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300 lines
8.8 KiB
300 lines
8.8 KiB
*> \brief \b CGBT05
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
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* LDX, XACT, LDXACT, FERR, BERR, RESLTS )
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*
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* .. Scalar Arguments ..
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* CHARACTER TRANS
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* INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
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* ..
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* .. Array Arguments ..
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* REAL BERR( * ), FERR( * ), RESLTS( * )
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* COMPLEX AB( LDAB, * ), B( LDB, * ), X( LDX, * ),
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* $ XACT( LDXACT, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> CGBT05 tests the error bounds from iterative refinement for the
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*> computed solution to a system of equations op(A)*X = B, where A is a
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*> general band matrix of order n with kl subdiagonals and ku
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*> superdiagonals and op(A) = A, A**T, or A**H, depending on TRANS.
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*>
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*> RESLTS(1) = test of the error bound
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*> = norm(X - XACT) / ( norm(X) * FERR )
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*>
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*> A large value is returned if this ratio is not less than one.
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*>
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*> RESLTS(2) = residual from the iterative refinement routine
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*> = the maximum of BERR / ( NZ*EPS + (*) ), where
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*> (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
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*> and NZ = max. number of nonzeros in any row of A, plus 1
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] TRANS
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*> \verbatim
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*> TRANS is CHARACTER*1
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*> Specifies the form of the system of equations.
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*> = 'N': A * X = B (No transpose)
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*> = 'T': A**T * X = B (Transpose)
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*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of rows of the matrices X, B, and XACT, and the
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*> order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KL
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*> \verbatim
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*> KL is INTEGER
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*> The number of subdiagonals within the band of A. KL >= 0.
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*> \endverbatim
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*>
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*> \param[in] KU
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*> \verbatim
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*> KU is INTEGER
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*> The number of superdiagonals within the band of A. KU >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of columns of the matrices X, B, and XACT.
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*> NRHS >= 0.
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*> \endverbatim
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*>
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*> \param[in] AB
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*> \verbatim
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*> AB is COMPLEX array, dimension (LDAB,N)
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*> The original band matrix A, stored in rows 1 to KL+KU+1.
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*> The j-th column of A is stored in the j-th column of the
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*> array AB as follows:
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*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array AB. LDAB >= KL+KU+1.
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*> \endverbatim
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*>
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*> \param[in] B
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*> \verbatim
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*> B is COMPLEX array, dimension (LDB,NRHS)
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*> The right hand side vectors for the system of linear
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*> equations.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] X
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*> \verbatim
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*> X is COMPLEX array, dimension (LDX,NRHS)
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*> The computed solution vectors. Each vector is stored as a
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*> column of the matrix X.
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*> \endverbatim
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*>
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*> \param[in] LDX
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*> \verbatim
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*> LDX is INTEGER
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*> The leading dimension of the array X. LDX >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] XACT
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*> \verbatim
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*> XACT is COMPLEX array, dimension (LDX,NRHS)
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*> The exact solution vectors. Each vector is stored as a
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*> column of the matrix XACT.
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*> \endverbatim
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*>
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*> \param[in] LDXACT
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*> \verbatim
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*> LDXACT is INTEGER
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*> The leading dimension of the array XACT. LDXACT >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] FERR
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*> \verbatim
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*> FERR is REAL array, dimension (NRHS)
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*> The estimated forward error bounds for each solution vector
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*> X. If XTRUE is the true solution, FERR bounds the magnitude
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*> of the largest entry in (X - XTRUE) divided by the magnitude
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*> of the largest entry in X.
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*> \endverbatim
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*>
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*> \param[in] BERR
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*> \verbatim
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*> BERR is REAL array, dimension (NRHS)
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*> The componentwise relative backward error of each solution
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*> vector (i.e., the smallest relative change in any entry of A
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*> or B that makes X an exact solution).
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*> \endverbatim
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*>
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*> \param[out] RESLTS
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*> \verbatim
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*> RESLTS is REAL array, dimension (2)
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*> The maximum over the NRHS solution vectors of the ratios:
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*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
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*> RESLTS(2) = BERR / ( NZ*EPS + (*) )
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complex_lin
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*
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* =====================================================================
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SUBROUTINE CGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
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$ LDX, XACT, LDXACT, FERR, BERR, RESLTS )
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*
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* -- LAPACK test routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER TRANS
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INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
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* ..
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* .. Array Arguments ..
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REAL BERR( * ), FERR( * ), RESLTS( * )
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COMPLEX AB( LDAB, * ), B( LDB, * ), X( LDX, * ),
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$ XACT( LDXACT, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN
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INTEGER I, IMAX, J, K, NZ
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REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
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COMPLEX ZDUM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ICAMAX
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REAL SLAMCH
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EXTERNAL LSAME, ICAMAX, SLAMCH
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, MAX, MIN, REAL
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* ..
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* .. Statement Functions ..
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REAL CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
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* ..
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* .. Executable Statements ..
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*
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* Quick exit if N = 0 or NRHS = 0.
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*
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IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
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RESLTS( 1 ) = ZERO
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RESLTS( 2 ) = ZERO
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RETURN
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END IF
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*
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EPS = SLAMCH( 'Epsilon' )
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UNFL = SLAMCH( 'Safe minimum' )
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OVFL = ONE / UNFL
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NOTRAN = LSAME( TRANS, 'N' )
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NZ = MIN( KL+KU+2, N+1 )
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*
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* Test 1: Compute the maximum of
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* norm(X - XACT) / ( norm(X) * FERR )
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* over all the vectors X and XACT using the infinity-norm.
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*
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ERRBND = ZERO
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DO 30 J = 1, NRHS
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IMAX = ICAMAX( N, X( 1, J ), 1 )
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XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
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DIFF = ZERO
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DO 10 I = 1, N
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DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
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10 CONTINUE
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*
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IF( XNORM.GT.ONE ) THEN
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GO TO 20
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ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
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GO TO 20
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ELSE
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ERRBND = ONE / EPS
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GO TO 30
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END IF
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*
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20 CONTINUE
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IF( DIFF / XNORM.LE.FERR( J ) ) THEN
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ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
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ELSE
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ERRBND = ONE / EPS
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END IF
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30 CONTINUE
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RESLTS( 1 ) = ERRBND
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*
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* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
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* (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
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*
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DO 70 K = 1, NRHS
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DO 60 I = 1, N
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TMP = CABS1( B( I, K ) )
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IF( NOTRAN ) THEN
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DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N )
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TMP = TMP + CABS1( AB( KU+1+I-J, J ) )*
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$ CABS1( X( J, K ) )
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40 CONTINUE
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ELSE
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DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N )
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TMP = TMP + CABS1( AB( KU+1+J-I, I ) )*
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$ CABS1( X( J, K ) )
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50 CONTINUE
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END IF
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IF( I.EQ.1 ) THEN
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AXBI = TMP
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ELSE
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AXBI = MIN( AXBI, TMP )
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END IF
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60 CONTINUE
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TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
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IF( K.EQ.1 ) THEN
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RESLTS( 2 ) = TMP
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ELSE
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RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
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END IF
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70 CONTINUE
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*
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RETURN
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*
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* End of CGBT05
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*
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END
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