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lapack/SRC/csprfs.f
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julie be91086043 Big commit before 3.2.1 release. 
Those are just cosmetic changes to update version number and various other minor change.
2009-04-16 18:10:16 +00:00

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SUBROUTINE CSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
$ FERR, BERR, WORK, RWORK, 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 CLACN2 in place of CLACON, 10 Feb 03, SJH.
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
REAL BERR( * ), FERR( * ), RWORK( * )
COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
$ X( LDX, * )
* ..
*
* Purpose
* =======
*
* CSPRFS improves the computed solution to a system of linear
* equations when the coefficient matrix is symmetric indefinite
* and packed, 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.
*
* AP (input) COMPLEX array, dimension (N*(N+1)/2)
* The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
* AFP (input) COMPLEX array, dimension (N*(N+1)/2)
* The factored form of the matrix A. AFP contains the block
* diagonal matrix D and the multipliers used to obtain the
* factor U or L from the factorization A = U*D*U**T or
* A = L*D*L**T as computed by CSPTRF, stored as a packed
* triangular matrix.
*
* IPIV (input) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D
* as determined by CSPTRF.
*
* 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 CSPTRS.
* 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, IK, J, K, KASE, KK, NZ
REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
COMPLEX ZDUM
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CCOPY, CLACN2, CSPMV, CSPTRS, 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( 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( 'CSPRFS', -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 CSPMV( UPLO, N, -ONE, AP, 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).
*
KK = 1
IF( UPPER ) THEN
DO 50 K = 1, N
S = ZERO
XK = CABS1( X( K, J ) )
IK = KK
DO 40 I = 1, K - 1
RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
IK = IK + 1
40 CONTINUE
RWORK( K ) = RWORK( K ) + CABS1( AP( KK+K-1 ) )*XK + S
KK = KK + K
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
XK = CABS1( X( K, J ) )
RWORK( K ) = RWORK( K ) + CABS1( AP( KK ) )*XK
IK = KK + 1
DO 60 I = K + 1, N
RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
IK = IK + 1
60 CONTINUE
RWORK( K ) = RWORK( K ) + S
KK = KK + ( N-K+1 )
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 CSPTRS( UPLO, N, 1, AFP, 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 CSPTRS( UPLO, N, 1, AFP, 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 CSPTRS( UPLO, N, 1, AFP, 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 CSPRFS
*
END
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