*> \brief \b CTPRFS * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CTPRFS + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, * FERR, BERR, WORK, RWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER DIAG, TRANS, UPLO * INTEGER INFO, LDB, LDX, N, NRHS * .. * .. Array Arguments .. * REAL BERR( * ), FERR( * ), RWORK( * ) * COMPLEX AP( * ), B( LDB, * ), WORK( * ), X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CTPRFS provides error bounds and backward error estimates for the *> solution to a system of linear equations with a triangular packed *> coefficient matrix. *> *> The solution matrix X must be computed by CTPTRS or some other *> means before entering this routine. CTPRFS 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] 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] AP *> \verbatim *> AP is COMPLEX array, dimension (N*(N+1)/2) *> The upper or lower triangular matrix A, packed columnwise in *> a linear array. The j-th column of A is stored in the array *> AP as follows: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. *> If DIAG = 'U', the diagonal elements of A are not referenced *> and are assumed to be 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 CTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, 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, LDB, LDX, N, NRHS * .. * .. Array Arguments .. REAL BERR( * ), FERR( * ), RWORK( * ) COMPLEX AP( * ), 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, KC, NZ REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK COMPLEX ZDUM * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Subroutines .. EXTERNAL CAXPY, CCOPY, CLACN2, CTPMV, CTPSV, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, AIMAG, MAX, 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( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'CTPRFS', -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 = N + 1 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 CTPMV( UPLO, TRANS, DIAG, N, AP, 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 KC = 1 IF( NOUNIT ) THEN DO 40 K = 1, N XK = CABS1( X( K, J ) ) DO 30 I = 1, K RWORK( I ) = RWORK( I ) + \$ CABS1( AP( KC+I-1 ) )*XK 30 CONTINUE KC = KC + K 40 CONTINUE ELSE DO 60 K = 1, N XK = CABS1( X( K, J ) ) DO 50 I = 1, K - 1 RWORK( I ) = RWORK( I ) + \$ CABS1( AP( KC+I-1 ) )*XK 50 CONTINUE RWORK( K ) = RWORK( K ) + XK KC = KC + K 60 CONTINUE END IF ELSE KC = 1 IF( NOUNIT ) THEN DO 80 K = 1, N XK = CABS1( X( K, J ) ) DO 70 I = K, N RWORK( I ) = RWORK( I ) + \$ CABS1( AP( KC+I-K ) )*XK 70 CONTINUE KC = KC + N - K + 1 80 CONTINUE ELSE DO 100 K = 1, N XK = CABS1( X( K, J ) ) DO 90 I = K + 1, N RWORK( I ) = RWORK( I ) + \$ CABS1( AP( KC+I-K ) )*XK 90 CONTINUE RWORK( K ) = RWORK( K ) + XK KC = KC + N - K + 1 100 CONTINUE END IF END IF ELSE * * Compute abs(A**H)*abs(X) + abs(B). * IF( UPPER ) THEN KC = 1 IF( NOUNIT ) THEN DO 120 K = 1, N S = ZERO DO 110 I = 1, K S = S + CABS1( AP( KC+I-1 ) )*CABS1( X( I, J ) ) 110 CONTINUE RWORK( K ) = RWORK( K ) + S KC = KC + K 120 CONTINUE ELSE DO 140 K = 1, N S = CABS1( X( K, J ) ) DO 130 I = 1, K - 1 S = S + CABS1( AP( KC+I-1 ) )*CABS1( X( I, J ) ) 130 CONTINUE RWORK( K ) = RWORK( K ) + S KC = KC + K 140 CONTINUE END IF ELSE KC = 1 IF( NOUNIT ) THEN DO 160 K = 1, N S = ZERO DO 150 I = K, N S = S + CABS1( AP( KC+I-K ) )*CABS1( X( I, J ) ) 150 CONTINUE RWORK( K ) = RWORK( K ) + S KC = KC + N - K + 1 160 CONTINUE ELSE DO 180 K = 1, N S = CABS1( X( K, J ) ) DO 170 I = K + 1, N S = S + CABS1( AP( KC+I-K ) )*CABS1( X( I, J ) ) 170 CONTINUE RWORK( K ) = RWORK( K ) + S KC = KC + N - K + 1 180 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 CTPSV( UPLO, TRANST, DIAG, N, AP, 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 CTPSV( UPLO, TRANSN, DIAG, N, AP, 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 CTPRFS * END