*> \brief \b CTBRFS
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CTBRFS + dependencies
*>
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*
* Definition:
* ===========
*
* SUBROUTINE CTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
* LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
* REAL BERR( * ), FERR( * ), RWORK( * )
* COMPLEX AB( LDAB, * ), B( LDB, * ), WORK( * ),
* $ X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CTBRFS 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 CTBTRS or some other
*> means before entering this routine. CTBRFS does not do iterative
*> refinement because doing so cannot improve the backward error.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': A is upper triangular;
*> = 'L': A is lower triangular.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is 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)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> = 'N': A is non-unit triangular;
*> = 'U': A is 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 B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*> AB is COMPLEX 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.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB,NRHS)
*> The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is COMPLEX array, dimension (LDX,NRHS)
*> The solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is REAL 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.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is REAL 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).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexOTHERcomputational
*
* =====================================================================
SUBROUTINE CTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
$ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
*
* -- LAPACK computational 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 INFO, KD, LDAB, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX AB( LDAB, * ), B( LDB, * ), WORK( * ),
$ X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
COMPLEX ONE
PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
CHARACTER TRANSN, TRANST
INTEGER I, J, K, KASE, NZ
REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
COMPLEX ZDUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CCOPY, CLACN2, CTBMV, CTBSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MAX, MIN, REAL
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH
EXTERNAL LSAME, SLAMCH
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. 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( 'CTBRFS', -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
TRANSN = 'N'
TRANST = 'C'
ELSE
TRANSN = 'C'
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = KD + 2
EPS = SLAMCH( 'Epsilon' )
SAFMIN = SLAMCH( '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, A**T, or A**H, depending on TRANS.
*
CALL CCOPY( N, X( 1, J ), 1, WORK, 1 )
CALL CTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
CALL CAXPY( N, -ONE, B( 1, J ), 1, WORK, 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
RWORK( I ) = CABS1( 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 = CABS1( X( K, J ) )
DO 30 I = MAX( 1, K-KD ), K
RWORK( I ) = RWORK( I ) +
$ CABS1( AB( KD+1+I-K, K ) )*XK
30 CONTINUE
40 CONTINUE
ELSE
DO 60 K = 1, N
XK = CABS1( X( K, J ) )
DO 50 I = MAX( 1, K-KD ), K - 1
RWORK( I ) = RWORK( I ) +
$ CABS1( AB( KD+1+I-K, K ) )*XK
50 CONTINUE
RWORK( K ) = RWORK( K ) + XK
60 CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 80 K = 1, N
XK = CABS1( X( K, J ) )
DO 70 I = K, MIN( N, K+KD )
RWORK( I ) = RWORK( I ) +
$ CABS1( AB( 1+I-K, K ) )*XK
70 CONTINUE
80 CONTINUE
ELSE
DO 100 K = 1, N
XK = CABS1( X( K, J ) )
DO 90 I = K + 1, MIN( N, K+KD )
RWORK( I ) = RWORK( I ) +
$ CABS1( AB( 1+I-K, K ) )*XK
90 CONTINUE
RWORK( K ) = RWORK( K ) + XK
100 CONTINUE
END IF
END IF
ELSE
*
* Compute abs(A**H)*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 + CABS1( AB( KD+1+I-K, K ) )*
$ CABS1( X( I, J ) )
110 CONTINUE
RWORK( K ) = RWORK( K ) + S
120 CONTINUE
ELSE
DO 140 K = 1, N
S = CABS1( X( K, J ) )
DO 130 I = MAX( 1, K-KD ), K - 1
S = S + CABS1( AB( KD+1+I-K, K ) )*
$ CABS1( X( I, J ) )
130 CONTINUE
RWORK( K ) = RWORK( 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 + CABS1( AB( 1+I-K, K ) )*
$ CABS1( X( I, J ) )
150 CONTINUE
RWORK( K ) = RWORK( K ) + S
160 CONTINUE
ELSE
DO 180 K = 1, N
S = CABS1( X( K, J ) )
DO 170 I = K + 1, MIN( N, K+KD )
S = S + CABS1( AB( 1+I-K, K ) )*
$ CABS1( X( I, J ) )
170 CONTINUE
RWORK( K ) = RWORK( K ) + S
180 CONTINUE
END IF
END IF
END IF
S = ZERO
DO 190 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
ELSE
S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
$ ( RWORK( 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 CLACN2 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( RWORK( I ).GT.SAFE2 ) THEN
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
ELSE
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
$ SAFE1
END IF
200 CONTINUE
*
KASE = 0
210 CONTINUE
CALL CLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**H).
*
CALL CTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, WORK,
$ 1 )
DO 220 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
220 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 230 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
230 CONTINUE
CALL CTBSV( UPLO, TRANSN, DIAG, N, KD, AB, LDAB, WORK,
$ 1 )
END IF
GO TO 210
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 240 I = 1, N
LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
240 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
250 CONTINUE
*
RETURN
*
* End of CTBRFS
*
END