Return to dgerfs.f CVS log

Up to [local] / rpl / lapack / lapack

Mise à jour de lapack vers la version 3.3.0.

    1:       SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
    2:      $                   X, LDX, FERR, BERR, WORK, IWORK, 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 DLACN2 in place of DLACON, 5 Feb 03, SJH.
   10: *
   11: *     .. Scalar Arguments ..
   12:       CHARACTER          TRANS
   13:       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
   14: *     ..
   15: *     .. Array Arguments ..
   16:       INTEGER            IPIV( * ), IWORK( * )
   17:       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
   18:      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
   19: *     ..
   20: *
   21: *  Purpose
   22: *  =======
   23: *
   24: *  DGERFS improves the computed solution to a system of linear
   25: *  equations and provides error bounds and backward error estimates for
   26: *  the solution.
   27: *
   28: *  Arguments
   29: *  =========
   30: *
   31: *  TRANS   (input) CHARACTER*1
   32: *          Specifies the form of the system of equations:
   33: *          = 'N':  A * X = B     (No transpose)
   34: *          = 'T':  A**T * X = B  (Transpose)
   35: *          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
   36: *
   37: *  N       (input) INTEGER
   38: *          The order of the matrix A.  N >= 0.
   39: *
   40: *  NRHS    (input) INTEGER
   41: *          The number of right hand sides, i.e., the number of columns
   42: *          of the matrices B and X.  NRHS >= 0.
   43: *
   44: *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
   45: *          The original N-by-N matrix A.
   46: *
   47: *  LDA     (input) INTEGER
   48: *          The leading dimension of the array A.  LDA >= max(1,N).
   49: *
   50: *  AF      (input) DOUBLE PRECISION array, dimension (LDAF,N)
   51: *          The factors L and U from the factorization A = P*L*U
   52: *          as computed by DGETRF.
   53: *
   54: *  LDAF    (input) INTEGER
   55: *          The leading dimension of the array AF.  LDAF >= max(1,N).
   56: *
   57: *  IPIV    (input) INTEGER array, dimension (N)
   58: *          The pivot indices from DGETRF; for 1<=i<=N, row i of the
   59: *          matrix was interchanged with row IPIV(i).
   60: *
   61: *  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
   62: *          The right hand side matrix B.
   63: *
   64: *  LDB     (input) INTEGER
   65: *          The leading dimension of the array B.  LDB >= max(1,N).
   66: *
   67: *  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
   68: *          On entry, the solution matrix X, as computed by DGETRS.
   69: *          On exit, the improved solution matrix X.
   70: *
   71: *  LDX     (input) INTEGER
   72: *          The leading dimension of the array X.  LDX >= max(1,N).
   73: *
   74: *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
   75: *          The estimated forward error bound for each solution vector
   76: *          X(j) (the j-th column of the solution matrix X).
   77: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
   78: *          is an estimated upper bound for the magnitude of the largest
   79: *          element in (X(j) - XTRUE) divided by the magnitude of the
   80: *          largest element in X(j).  The estimate is as reliable as
   81: *          the estimate for RCOND, and is almost always a slight
   82: *          overestimate of the true error.
   83: *
   84: *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
   85: *          The componentwise relative backward error of each solution
   86: *          vector X(j) (i.e., the smallest relative change in
   87: *          any element of A or B that makes X(j) an exact solution).
   88: *
   89: *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
   90: *
   91: *  IWORK   (workspace) INTEGER array, dimension (N)
   92: *
   93: *  INFO    (output) INTEGER
   94: *          = 0:  successful exit
   95: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   96: *
   97: *  Internal Parameters
   98: *  ===================
   99: *
  100: *  ITMAX is the maximum number of steps of iterative refinement.
  101: *
  102: *  =====================================================================
  103: *
  104: *     .. Parameters ..
  105:       INTEGER            ITMAX
  106:       PARAMETER          ( ITMAX = 5 )
  107:       DOUBLE PRECISION   ZERO
  108:       PARAMETER          ( ZERO = 0.0D+0 )
  109:       DOUBLE PRECISION   ONE
  110:       PARAMETER          ( ONE = 1.0D+0 )
  111:       DOUBLE PRECISION   TWO
  112:       PARAMETER          ( TWO = 2.0D+0 )
  113:       DOUBLE PRECISION   THREE
  114:       PARAMETER          ( THREE = 3.0D+0 )
  115: *     ..
  116: *     .. Local Scalars ..
  117:       LOGICAL            NOTRAN
  118:       CHARACTER          TRANST
  119:       INTEGER            COUNT, I, J, K, KASE, NZ
  120:       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
  121: *     ..
  122: *     .. Local Arrays ..
  123:       INTEGER            ISAVE( 3 )
  124: *     ..
  125: *     .. External Subroutines ..
  126:       EXTERNAL           DAXPY, DCOPY, DGEMV, DGETRS, DLACN2, XERBLA
  127: *     ..
  128: *     .. Intrinsic Functions ..
  129:       INTRINSIC          ABS, MAX
  130: *     ..
  131: *     .. External Functions ..
  132:       LOGICAL            LSAME
  133:       DOUBLE PRECISION   DLAMCH
  134:       EXTERNAL           LSAME, DLAMCH
  135: *     ..
  136: *     .. Executable Statements ..
  137: *
  138: *     Test the input parameters.
  139: *
  140:       INFO = 0
  141:       NOTRAN = LSAME( TRANS, 'N' )
  142:       IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
  143:      $    LSAME( TRANS, 'C' ) ) THEN
  144:          INFO = -1
  145:       ELSE IF( N.LT.0 ) THEN
  146:          INFO = -2
  147:       ELSE IF( NRHS.LT.0 ) THEN
  148:          INFO = -3
  149:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  150:          INFO = -5
  151:       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
  152:          INFO = -7
  153:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  154:          INFO = -10
  155:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
  156:          INFO = -12
  157:       END IF
  158:       IF( INFO.NE.0 ) THEN
  159:          CALL XERBLA( 'DGERFS', -INFO )
  160:          RETURN
  161:       END IF
  162: *
  163: *     Quick return if possible
  164: *
  165:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
  166:          DO 10 J = 1, NRHS
  167:             FERR( J ) = ZERO
  168:             BERR( J ) = ZERO
  169:    10    CONTINUE
  170:          RETURN
  171:       END IF
  172: *
  173:       IF( NOTRAN ) THEN
  174:          TRANST = 'T'
  175:       ELSE
  176:          TRANST = 'N'
  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 - op(A) * X,
  198: *        where op(A) = A, A**T, or A**H, depending on TRANS.
  199: *
  200:          CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
  201:          CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
  202:      $               WORK( N+1 ), 1 )
  203: *
  204: *        Compute componentwise relative backward error from formula
  205: *
  206: *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
  207: *
  208: *        where abs(Z) is the componentwise absolute value of the matrix
  209: *        or vector Z.  If the i-th component of the denominator is less
  210: *        than SAFE2, then SAFE1 is added to the i-th components of the
  211: *        numerator and denominator before dividing.
  212: *
  213:          DO 30 I = 1, N
  214:             WORK( I ) = ABS( B( I, J ) )
  215:    30    CONTINUE
  216: *
  217: *        Compute abs(op(A))*abs(X) + abs(B).
  218: *
  219:          IF( NOTRAN ) THEN
  220:             DO 50 K = 1, N
  221:                XK = ABS( X( K, J ) )
  222:                DO 40 I = 1, N
  223:                   WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
  224:    40          CONTINUE
  225:    50       CONTINUE
  226:          ELSE
  227:             DO 70 K = 1, N
  228:                S = ZERO
  229:                DO 60 I = 1, N
  230:                   S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
  231:    60          CONTINUE
  232:                WORK( K ) = WORK( K ) + S
  233:    70       CONTINUE
  234:          END IF
  235:          S = ZERO
  236:          DO 80 I = 1, N
  237:             IF( WORK( I ).GT.SAFE2 ) THEN
  238:                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
  239:             ELSE
  240:                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
  241:      $             ( WORK( I )+SAFE1 ) )
  242:             END IF
  243:    80    CONTINUE
  244:          BERR( J ) = S
  245: *
  246: *        Test stopping criterion. Continue iterating if
  247: *           1) The residual BERR(J) is larger than machine epsilon, and
  248: *           2) BERR(J) decreased by at least a factor of 2 during the
  249: *              last iteration, and
  250: *           3) At most ITMAX iterations tried.
  251: *
  252:          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
  253:      $       COUNT.LE.ITMAX ) THEN
  254: *
  255: *           Update solution and try again.
  256: *
  257:             CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
  258:      $                   INFO )
  259:             CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
  260:             LSTRES = BERR( J )
  261:             COUNT = COUNT + 1
  262:             GO TO 20
  263:          END IF
  264: *
  265: *        Bound error from formula
  266: *
  267: *        norm(X - XTRUE) / norm(X) .le. FERR =
  268: *        norm( abs(inv(op(A)))*
  269: *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
  270: *
  271: *        where
  272: *          norm(Z) is the magnitude of the largest component of Z
  273: *          inv(op(A)) is the inverse of op(A)
  274: *          abs(Z) is the componentwise absolute value of the matrix or
  275: *             vector Z
  276: *          NZ is the maximum number of nonzeros in any row of A, plus 1
  277: *          EPS is machine epsilon
  278: *
  279: *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
  280: *        is incremented by SAFE1 if the i-th component of
  281: *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
  282: *
  283: *        Use DLACN2 to estimate the infinity-norm of the matrix
  284: *           inv(op(A)) * diag(W),
  285: *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
  286: *
  287:          DO 90 I = 1, N
  288:             IF( WORK( I ).GT.SAFE2 ) THEN
  289:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
  290:             ELSE
  291:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
  292:             END IF
  293:    90    CONTINUE
  294: *
  295:          KASE = 0
  296:   100    CONTINUE
  297:          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
  298:      $                KASE, ISAVE )
  299:          IF( KASE.NE.0 ) THEN
  300:             IF( KASE.EQ.1 ) THEN
  301: *
  302: *              Multiply by diag(W)*inv(op(A)**T).
  303: *
  304:                CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
  305:      $                      N, INFO )
  306:                DO 110 I = 1, N
  307:                   WORK( N+I ) = WORK( I )*WORK( N+I )
  308:   110          CONTINUE
  309:             ELSE
  310: *
  311: *              Multiply by inv(op(A))*diag(W).
  312: *
  313:                DO 120 I = 1, N
  314:                   WORK( N+I ) = WORK( I )*WORK( N+I )
  315:   120          CONTINUE
  316:                CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
  317:      $                      INFO )
  318:             END IF
  319:             GO TO 100
  320:          END IF
  321: *
  322: *        Normalize error.
  323: *
  324:          LSTRES = ZERO
  325:          DO 130 I = 1, N
  326:             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  327:   130    CONTINUE
  328:          IF( LSTRES.NE.ZERO )
  329:      $      FERR( J ) = FERR( J ) / LSTRES
  330: *
  331:   140 CONTINUE
  332: *
  333:       RETURN
  334: *
  335: *     End of DGERFS
  336: *
  337:       END