numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
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| lapack/TESTING/LIN/ztbt03.f | 8482B | -rw-r--r-- |
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*> \brief \b ZTBT03 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE ZTBT03( 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 CNORM( * ) * COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ), * $ X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZTBT03 computes the residual for the solution to a scaled triangular *> system of equations A*x = s*b, A**T *x = s*b, or A**H *x = s*b *> when A is a triangular band matrix. Here A**T denotes the transpose *> of A, A**H denotes the conjugate 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, A**T, or A**H, 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 = s*b (No transpose) *> = 'T': A**T *x = s*b (Transpose) *> = 'C': A**H *x = s*b (Conjugate 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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. * *> \ingroup complex16_lin * * ===================================================================== SUBROUTINE ZTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, $ SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK, $ RESID ) * * -- 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 DOUBLE PRECISION RESID, SCALE, TSCAL * .. * .. Array Arguments .. DOUBLE PRECISION CNORM( * ) COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ), $ X( LDX, * ) * .. * * ===================================================================== * * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. INTEGER IX, J DOUBLE PRECISION EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL * .. * .. External Functions .. LOGICAL LSAME INTEGER IZAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IZAMAX, DLAMCH * .. * .. External Subroutines .. EXTERNAL ZAXPY, ZCOPY, ZDSCAL, ZTBMV * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, 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' ) * * Compute the norm of the triangular matrix A using the column * norms already computed by ZLATBS. * 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 ZCOPY( N, X( 1, J ), 1, WORK, 1 ) IX = IZAMAX( N, WORK, 1 ) XNORM = MAX( ONE, ABS( X( IX, J ) ) ) XSCAL = ( ONE / XNORM ) / DBLE( KD+1 ) CALL ZDSCAL( N, XSCAL, WORK, 1 ) CALL ZTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 ) CALL ZAXPY( N, DCMPLX( -SCALE*XSCAL ), B( 1, J ), 1, WORK, 1 ) IX = IZAMAX( N, WORK, 1 ) ERR = TSCAL*ABS( WORK( IX ) ) IX = IZAMAX( 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 ZTBT03 * END