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    1:       SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
    2:      $                   LDX, FERR, BERR, WORK, RWORK, INFO )
    3: *
    4: *  -- LAPACK routine (version 3.2) --
    5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    7: *     November 2006
    8: *
    9: *     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
   10: *
   11: *     .. Scalar Arguments ..
   12:       CHARACTER          UPLO
   13:       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
   14: *     ..
   15: *     .. Array Arguments ..
   16:       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
   17:       COMPLEX*16         A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
   18:      $                   WORK( * ), X( LDX, * )
   19: *     ..
   20: *
   21: *  Purpose
   22: *  =======
   23: *
   24: *  ZPORFS improves the computed solution to a system of linear
   25: *  equations when the coefficient matrix is Hermitian positive definite,
   26: *  and provides error bounds and backward error estimates for the
   27: *  solution.
   28: *
   29: *  Arguments
   30: *  =========
   31: *
   32: *  UPLO    (input) CHARACTER*1
   33: *          = 'U':  Upper triangle of A is stored;
   34: *          = 'L':  Lower triangle of A is stored.
   35: *
   36: *  N       (input) INTEGER
   37: *          The order of the matrix A.  N >= 0.
   38: *
   39: *  NRHS    (input) INTEGER
   40: *          The number of right hand sides, i.e., the number of columns
   41: *          of the matrices B and X.  NRHS >= 0.
   42: *
   43: *  A       (input) COMPLEX*16 array, dimension (LDA,N)
   44: *          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N
   45: *          upper triangular part of A contains the upper triangular part
   46: *          of the matrix A, and the strictly lower triangular part of A
   47: *          is not referenced.  If UPLO = 'L', the leading N-by-N lower
   48: *          triangular part of A contains the lower triangular part of
   49: *          the matrix A, and the strictly upper triangular part of A is
   50: *          not referenced.
   51: *
   52: *  LDA     (input) INTEGER
   53: *          The leading dimension of the array A.  LDA >= max(1,N).
   54: *
   55: *  AF      (input) COMPLEX*16 array, dimension (LDAF,N)
   56: *          The triangular factor U or L from the Cholesky factorization
   57: *          A = U**H*U or A = L*L**H, as computed by ZPOTRF.
   58: *
   59: *  LDAF    (input) INTEGER
   60: *          The leading dimension of the array AF.  LDAF >= max(1,N).
   61: *
   62: *  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
   63: *          The right hand side matrix B.
   64: *
   65: *  LDB     (input) INTEGER
   66: *          The leading dimension of the array B.  LDB >= max(1,N).
   67: *
   68: *  X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
   69: *          On entry, the solution matrix X, as computed by ZPOTRS.
   70: *          On exit, the improved solution matrix X.
   71: *
   72: *  LDX     (input) INTEGER
   73: *          The leading dimension of the array X.  LDX >= max(1,N).
   74: *
   75: *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
   76: *          The estimated forward error bound for each solution vector
   77: *          X(j) (the j-th column of the solution matrix X).
   78: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
   79: *          is an estimated upper bound for the magnitude of the largest
   80: *          element in (X(j) - XTRUE) divided by the magnitude of the
   81: *          largest element in X(j).  The estimate is as reliable as
   82: *          the estimate for RCOND, and is almost always a slight
   83: *          overestimate of the true error.
   84: *
   85: *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
   86: *          The componentwise relative backward error of each solution
   87: *          vector X(j) (i.e., the smallest relative change in
   88: *          any element of A or B that makes X(j) an exact solution).
   89: *
   90: *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
   91: *
   92: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
   93: *
   94: *  INFO    (output) INTEGER
   95: *          = 0:  successful exit
   96: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   97: *
   98: *  Internal Parameters
   99: *  ===================
  100: *
  101: *  ITMAX is the maximum number of steps of iterative refinement.
  102: *
  103: *  ====================================================================
  104: *
  105: *     .. Parameters ..
  106:       INTEGER            ITMAX
  107:       PARAMETER          ( ITMAX = 5 )
  108:       DOUBLE PRECISION   ZERO
  109:       PARAMETER          ( ZERO = 0.0D+0 )
  110:       COMPLEX*16         ONE
  111:       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
  112:       DOUBLE PRECISION   TWO
  113:       PARAMETER          ( TWO = 2.0D+0 )
  114:       DOUBLE PRECISION   THREE
  115:       PARAMETER          ( THREE = 3.0D+0 )
  116: *     ..
  117: *     .. Local Scalars ..
  118:       LOGICAL            UPPER
  119:       INTEGER            COUNT, I, J, K, KASE, NZ
  120:       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
  121:       COMPLEX*16         ZDUM
  122: *     ..
  123: *     .. Local Arrays ..
  124:       INTEGER            ISAVE( 3 )
  125: *     ..
  126: *     .. External Subroutines ..
  127:       EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
  128: *     ..
  129: *     .. Intrinsic Functions ..
  130:       INTRINSIC          ABS, DBLE, DIMAG, MAX
  131: *     ..
  132: *     .. External Functions ..
  133:       LOGICAL            LSAME
  134:       DOUBLE PRECISION   DLAMCH
  135:       EXTERNAL           LSAME, DLAMCH
  136: *     ..
  137: *     .. Statement Functions ..
  138:       DOUBLE PRECISION   CABS1
  139: *     ..
  140: *     .. Statement Function definitions ..
  141:       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
  142: *     ..
  143: *     .. Executable Statements ..
  144: *
  145: *     Test the input parameters.
  146: *
  147:       INFO = 0
  148:       UPPER = LSAME( UPLO, 'U' )
  149:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  150:          INFO = -1
  151:       ELSE IF( N.LT.0 ) THEN
  152:          INFO = -2
  153:       ELSE IF( NRHS.LT.0 ) THEN
  154:          INFO = -3
  155:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  156:          INFO = -5
  157:       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
  158:          INFO = -7
  159:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  160:          INFO = -9
  161:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
  162:          INFO = -11
  163:       END IF
  164:       IF( INFO.NE.0 ) THEN
  165:          CALL XERBLA( 'ZPORFS', -INFO )
  166:          RETURN
  167:       END IF
  168: *
  169: *     Quick return if possible
  170: *
  171:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
  172:          DO 10 J = 1, NRHS
  173:             FERR( J ) = ZERO
  174:             BERR( J ) = ZERO
  175:    10    CONTINUE
  176:          RETURN
  177:       END IF
  178: *
  179: *     NZ = maximum number of nonzero elements in each row of A, plus 1
  180: *
  181:       NZ = N + 1
  182:       EPS = DLAMCH( 'Epsilon' )
  183:       SAFMIN = DLAMCH( 'Safe minimum' )
  184:       SAFE1 = NZ*SAFMIN
  185:       SAFE2 = SAFE1 / EPS
  186: *
  187: *     Do for each right hand side
  188: *
  189:       DO 140 J = 1, NRHS
  190: *
  191:          COUNT = 1
  192:          LSTRES = THREE
  193:    20    CONTINUE
  194: *
  195: *        Loop until stopping criterion is satisfied.
  196: *
  197: *        Compute residual R = B - A * X
  198: *
  199:          CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
  200:          CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
  201: *
  202: *        Compute componentwise relative backward error from formula
  203: *
  204: *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
  205: *
  206: *        where abs(Z) is the componentwise absolute value of the matrix
  207: *        or vector Z.  If the i-th component of the denominator is less
  208: *        than SAFE2, then SAFE1 is added to the i-th components of the
  209: *        numerator and denominator before dividing.
  210: *
  211:          DO 30 I = 1, N
  212:             RWORK( I ) = CABS1( B( I, J ) )
  213:    30    CONTINUE
  214: *
  215: *        Compute abs(A)*abs(X) + abs(B).
  216: *
  217:          IF( UPPER ) THEN
  218:             DO 50 K = 1, N
  219:                S = ZERO
  220:                XK = CABS1( X( K, J ) )
  221:                DO 40 I = 1, K - 1
  222:                   RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
  223:                   S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
  224:    40          CONTINUE
  225:                RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
  226:    50       CONTINUE
  227:          ELSE
  228:             DO 70 K = 1, N
  229:                S = ZERO
  230:                XK = CABS1( X( K, J ) )
  231:                RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
  232:                DO 60 I = K + 1, N
  233:                   RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
  234:                   S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
  235:    60          CONTINUE
  236:                RWORK( K ) = RWORK( K ) + S
  237:    70       CONTINUE
  238:          END IF
  239:          S = ZERO
  240:          DO 80 I = 1, N
  241:             IF( RWORK( I ).GT.SAFE2 ) THEN
  242:                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
  243:             ELSE
  244:                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
  245:      $             ( RWORK( I )+SAFE1 ) )
  246:             END IF
  247:    80    CONTINUE
  248:          BERR( J ) = S
  249: *
  250: *        Test stopping criterion. Continue iterating if
  251: *           1) The residual BERR(J) is larger than machine epsilon, and
  252: *           2) BERR(J) decreased by at least a factor of 2 during the
  253: *              last iteration, and
  254: *           3) At most ITMAX iterations tried.
  255: *
  256:          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
  257:      $       COUNT.LE.ITMAX ) THEN
  258: *
  259: *           Update solution and try again.
  260: *
  261:             CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
  262:             CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
  263:             LSTRES = BERR( J )
  264:             COUNT = COUNT + 1
  265:             GO TO 20
  266:          END IF
  267: *
  268: *        Bound error from formula
  269: *
  270: *        norm(X - XTRUE) / norm(X) .le. FERR =
  271: *        norm( abs(inv(A))*
  272: *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
  273: *
  274: *        where
  275: *          norm(Z) is the magnitude of the largest component of Z
  276: *          inv(A) is the inverse of A
  277: *          abs(Z) is the componentwise absolute value of the matrix or
  278: *             vector Z
  279: *          NZ is the maximum number of nonzeros in any row of A, plus 1
  280: *          EPS is machine epsilon
  281: *
  282: *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
  283: *        is incremented by SAFE1 if the i-th component of
  284: *        abs(A)*abs(X) + abs(B) is less than SAFE2.
  285: *
  286: *        Use ZLACN2 to estimate the infinity-norm of the matrix
  287: *           inv(A) * diag(W),
  288: *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
  289: *
  290:          DO 90 I = 1, N
  291:             IF( RWORK( I ).GT.SAFE2 ) THEN
  292:                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
  293:             ELSE
  294:                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
  295:      $                      SAFE1
  296:             END IF
  297:    90    CONTINUE
  298: *
  299:          KASE = 0
  300:   100    CONTINUE
  301:          CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
  302:          IF( KASE.NE.0 ) THEN
  303:             IF( KASE.EQ.1 ) THEN
  304: *
  305: *              Multiply by diag(W)*inv(A').
  306: *
  307:                CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
  308:                DO 110 I = 1, N
  309:                   WORK( I ) = RWORK( I )*WORK( I )
  310:   110          CONTINUE
  311:             ELSE IF( KASE.EQ.2 ) THEN
  312: *
  313: *              Multiply by inv(A)*diag(W).
  314: *
  315:                DO 120 I = 1, N
  316:                   WORK( I ) = RWORK( I )*WORK( I )
  317:   120          CONTINUE
  318:                CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
  319:             END IF
  320:             GO TO 100
  321:          END IF
  322: *
  323: *        Normalize error.
  324: *
  325:          LSTRES = ZERO
  326:          DO 130 I = 1, N
  327:             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
  328:   130    CONTINUE
  329:          IF( LSTRES.NE.ZERO )
  330:      $      FERR( J ) = FERR( J ) / LSTRES
  331: *
  332:   140 CONTINUE
  333: *
  334:       RETURN
  335: *
  336: *     End of ZPORFS
  337: *
  338:       END