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Diffstat (limited to 'SRC/cherfs.f')
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diff --git a/SRC/cherfs.f b/SRC/cherfs.f new file mode 100644 index 00000000..673026dc --- /dev/null +++ b/SRC/cherfs.f @@ -0,0 +1,343 @@ + SUBROUTINE CHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, 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 UPLO + INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS +* .. +* .. Array Arguments .. + INTEGER IPIV( * ) + REAL BERR( * ), FERR( * ), RWORK( * ) + COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), + $ WORK( * ), X( LDX, * ) +* .. +* +* Purpose +* ======= +* +* CHERFS improves the computed solution to a system of linear +* equations when the coefficient matrix is Hermitian indefinite, and +* provides error bounds and backward error estimates for the solution. +* +* Arguments +* ========= +* +* UPLO (input) CHARACTER*1 +* = 'U': Upper triangle of A is stored; +* = 'L': Lower triangle of A is stored. +* +* 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. +* +* A (input) COMPLEX array, dimension (LDA,N) +* The Hermitian matrix A. If UPLO = 'U', the leading N-by-N +* upper triangular part of A contains the upper triangular part +* of the matrix A, and the strictly lower triangular part of A +* is not referenced. If UPLO = 'L', the leading N-by-N lower +* triangular part of A contains the lower triangular part of +* the matrix A, and the strictly upper triangular part of A is +* not referenced. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* AF (input) COMPLEX array, dimension (LDAF,N) +* The factored form of the matrix A. AF contains the block +* diagonal matrix D and the multipliers used to obtain the +* factor U or L from the factorization A = U*D*U**H or +* A = L*D*L**H as computed by CHETRF. +* +* LDAF (input) INTEGER +* The leading dimension of the array AF. LDAF >= max(1,N). +* +* IPIV (input) INTEGER array, dimension (N) +* Details of the interchanges and the block structure of D +* as determined by CHETRF. +* +* 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/output) COMPLEX array, dimension (LDX,NRHS) +* On entry, the solution matrix X, as computed by CHETRS. +* On exit, the improved 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 +* +* Internal Parameters +* =================== +* +* ITMAX is the maximum number of steps of iterative refinement. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER ITMAX + PARAMETER ( ITMAX = 5 ) + REAL ZERO + PARAMETER ( ZERO = 0.0E+0 ) + COMPLEX ONE + PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) ) + REAL TWO + PARAMETER ( TWO = 2.0E+0 ) + REAL THREE + PARAMETER ( THREE = 3.0E+0 ) +* .. +* .. Local Scalars .. + LOGICAL UPPER + INTEGER COUNT, 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, CHEMV, CHETRS, CLACN2, 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' ) + IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( NRHS.LT.0 ) THEN + INFO = -3 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -5 + ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN + INFO = -7 + 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( 'CHERFS', -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 +* +* 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 140 J = 1, NRHS +* + COUNT = 1 + LSTRES = THREE + 20 CONTINUE +* +* Loop until stopping criterion is satisfied. +* +* Compute residual R = B - A * X +* + CALL CCOPY( N, B( 1, J ), 1, WORK, 1 ) + CALL CHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 ) +* +* Compute componentwise relative backward error from formula +* +* max(i) ( abs(R(i)) / ( abs(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 30 I = 1, N + RWORK( I ) = CABS1( B( I, J ) ) + 30 CONTINUE +* +* Compute abs(A)*abs(X) + abs(B). +* + IF( UPPER ) THEN + DO 50 K = 1, N + S = ZERO + XK = CABS1( X( K, J ) ) + DO 40 I = 1, K - 1 + RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK + S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) + 40 CONTINUE + RWORK( K ) = RWORK( K ) + ABS( REAL( A( K, K ) ) )*XK + S + 50 CONTINUE + ELSE + DO 70 K = 1, N + S = ZERO + XK = CABS1( X( K, J ) ) + RWORK( K ) = RWORK( K ) + ABS( REAL( A( K, K ) ) )*XK + DO 60 I = K + 1, N + RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK + S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) + 60 CONTINUE + RWORK( K ) = RWORK( K ) + S + 70 CONTINUE + END IF + S = ZERO + DO 80 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 + 80 CONTINUE + BERR( J ) = S +* +* Test stopping criterion. Continue iterating if +* 1) The residual BERR(J) is larger than machine epsilon, and +* 2) BERR(J) decreased by at least a factor of 2 during the +* last iteration, and +* 3) At most ITMAX iterations tried. +* + IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. + $ COUNT.LE.ITMAX ) THEN +* +* Update solution and try again. +* + CALL CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) + CALL CAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) + LSTRES = BERR( J ) + COUNT = COUNT + 1 + GO TO 20 + END IF +* +* Bound error from formula +* +* norm(X - XTRUE) / norm(X) .le. FERR = +* norm( abs(inv(A))* +* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) +* +* where +* norm(Z) is the magnitude of the largest component of Z +* inv(A) is the inverse of 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(A)*abs(X)+abs(B)) +* is incremented by SAFE1 if the i-th component of +* abs(A)*abs(X) + abs(B) is less than SAFE2. +* +* Use CLACN2 to estimate the infinity-norm of the matrix +* inv(A) * diag(W), +* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) +* + DO 90 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 + 90 CONTINUE +* + KASE = 0 + 100 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(A'). +* + CALL CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) + DO 110 I = 1, N + WORK( I ) = RWORK( I )*WORK( I ) + 110 CONTINUE + ELSE IF( KASE.EQ.2 ) THEN +* +* Multiply by inv(A)*diag(W). +* + DO 120 I = 1, N + WORK( I ) = RWORK( I )*WORK( I ) + 120 CONTINUE + CALL CHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) + END IF + GO TO 100 + END IF +* +* Normalize error. +* + LSTRES = ZERO + DO 130 I = 1, N + LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) + 130 CONTINUE + IF( LSTRES.NE.ZERO ) + $ FERR( J ) = FERR( J ) / LSTRES +* + 140 CONTINUE +* + RETURN +* +* End of CHERFS +* + END |