summaryrefslogtreecommitdiff
path: root/SRC/dporfs.f
diff options
context:
space:
mode:
authorjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
committerjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
commitbaba851215b44ac3b60b9248eb02bcce7eb76247 (patch)
tree8c0f5c006875532a30d4409f5e94b0f310ff00a7 /SRC/dporfs.f
downloadlapack-baba851215b44ac3b60b9248eb02bcce7eb76247.tar.gz
lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.tar.bz2
lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.zip
Move LAPACK trunk into position.
Diffstat (limited to 'SRC/dporfs.f')
-rw-r--r--SRC/dporfs.f331
1 files changed, 331 insertions, 0 deletions
diff --git a/SRC/dporfs.f b/SRC/dporfs.f
new file mode 100644
index 00000000..5a34b611
--- /dev/null
+++ b/SRC/dporfs.f
@@ -0,0 +1,331 @@
+ SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, 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 UPLO
+ INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
+* ..
+* .. Array Arguments ..
+ INTEGER IWORK( * )
+ DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
+ $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
+* ..
+*
+* Purpose
+* =======
+*
+* DPORFS improves the computed solution to a system of linear
+* equations when the coefficient matrix is symmetric positive definite,
+* 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) DOUBLE PRECISION array, dimension (LDA,N)
+* The symmetric 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) DOUBLE PRECISION array, dimension (LDAF,N)
+* The triangular factor U or L from the Cholesky factorization
+* A = U**T*U or A = L*L**T, as computed by DPOTRF.
+*
+* LDAF (input) INTEGER
+* The leading dimension of the array AF. LDAF >= max(1,N).
+*
+* 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 DPOTRS.
+* 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
+ PARAMETER ( ZERO = 0.0D+0 )
+ DOUBLE PRECISION ONE
+ PARAMETER ( ONE = 1.0D+0 )
+ DOUBLE PRECISION TWO
+ PARAMETER ( TWO = 2.0D+0 )
+ DOUBLE PRECISION THREE
+ PARAMETER ( THREE = 3.0D+0 )
+* ..
+* .. Local Scalars ..
+ LOGICAL UPPER
+ INTEGER COUNT, 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, DPOTRS, DSYMV, XERBLA
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, MAX
+* ..
+* .. External Functions ..
+ LOGICAL LSAME
+ DOUBLE PRECISION DLAMCH
+ EXTERNAL LSAME, DLAMCH
+* ..
+* .. 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 = -9
+ ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
+ INFO = -11
+ END IF
+ IF( INFO.NE.0 ) THEN
+ CALL XERBLA( 'DPORFS', -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 = DLAMCH( 'Epsilon' )
+ SAFMIN = DLAMCH( '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 DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
+ CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
+ $ WORK( N+1 ), 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
+ WORK( I ) = ABS( B( I, J ) )
+ 30 CONTINUE
+*
+* Compute abs(A)*abs(X) + abs(B).
+*
+ IF( UPPER ) THEN
+ DO 50 K = 1, N
+ S = ZERO
+ XK = ABS( X( K, J ) )
+ DO 40 I = 1, K - 1
+ WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
+ S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
+ 40 CONTINUE
+ WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
+ 50 CONTINUE
+ ELSE
+ DO 70 K = 1, N
+ S = ZERO
+ XK = ABS( X( K, J ) )
+ WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
+ DO 60 I = K + 1, N
+ WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
+ S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
+ 60 CONTINUE
+ WORK( K ) = WORK( K ) + S
+ 70 CONTINUE
+ END IF
+ S = ZERO
+ DO 80 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
+ 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 DPOTRS( UPLO, N, 1, AF, LDAF, 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(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 DLACN2 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( 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
+ 90 CONTINUE
+*
+ KASE = 0
+ 100 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(A').
+*
+ CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
+ DO 110 I = 1, N
+ WORK( N+I ) = WORK( I )*WORK( N+I )
+ 110 CONTINUE
+ ELSE IF( KASE.EQ.2 ) THEN
+*
+* Multiply by inv(A)*diag(W).
+*
+ DO 120 I = 1, N
+ WORK( N+I ) = WORK( I )*WORK( N+I )
+ 120 CONTINUE
+ CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
+ END IF
+ GO TO 100
+ END IF
+*
+* Normalize error.
+*
+ LSTRES = ZERO
+ DO 130 I = 1, N
+ LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
+ 130 CONTINUE
+ IF( LSTRES.NE.ZERO )
+ $ FERR( J ) = FERR( J ) / LSTRES
+*
+ 140 CONTINUE
+*
+ RETURN
+*
+* End of DPORFS
+*
+ END