SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, \$ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), \$ FERR( * ), WORK( * ), X( LDX, * ) * .. * * Purpose * ======= * * DTBRFS 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 DTBTRS or some other * means before entering this routine. DTBRFS does not do iterative * refinement because doing so cannot improve the backward error. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': A is upper triangular; * = 'L': A is lower triangular. * * TRANS (input) 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 = Transpose) * * DIAG (input) CHARACTER*1 * = 'N': A is non-unit triangular; * = 'U': A is unit triangular. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals or subdiagonals of the * triangular band matrix A. KD >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices B and X. NRHS >= 0. * * AB (input) 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). * If DIAG = 'U', the diagonal elements of A are not referenced * and are assumed to be 1. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD+1. * * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) * The right hand side matrix B. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) * The solution matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * FERR (output) DOUBLE PRECISION 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. * * BERR (output) DOUBLE PRECISION 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). * * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) * * IWORK (workspace) INTEGER array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER CHARACTER TRANST INTEGER I, J, K, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACN2, DTBMV, DTBSV, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH * .. * .. 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( 'DTBRFS', -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 TRANST = 'T' ELSE TRANST = 'N' END IF * * NZ = maximum number of nonzero elements in each row of A, plus 1 * NZ = KD + 2 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( '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 or A', depending on TRANS. * CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), \$ 1 ) CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 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 WORK( I ) = ABS( 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 = ABS( X( K, J ) ) DO 30 I = MAX( 1, K-KD ), K WORK( I ) = WORK( I ) + \$ ABS( AB( KD+1+I-K, K ) )*XK 30 CONTINUE 40 CONTINUE ELSE DO 60 K = 1, N XK = ABS( X( K, J ) ) DO 50 I = MAX( 1, K-KD ), K - 1 WORK( I ) = WORK( I ) + \$ ABS( AB( KD+1+I-K, K ) )*XK 50 CONTINUE WORK( K ) = WORK( K ) + XK 60 CONTINUE END IF ELSE IF( NOUNIT ) THEN DO 80 K = 1, N XK = ABS( X( K, J ) ) DO 70 I = K, MIN( N, K+KD ) WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK 70 CONTINUE 80 CONTINUE ELSE DO 100 K = 1, N XK = ABS( X( K, J ) ) DO 90 I = K + 1, MIN( N, K+KD ) WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK 90 CONTINUE WORK( K ) = WORK( K ) + XK 100 CONTINUE END IF END IF ELSE * * Compute abs(A')*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 + ABS( AB( KD+1+I-K, K ) )* \$ ABS( X( I, J ) ) 110 CONTINUE WORK( K ) = WORK( K ) + S 120 CONTINUE ELSE DO 140 K = 1, N S = ABS( X( K, J ) ) DO 130 I = MAX( 1, K-KD ), K - 1 S = S + ABS( AB( KD+1+I-K, K ) )* \$ ABS( X( I, J ) ) 130 CONTINUE WORK( K ) = WORK( 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 + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) ) 150 CONTINUE WORK( K ) = WORK( K ) + S 160 CONTINUE ELSE DO 180 K = 1, N S = ABS( X( K, J ) ) DO 170 I = K + 1, MIN( N, K+KD ) S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) ) 170 CONTINUE WORK( K ) = WORK( K ) + S 180 CONTINUE END IF END IF END IF S = ZERO DO 190 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / \$ ( WORK( 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 DLACN2 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( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 200 CONTINUE * KASE = 0 210 CONTINUE CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), \$ KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Multiply by diag(W)*inv(op(A)'). * CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, \$ WORK( N+1 ), 1 ) DO 220 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 220 CONTINUE ELSE * * Multiply by inv(op(A))*diag(W). * DO 230 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 230 CONTINUE CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, \$ WORK( N+1 ), 1 ) END IF GO TO 210 END IF * * Normalize error. * LSTRES = ZERO DO 240 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 240 CONTINUE IF( LSTRES.NE.ZERO ) \$ FERR( J ) = FERR( J ) / LSTRES * 250 CONTINUE * RETURN * * End of DTBRFS * END