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB,
         $                     WORK( N+1 ), 1 )
                   DO 220 I = 1, N
                      WORK( N+I ) = WORK( I )*WORK( N+I )
      220          CONTINUE
                ELSE
    *
    *              Multiply by inv(op(A))*diag(W).
    *
                   DO 230 I = 1, N
                      WORK( N+I ) = WORK( I )*WORK( N+I )
      230          CONTINUE
                   CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB,
         $                     WORK( N+1 ), 1 )
                END IF
                GO TO 210
             END IF
    *
    *        Normalize error.
    *
             LSTRES = ZERO
             DO 240 I = 1, N
                LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
      240    CONTINUE
             IF( LSTRES.NE.ZERO )
         $      FERR( J ) = FERR( J ) / LSTRES
    *
      250 CONTINUE
    *
          RETURN
    *
    *     End of DTBRFS
    *
          END