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Diffstat (limited to 'SRC/dgtrfs.f')
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diff --git a/SRC/dgtrfs.f b/SRC/dgtrfs.f new file mode 100644 index 00000000..4150e294 --- /dev/null +++ b/SRC/dgtrfs.f @@ -0,0 +1,361 @@ + SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, + $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, + $ INFO ) +* +* -- LAPACK routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. +* +* .. Scalar Arguments .. + CHARACTER TRANS + INTEGER INFO, LDB, LDX, N, NRHS +* .. +* .. Array Arguments .. + INTEGER IPIV( * ), IWORK( * ) + DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), + $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), + $ FERR( * ), WORK( * ), X( LDX, * ) +* .. +* +* Purpose +* ======= +* +* DGTRFS improves the computed solution to a system of linear +* equations when the coefficient matrix is tridiagonal, and provides +* error bounds and backward error estimates for the solution. +* +* Arguments +* ========= +* +* 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) +* +* 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 matrix B. NRHS >= 0. +* +* DL (input) DOUBLE PRECISION array, dimension (N-1) +* The (n-1) subdiagonal elements of A. +* +* D (input) DOUBLE PRECISION array, dimension (N) +* The diagonal elements of A. +* +* DU (input) DOUBLE PRECISION array, dimension (N-1) +* The (n-1) superdiagonal elements of A. +* +* DLF (input) DOUBLE PRECISION array, dimension (N-1) +* The (n-1) multipliers that define the matrix L from the +* LU factorization of A as computed by DGTTRF. +* +* DF (input) DOUBLE PRECISION array, dimension (N) +* The n diagonal elements of the upper triangular matrix U from +* the LU factorization of A. +* +* DUF (input) DOUBLE PRECISION array, dimension (N-1) +* The (n-1) elements of the first superdiagonal of U. +* +* DU2 (input) DOUBLE PRECISION array, dimension (N-2) +* The (n-2) elements of the second superdiagonal of U. +* +* IPIV (input) INTEGER array, dimension (N) +* The pivot indices; for 1 <= i <= n, row i of the matrix was +* interchanged with row IPIV(i). IPIV(i) will always be either +* i or i+1; IPIV(i) = i indicates a row interchange was not +* required. +* +* 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/output) DOUBLE PRECISION array, dimension (LDX,NRHS) +* On entry, the solution matrix X, as computed by DGTTRS. +* On exit, the improved 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 +* +* Internal Parameters +* =================== +* +* ITMAX is the maximum number of steps of iterative refinement. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER ITMAX + PARAMETER ( ITMAX = 5 ) + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) + DOUBLE PRECISION TWO + PARAMETER ( TWO = 2.0D+0 ) + DOUBLE PRECISION THREE + PARAMETER ( THREE = 3.0D+0 ) +* .. +* .. Local Scalars .. + LOGICAL NOTRAN + CHARACTER TRANSN, TRANST + INTEGER COUNT, I, J, KASE, NZ + DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN +* .. +* .. Local Arrays .. + INTEGER ISAVE( 3 ) +* .. +* .. External Subroutines .. + EXTERNAL DAXPY, DCOPY, DGTTRS, DLACN2, DLAGTM, XERBLA +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, MAX +* .. +* .. External Functions .. + LOGICAL LSAME + DOUBLE PRECISION DLAMCH + EXTERNAL LSAME, DLAMCH +* .. +* .. Executable Statements .. +* +* Test the input parameters. +* + INFO = 0 + NOTRAN = LSAME( TRANS, 'N' ) + IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. + $ LSAME( TRANS, 'C' ) ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -2 + ELSE IF( NRHS.LT.0 ) THEN + INFO = -3 + ELSE IF( LDB.LT.MAX( 1, N ) ) THEN + INFO = -13 + ELSE IF( LDX.LT.MAX( 1, N ) ) THEN + INFO = -15 + END IF + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DGTRFS', -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 = 'T' + ELSE + TRANSN = 'T' + TRANST = 'N' + END IF +* +* NZ = maximum number of nonzero elements in each row of A, plus 1 +* + NZ = 4 + EPS = DLAMCH( 'Epsilon' ) + SAFMIN = DLAMCH( 'Safe minimum' ) + SAFE1 = NZ*SAFMIN + SAFE2 = SAFE1 / EPS +* +* Do for each right hand side +* + DO 110 J = 1, NRHS +* + COUNT = 1 + LSTRES = THREE + 20 CONTINUE +* +* Loop until stopping criterion is satisfied. +* +* Compute residual R = B - op(A) * X, +* where op(A) = A, A**T, or A**H, depending on TRANS. +* + CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) + CALL DLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE, + $ WORK( N+1 ), N ) +* +* Compute abs(op(A))*abs(x) + abs(b) for use in the backward +* error bound. +* + IF( NOTRAN ) THEN + IF( N.EQ.1 ) THEN + WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + ELSE + WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + + $ ABS( DU( 1 )*X( 2, J ) ) + DO 30 I = 2, N - 1 + WORK( I ) = ABS( B( I, J ) ) + + $ ABS( DL( I-1 )*X( I-1, J ) ) + + $ ABS( D( I )*X( I, J ) ) + + $ ABS( DU( I )*X( I+1, J ) ) + 30 CONTINUE + WORK( N ) = ABS( B( N, J ) ) + + $ ABS( DL( N-1 )*X( N-1, J ) ) + + $ ABS( D( N )*X( N, J ) ) + END IF + ELSE + IF( N.EQ.1 ) THEN + WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + ELSE + WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + + $ ABS( DL( 1 )*X( 2, J ) ) + DO 40 I = 2, N - 1 + WORK( I ) = ABS( B( I, J ) ) + + $ ABS( DU( I-1 )*X( I-1, J ) ) + + $ ABS( D( I )*X( I, J ) ) + + $ ABS( DL( I )*X( I+1, J ) ) + 40 CONTINUE + WORK( N ) = ABS( B( N, J ) ) + + $ ABS( DU( N-1 )*X( N-1, J ) ) + + $ ABS( D( N )*X( N, J ) ) + END IF + END IF +* +* 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. +* + S = ZERO + DO 50 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 + 50 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 DGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, + $ WORK( N+1 ), N, INFO ) + CALL DAXPY( N, ONE, WORK( N+1 ), 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(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 60 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 + 60 CONTINUE +* + KASE = 0 + 70 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)**T). +* + CALL DGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, + $ WORK( N+1 ), N, INFO ) + DO 80 I = 1, N + WORK( N+I ) = WORK( I )*WORK( N+I ) + 80 CONTINUE + ELSE +* +* Multiply by inv(op(A))*diag(W). +* + DO 90 I = 1, N + WORK( N+I ) = WORK( I )*WORK( N+I ) + 90 CONTINUE + CALL DGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, + $ WORK( N+1 ), N, INFO ) + END IF + GO TO 70 + END IF +* +* Normalize error. +* + LSTRES = ZERO + DO 100 I = 1, N + LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) + 100 CONTINUE + IF( LSTRES.NE.ZERO ) + $ FERR( J ) = FERR( J ) / LSTRES +* + 110 CONTINUE +* + RETURN +* +* End of DGTRFS +* + END |