stbt03()

subroutine stbt03	(	character	uplo,
		character	trans,
		character	diag,
		integer	n,
		integer	kd,
		integer	nrhs,
		real, dimension( ldab, * )	ab,
		integer	ldab,
		real	scale,
		real, dimension( * )	cnorm,
		real	tscal,
		real, dimension( ldx, * )	x,
		integer	ldx,
		real, dimension( ldb, * )	b,
		integer	ldb,
		real, dimension( * )	work,
		real	resid
	)

STBT03

Purpose:

 STBT03 computes the residual for the solution to a scaled triangular
 system of equations  A*x = s*b  or  A'*x = s*b  when A is a
 triangular band matrix. Here A' is the transpose of A, s is a scalar,
 and x and b are N by NRHS matrices.  The test ratio is the maximum
 over the number of right hand sides of
    norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
 where op(A) denotes A or A' and EPS is the machine epsilon.

Parameters

[in]	UPLO	UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular
[in]	TRANS	TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': A *x = b (No transpose) = 'T': A'*x = b (Transpose) = 'C': A'*x = b (Conjugate transpose = Transpose)
[in]	DIAG	DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular
[in]	N	N is INTEGER The order of the matrix A. N >= 0.
[in]	KD	KD is INTEGER The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 0.
[in]	NRHS	NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices X and B. NRHS >= 0.
[in]	AB	AB is REAL 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).
[in]	LDAB	LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.
[in]	SCALE	SCALE is REAL The scaling factor s used in solving the triangular system.
[in]	CNORM	CNORM is REAL array, dimension (N) The 1-norms of the columns of A, not counting the diagonal.
[in]	TSCAL	TSCAL is REAL The scaling factor used in computing the 1-norms in CNORM. CNORM actually contains the column norms of TSCAL*A.
[in]	X	X is REAL array, dimension (LDX,NRHS) The computed solution vectors for the system of linear equations.
[in]	LDX	LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).
[in]	B	B is REAL array, dimension (LDB,NRHS) The right hand side vectors for the system of linear equations.
[in]	LDB	LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
[out]	WORK	WORK is REAL array, dimension (N)
[out]	RESID	RESID is REAL The maximum over the number of right hand sides of norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 172 of file stbt03.f.

*
*  -- LAPACK test routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            KD, LDAB, LDB, LDX, N, NRHS
      REAL               RESID, SCALE, TSCAL
*     ..
*     .. Array Arguments ..
      REAL               AB( LDAB, * ), B( LDB, * ), CNORM( * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IX, J
      REAL               BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SLAMCH
      EXTERNAL           lsame, isamax, slamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, scopy, sscal, stbmv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, real
*     ..
*     .. Executable Statements ..
*
*     Quick exit if N = 0
*
      IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
         resid = zero
         RETURN
      END IF
      eps = slamch( 'Epsilon' )
      smlnum = slamch( 'Safe minimum' )
      bignum = one / smlnum
*
*     Compute the norm of the triangular matrix A using the column
*     norms already computed by SLATBS.
*
      tnorm = zero
      IF( lsame( diag, 'N' ) ) THEN
         IF( lsame( uplo, 'U' ) ) THEN
            DO 10 j = 1, n
               tnorm = max( tnorm, tscal*abs( ab( kd+1, j ) )+
     $                 cnorm( j ) )
   10       CONTINUE
         ELSE
            DO 20 j = 1, n
               tnorm = max( tnorm, tscal*abs( ab( 1, j ) )+cnorm( j ) )
   20       CONTINUE
         END IF
      ELSE
         DO 30 j = 1, n
            tnorm = max( tnorm, tscal+cnorm( j ) )
   30    CONTINUE
      END IF
*
*     Compute the maximum over the number of right hand sides of
*        norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
*
      resid = zero
      DO 40 j = 1, nrhs
         CALL scopy( n, x( 1, j ), 1, work, 1 )
         ix = isamax( n, work, 1 )
         xnorm = max( one, abs( x( ix, j ) ) )
         xscal = ( one / xnorm ) / real( kd+1 )
         CALL sscal( n, xscal, work, 1 )
         CALL stbmv( uplo, trans, diag, n, kd, ab, ldab, work, 1 )
         CALL saxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
         ix = isamax( n, work, 1 )
         err = tscal*abs( work( ix ) )
         ix = isamax( n, x( 1, j ), 1 )
         xnorm = abs( x( ix, j ) )
         IF( err*smlnum.LE.xnorm ) THEN
            IF( xnorm.GT.zero )
     $         err = err / xnorm
         ELSE
            IF( err.GT.zero )
     $         err = one / eps
         END IF
         IF( err*smlnum.LE.tnorm ) THEN
            IF( tnorm.GT.zero )
     $         err = err / tnorm
         ELSE
            IF( err.GT.zero )
     $         err = one / eps
         END IF
         resid = max( resid, err )
   40 CONTINUE
*
      RETURN
*
*     End of STBT03
*

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