LAPACK: SRC/zpprfs.f Source File

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00001       SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
00002      $                   BERR, WORK, RWORK, INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
00010 *
00011 *     .. Scalar Arguments ..
00012       CHARACTER          UPLO
00013       INTEGER            INFO, LDB, LDX, N, NRHS
00014 *     ..
00015 *     .. Array Arguments ..
00016       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
00017       COMPLEX*16         AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
00018      $                   X( LDX, * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  ZPPRFS improves the computed solution to a system of linear
00025 *  equations when the coefficient matrix is Hermitian positive definite
00026 *  and packed, and provides error bounds and backward error estimates
00027 *  for the solution.
00028 *
00029 *  Arguments
00030 *  =========
00031 *
00032 *  UPLO    (input) CHARACTER*1
00033 *          = 'U':  Upper triangle of A is stored;
00034 *          = 'L':  Lower triangle of A is stored.
00035 *
00036 *  N       (input) INTEGER
00037 *          The order of the matrix A.  N >= 0.
00038 *
00039 *  NRHS    (input) INTEGER
00040 *          The number of right hand sides, i.e., the number of columns
00041 *          of the matrices B and X.  NRHS >= 0.
00042 *
00043 *  AP      (input) COMPLEX*16 array, dimension (N*(N+1)/2)
00044 *          The upper or lower triangle of the Hermitian matrix A, packed
00045 *          columnwise in a linear array.  The j-th column of A is stored
00046 *          in the array AP as follows:
00047 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00048 *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00049 *
00050 *  AFP     (input) COMPLEX*16 array, dimension (N*(N+1)/2)
00051 *          The triangular factor U or L from the Cholesky factorization
00052 *          A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF,
00053 *          packed columnwise in a linear array in the same format as A
00054 *          (see AP).
00055 *
00056 *  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
00057 *          The right hand side matrix B.
00058 *
00059 *  LDB     (input) INTEGER
00060 *          The leading dimension of the array B.  LDB >= max(1,N).
00061 *
00062 *  X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
00063 *          On entry, the solution matrix X, as computed by ZPPTRS.
00064 *          On exit, the improved solution matrix X.
00065 *
00066 *  LDX     (input) INTEGER
00067 *          The leading dimension of the array X.  LDX >= max(1,N).
00068 *
00069 *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00070 *          The estimated forward error bound for each solution vector
00071 *          X(j) (the j-th column of the solution matrix X).
00072 *          If XTRUE is the true solution corresponding to X(j), FERR(j)
00073 *          is an estimated upper bound for the magnitude of the largest
00074 *          element in (X(j) - XTRUE) divided by the magnitude of the
00075 *          largest element in X(j).  The estimate is as reliable as
00076 *          the estimate for RCOND, and is almost always a slight
00077 *          overestimate of the true error.
00078 *
00079 *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00080 *          The componentwise relative backward error of each solution
00081 *          vector X(j) (i.e., the smallest relative change in
00082 *          any element of A or B that makes X(j) an exact solution).
00083 *
00084 *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
00085 *
00086 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
00087 *
00088 *  INFO    (output) INTEGER
00089 *          = 0:  successful exit
00090 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00091 *
00092 *  Internal Parameters
00093 *  ===================
00094 *
00095 *  ITMAX is the maximum number of steps of iterative refinement.
00096 *
00097 *  ====================================================================
00098 *
00099 *     .. Parameters ..
00100       INTEGER            ITMAX
00101       PARAMETER          ( ITMAX = 5 )
00102       DOUBLE PRECISION   ZERO
00103       PARAMETER          ( ZERO = 0.0D+0 )
00104       COMPLEX*16         CONE
00105       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
00106       DOUBLE PRECISION   TWO
00107       PARAMETER          ( TWO = 2.0D+0 )
00108       DOUBLE PRECISION   THREE
00109       PARAMETER          ( THREE = 3.0D+0 )
00110 *     ..
00111 *     .. Local Scalars ..
00112       LOGICAL            UPPER
00113       INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
00114       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00115       COMPLEX*16         ZDUM
00116 *     ..
00117 *     .. Local Arrays ..
00118       INTEGER            ISAVE( 3 )
00119 *     ..
00120 *     .. External Subroutines ..
00121       EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS
00122 *     ..
00123 *     .. Intrinsic Functions ..
00124       INTRINSIC          ABS, DBLE, DIMAG, MAX
00125 *     ..
00126 *     .. External Functions ..
00127       LOGICAL            LSAME
00128       DOUBLE PRECISION   DLAMCH
00129       EXTERNAL           LSAME, DLAMCH
00130 *     ..
00131 *     .. Statement Functions ..
00132       DOUBLE PRECISION   CABS1
00133 *     ..
00134 *     .. Statement Function definitions ..
00135       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00136 *     ..
00137 *     .. Executable Statements ..
00138 *
00139 *     Test the input parameters.
00140 *
00141       INFO = 0
00142       UPPER = LSAME( UPLO, 'U' )
00143       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00144          INFO = -1
00145       ELSE IF( N.LT.0 ) THEN
00146          INFO = -2
00147       ELSE IF( NRHS.LT.0 ) THEN
00148          INFO = -3
00149       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00150          INFO = -7
00151       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00152          INFO = -9
00153       END IF
00154       IF( INFO.NE.0 ) THEN
00155          CALL XERBLA( 'ZPPRFS', -INFO )
00156          RETURN
00157       END IF
00158 *
00159 *     Quick return if possible
00160 *
00161       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00162          DO 10 J = 1, NRHS
00163             FERR( J ) = ZERO
00164             BERR( J ) = ZERO
00165    10    CONTINUE
00166          RETURN
00167       END IF
00168 *
00169 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00170 *
00171       NZ = N + 1
00172       EPS = DLAMCH( 'Epsilon' )
00173       SAFMIN = DLAMCH( 'Safe minimum' )
00174       SAFE1 = NZ*SAFMIN
00175       SAFE2 = SAFE1 / EPS
00176 *
00177 *     Do for each right hand side
00178 *
00179       DO 140 J = 1, NRHS
00180 *
00181          COUNT = 1
00182          LSTRES = THREE
00183    20    CONTINUE
00184 *
00185 *        Loop until stopping criterion is satisfied.
00186 *
00187 *        Compute residual R = B - A * X
00188 *
00189          CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
00190          CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 )
00191 *
00192 *        Compute componentwise relative backward error from formula
00193 *
00194 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00195 *
00196 *        where abs(Z) is the componentwise absolute value of the matrix
00197 *        or vector Z.  If the i-th component of the denominator is less
00198 *        than SAFE2, then SAFE1 is added to the i-th components of the
00199 *        numerator and denominator before dividing.
00200 *
00201          DO 30 I = 1, N
00202             RWORK( I ) = CABS1( B( I, J ) )
00203    30    CONTINUE
00204 *
00205 *        Compute abs(A)*abs(X) + abs(B).
00206 *
00207          KK = 1
00208          IF( UPPER ) THEN
00209             DO 50 K = 1, N
00210                S = ZERO
00211                XK = CABS1( X( K, J ) )
00212                IK = KK
00213                DO 40 I = 1, K - 1
00214                   RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
00215                   S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
00216                   IK = IK + 1
00217    40          CONTINUE
00218                RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
00219      $                      XK + S
00220                KK = KK + K
00221    50       CONTINUE
00222          ELSE
00223             DO 70 K = 1, N
00224                S = ZERO
00225                XK = CABS1( X( K, J ) )
00226                RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
00227                IK = KK + 1
00228                DO 60 I = K + 1, N
00229                   RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
00230                   S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
00231                   IK = IK + 1
00232    60          CONTINUE
00233                RWORK( K ) = RWORK( K ) + S
00234                KK = KK + ( N-K+1 )
00235    70       CONTINUE
00236          END IF
00237          S = ZERO
00238          DO 80 I = 1, N
00239             IF( RWORK( I ).GT.SAFE2 ) THEN
00240                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
00241             ELSE
00242                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
00243      $             ( RWORK( I )+SAFE1 ) )
00244             END IF
00245    80    CONTINUE
00246          BERR( J ) = S
00247 *
00248 *        Test stopping criterion. Continue iterating if
00249 *           1) The residual BERR(J) is larger than machine epsilon, and
00250 *           2) BERR(J) decreased by at least a factor of 2 during the
00251 *              last iteration, and
00252 *           3) At most ITMAX iterations tried.
00253 *
00254          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00255      $       COUNT.LE.ITMAX ) THEN
00256 *
00257 *           Update solution and try again.
00258 *
00259             CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
00260             CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
00261             LSTRES = BERR( J )
00262             COUNT = COUNT + 1
00263             GO TO 20
00264          END IF
00265 *
00266 *        Bound error from formula
00267 *
00268 *        norm(X - XTRUE) / norm(X) .le. FERR =
00269 *        norm( abs(inv(A))*
00270 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00271 *
00272 *        where
00273 *          norm(Z) is the magnitude of the largest component of Z
00274 *          inv(A) is the inverse of A
00275 *          abs(Z) is the componentwise absolute value of the matrix or
00276 *             vector Z
00277 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00278 *          EPS is machine epsilon
00279 *
00280 *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
00281 *        is incremented by SAFE1 if the i-th component of
00282 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00283 *
00284 *        Use ZLACN2 to estimate the infinity-norm of the matrix
00285 *           inv(A) * diag(W),
00286 *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
00287 *
00288          DO 90 I = 1, N
00289             IF( RWORK( I ).GT.SAFE2 ) THEN
00290                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
00291             ELSE
00292                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
00293      $                      SAFE1
00294             END IF
00295    90    CONTINUE
00296 *
00297          KASE = 0
00298   100    CONTINUE
00299          CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
00300          IF( KASE.NE.0 ) THEN
00301             IF( KASE.EQ.1 ) THEN
00302 *
00303 *              Multiply by diag(W)*inv(A').
00304 *
00305                CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
00306                DO 110 I = 1, N
00307                   WORK( I ) = RWORK( I )*WORK( I )
00308   110          CONTINUE
00309             ELSE IF( KASE.EQ.2 ) THEN
00310 *
00311 *              Multiply by inv(A)*diag(W).
00312 *
00313                DO 120 I = 1, N
00314                   WORK( I ) = RWORK( I )*WORK( I )
00315   120          CONTINUE
00316                CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
00317             END IF
00318             GO TO 100
00319          END IF
00320 *
00321 *        Normalize error.
00322 *
00323          LSTRES = ZERO
00324          DO 130 I = 1, N
00325             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
00326   130    CONTINUE
00327          IF( LSTRES.NE.ZERO )
00328      $      FERR( J ) = FERR( J ) / LSTRES
00329 *
00330   140 CONTINUE
00331 *
00332       RETURN
00333 *
00334 *     End of ZPPRFS
00335 *
00336       END