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Diffstat (limited to 'SRC/ctprfs.f')
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diff --git a/SRC/ctprfs.f b/SRC/ctprfs.f new file mode 100644 index 00000000..d9cb2283 --- /dev/null +++ b/SRC/ctprfs.f @@ -0,0 +1,391 @@ + SUBROUTINE CTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, + $ FERR, BERR, WORK, RWORK, INFO ) +* +* -- LAPACK routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. +* +* .. 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, * ) +* .. +* +* Purpose +* ======= +* +* 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. +* +* 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) +* +* 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. +* +* NRHS (input) INTEGER +* The number of right hand sides, i.e., the number of columns +* of the matrices B and X. NRHS >= 0. +* +* AP (input) 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. +* +* B (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS) +* The solution matrix X. +* +* LDX (input) INTEGER +* The leading dimension of the array X. LDX >= max(1,N). +* +* FERR (output) 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. +* +* BERR (output) 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). +* +* WORK (workspace) COMPLEX array, dimension (2*N) +* +* RWORK (workspace) REAL array, dimension (N) +* +* INFO (output) INTEGER +* = 0: successful exit +* < 0: if INFO = -i, the i-th argument had an illegal value +* +* ===================================================================== +* +* .. 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 |