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*> \brief \b DTBT03
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
* SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
* RESID )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER KD, LDAB, LDB, LDX, N, NRHS
* DOUBLE PRECISION RESID, SCALE, TSCAL
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), CNORM( * ),
* $ WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTBT03 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> 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)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of superdiagonals or subdiagonals of the
*> triangular band matrix A. KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices X and B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is 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).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The scaling factor s used in solving the triangular system.
*> \endverbatim
*>
*> \param[in] CNORM
*> \verbatim
*> CNORM is DOUBLE PRECISION array, dimension (N)
*> The 1-norms of the columns of A, not counting the diagonal.
*> \endverbatim
*>
*> \param[in] TSCAL
*> \verbatim
*> TSCAL is DOUBLE PRECISION
*> The scaling factor used in computing the 1-norms in CNORM.
*> CNORM actually contains the column norms of TSCAL*A.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The computed solution vectors for the system of linear
*> equations.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side vectors for the system of linear
*> equations.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is DOUBLE PRECISION
*> The maximum over the number of right hand sides of
*> norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_lin
*
* =====================================================================
SUBROUTINE DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
$ SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
$ RESID )
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER KD, LDAB, LDB, LDX, N, NRHS
DOUBLE PRECISION RESID, SCALE, TSCAL
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), CNORM( * ),
$ WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER IX, J
DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLABAD, DSCAL, DTBMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0
*
IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
EPS = DLAMCH( 'Epsilon' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
*
* Compute the norm of the triangular matrix A using the column
* norms already computed by DLATBS.
*
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 DCOPY( N, X( 1, J ), 1, WORK, 1 )
IX = IDAMAX( N, WORK, 1 )
XNORM = MAX( ONE, ABS( X( IX, J ) ) )
XSCAL = ( ONE / XNORM ) / DBLE( KD+1 )
CALL DSCAL( N, XSCAL, WORK, 1 )
CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
CALL DAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
IX = IDAMAX( N, WORK, 1 )
ERR = TSCAL*ABS( WORK( IX ) )
IX = IDAMAX( 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 DTBT03
*
END
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