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    1: *> \brief \b ZTRRFS
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download ZTRRFS + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztrrfs.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztrrfs.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztrrfs.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
   22: *                          LDX, FERR, BERR, WORK, RWORK, INFO )
   23: * 
   24: *       .. Scalar Arguments ..
   25: *       CHARACTER          DIAG, TRANS, UPLO
   26: *       INTEGER            INFO, LDA, LDB, LDX, N, NRHS
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
   30: *       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
   31: *      $                   X( LDX, * )
   32: *       ..
   33: *  
   34: *
   35: *> \par Purpose:
   36: *  =============
   37: *>
   38: *> \verbatim
   39: *>
   40: *> ZTRRFS provides error bounds and backward error estimates for the
   41: *> solution to a system of linear equations with a triangular
   42: *> coefficient matrix.
   43: *>
   44: *> The solution matrix X must be computed by ZTRTRS or some other
   45: *> means before entering this routine.  ZTRRFS does not do iterative
   46: *> refinement because doing so cannot improve the backward error.
   47: *> \endverbatim
   48: *
   49: *  Arguments:
   50: *  ==========
   51: *
   52: *> \param[in] UPLO
   53: *> \verbatim
   54: *>          UPLO is CHARACTER*1
   55: *>          = 'U':  A is upper triangular;
   56: *>          = 'L':  A is lower triangular.
   57: *> \endverbatim
   58: *>
   59: *> \param[in] TRANS
   60: *> \verbatim
   61: *>          TRANS is CHARACTER*1
   62: *>          Specifies the form of the system of equations:
   63: *>          = 'N':  A * X = B     (No transpose)
   64: *>          = 'T':  A**T * X = B  (Transpose)
   65: *>          = 'C':  A**H * X = B  (Conjugate transpose)
   66: *> \endverbatim
   67: *>
   68: *> \param[in] DIAG
   69: *> \verbatim
   70: *>          DIAG is CHARACTER*1
   71: *>          = 'N':  A is non-unit triangular;
   72: *>          = 'U':  A is unit triangular.
   73: *> \endverbatim
   74: *>
   75: *> \param[in] N
   76: *> \verbatim
   77: *>          N is INTEGER
   78: *>          The order of the matrix A.  N >= 0.
   79: *> \endverbatim
   80: *>
   81: *> \param[in] NRHS
   82: *> \verbatim
   83: *>          NRHS is INTEGER
   84: *>          The number of right hand sides, i.e., the number of columns
   85: *>          of the matrices B and X.  NRHS >= 0.
   86: *> \endverbatim
   87: *>
   88: *> \param[in] A
   89: *> \verbatim
   90: *>          A is COMPLEX*16 array, dimension (LDA,N)
   91: *>          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
   92: *>          upper triangular part of the array A contains the upper
   93: *>          triangular matrix, and the strictly lower triangular part of
   94: *>          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
   95: *>          triangular part of the array A contains the lower triangular
   96: *>          matrix, and the strictly upper triangular part of A is not
   97: *>          referenced.  If DIAG = 'U', the diagonal elements of A are
   98: *>          also not referenced and are assumed to be 1.
   99: *> \endverbatim
  100: *>
  101: *> \param[in] LDA
  102: *> \verbatim
  103: *>          LDA is INTEGER
  104: *>          The leading dimension of the array A.  LDA >= max(1,N).
  105: *> \endverbatim
  106: *>
  107: *> \param[in] B
  108: *> \verbatim
  109: *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
  110: *>          The right hand side matrix B.
  111: *> \endverbatim
  112: *>
  113: *> \param[in] LDB
  114: *> \verbatim
  115: *>          LDB is INTEGER
  116: *>          The leading dimension of the array B.  LDB >= max(1,N).
  117: *> \endverbatim
  118: *>
  119: *> \param[in] X
  120: *> \verbatim
  121: *>          X is COMPLEX*16 array, dimension (LDX,NRHS)
  122: *>          The solution matrix X.
  123: *> \endverbatim
  124: *>
  125: *> \param[in] LDX
  126: *> \verbatim
  127: *>          LDX is INTEGER
  128: *>          The leading dimension of the array X.  LDX >= max(1,N).
  129: *> \endverbatim
  130: *>
  131: *> \param[out] FERR
  132: *> \verbatim
  133: *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
  134: *>          The estimated forward error bound for each solution vector
  135: *>          X(j) (the j-th column of the solution matrix X).
  136: *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
  137: *>          is an estimated upper bound for the magnitude of the largest
  138: *>          element in (X(j) - XTRUE) divided by the magnitude of the
  139: *>          largest element in X(j).  The estimate is as reliable as
  140: *>          the estimate for RCOND, and is almost always a slight
  141: *>          overestimate of the true error.
  142: *> \endverbatim
  143: *>
  144: *> \param[out] BERR
  145: *> \verbatim
  146: *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
  147: *>          The componentwise relative backward error of each solution
  148: *>          vector X(j) (i.e., the smallest relative change in
  149: *>          any element of A or B that makes X(j) an exact solution).
  150: *> \endverbatim
  151: *>
  152: *> \param[out] WORK
  153: *> \verbatim
  154: *>          WORK is COMPLEX*16 array, dimension (2*N)
  155: *> \endverbatim
  156: *>
  157: *> \param[out] RWORK
  158: *> \verbatim
  159: *>          RWORK is DOUBLE PRECISION array, dimension (N)
  160: *> \endverbatim
  161: *>
  162: *> \param[out] INFO
  163: *> \verbatim
  164: *>          INFO is INTEGER
  165: *>          = 0:  successful exit
  166: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  167: *> \endverbatim
  168: *
  169: *  Authors:
  170: *  ========
  171: *
  172: *> \author Univ. of Tennessee 
  173: *> \author Univ. of California Berkeley 
  174: *> \author Univ. of Colorado Denver 
  175: *> \author NAG Ltd. 
  176: *
  177: *> \date November 2011
  178: *
  179: *> \ingroup complex16OTHERcomputational
  180: *
  181: *  =====================================================================
  182:       SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
  183:      $                   LDX, FERR, BERR, WORK, RWORK, INFO )
  184: *
  185: *  -- LAPACK computational routine (version 3.4.0) --
  186: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  187: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  188: *     November 2011
  189: *
  190: *     .. Scalar Arguments ..
  191:       CHARACTER          DIAG, TRANS, UPLO
  192:       INTEGER            INFO, LDA, LDB, LDX, N, NRHS
  193: *     ..
  194: *     .. Array Arguments ..
  195:       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
  196:       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
  197:      $                   X( LDX, * )
  198: *     ..
  199: *
  200: *  =====================================================================
  201: *
  202: *     .. Parameters ..
  203:       DOUBLE PRECISION   ZERO
  204:       PARAMETER          ( ZERO = 0.0D+0 )
  205:       COMPLEX*16         ONE
  206:       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
  207: *     ..
  208: *     .. Local Scalars ..
  209:       LOGICAL            NOTRAN, NOUNIT, UPPER
  210:       CHARACTER          TRANSN, TRANST
  211:       INTEGER            I, J, K, KASE, NZ
  212:       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
  213:       COMPLEX*16         ZDUM
  214: *     ..
  215: *     .. Local Arrays ..
  216:       INTEGER            ISAVE( 3 )
  217: *     ..
  218: *     .. External Subroutines ..
  219:       EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTRMV, ZTRSV
  220: *     ..
  221: *     .. Intrinsic Functions ..
  222:       INTRINSIC          ABS, DBLE, DIMAG, MAX
  223: *     ..
  224: *     .. External Functions ..
  225:       LOGICAL            LSAME
  226:       DOUBLE PRECISION   DLAMCH
  227:       EXTERNAL           LSAME, DLAMCH
  228: *     ..
  229: *     .. Statement Functions ..
  230:       DOUBLE PRECISION   CABS1
  231: *     ..
  232: *     .. Statement Function definitions ..
  233:       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
  234: *     ..
  235: *     .. Executable Statements ..
  236: *
  237: *     Test the input parameters.
  238: *
  239:       INFO = 0
  240:       UPPER = LSAME( UPLO, 'U' )
  241:       NOTRAN = LSAME( TRANS, 'N' )
  242:       NOUNIT = LSAME( DIAG, 'N' )
  243: *
  244:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  245:          INFO = -1
  246:       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
  247:      $         LSAME( TRANS, 'C' ) ) THEN
  248:          INFO = -2
  249:       ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
  250:          INFO = -3
  251:       ELSE IF( N.LT.0 ) THEN
  252:          INFO = -4
  253:       ELSE IF( NRHS.LT.0 ) THEN
  254:          INFO = -5
  255:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  256:          INFO = -7
  257:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  258:          INFO = -9
  259:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
  260:          INFO = -11
  261:       END IF
  262:       IF( INFO.NE.0 ) THEN
  263:          CALL XERBLA( 'ZTRRFS', -INFO )
  264:          RETURN
  265:       END IF
  266: *
  267: *     Quick return if possible
  268: *
  269:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
  270:          DO 10 J = 1, NRHS
  271:             FERR( J ) = ZERO
  272:             BERR( J ) = ZERO
  273:    10    CONTINUE
  274:          RETURN
  275:       END IF
  276: *
  277:       IF( NOTRAN ) THEN
  278:          TRANSN = 'N'
  279:          TRANST = 'C'
  280:       ELSE
  281:          TRANSN = 'C'
  282:          TRANST = 'N'
  283:       END IF
  284: *
  285: *     NZ = maximum number of nonzero elements in each row of A, plus 1
  286: *
  287:       NZ = N + 1
  288:       EPS = DLAMCH( 'Epsilon' )
  289:       SAFMIN = DLAMCH( 'Safe minimum' )
  290:       SAFE1 = NZ*SAFMIN
  291:       SAFE2 = SAFE1 / EPS
  292: *
  293: *     Do for each right hand side
  294: *
  295:       DO 250 J = 1, NRHS
  296: *
  297: *        Compute residual R = B - op(A) * X,
  298: *        where op(A) = A, A**T, or A**H, depending on TRANS.
  299: *
  300:          CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
  301:          CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
  302:          CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
  303: *
  304: *        Compute componentwise relative backward error from formula
  305: *
  306: *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
  307: *
  308: *        where abs(Z) is the componentwise absolute value of the matrix
  309: *        or vector Z.  If the i-th component of the denominator is less
  310: *        than SAFE2, then SAFE1 is added to the i-th components of the
  311: *        numerator and denominator before dividing.
  312: *
  313:          DO 20 I = 1, N
  314:             RWORK( I ) = CABS1( B( I, J ) )
  315:    20    CONTINUE
  316: *
  317:          IF( NOTRAN ) THEN
  318: *
  319: *           Compute abs(A)*abs(X) + abs(B).
  320: *
  321:             IF( UPPER ) THEN
  322:                IF( NOUNIT ) THEN
  323:                   DO 40 K = 1, N
  324:                      XK = CABS1( X( K, J ) )
  325:                      DO 30 I = 1, K
  326:                         RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
  327:    30                CONTINUE
  328:    40             CONTINUE
  329:                ELSE
  330:                   DO 60 K = 1, N
  331:                      XK = CABS1( X( K, J ) )
  332:                      DO 50 I = 1, K - 1
  333:                         RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
  334:    50                CONTINUE
  335:                      RWORK( K ) = RWORK( K ) + XK
  336:    60             CONTINUE
  337:                END IF
  338:             ELSE
  339:                IF( NOUNIT ) THEN
  340:                   DO 80 K = 1, N
  341:                      XK = CABS1( X( K, J ) )
  342:                      DO 70 I = K, N
  343:                         RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
  344:    70                CONTINUE
  345:    80             CONTINUE
  346:                ELSE
  347:                   DO 100 K = 1, N
  348:                      XK = CABS1( X( K, J ) )
  349:                      DO 90 I = K + 1, N
  350:                         RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
  351:    90                CONTINUE
  352:                      RWORK( K ) = RWORK( K ) + XK
  353:   100             CONTINUE
  354:                END IF
  355:             END IF
  356:          ELSE
  357: *
  358: *           Compute abs(A**H)*abs(X) + abs(B).
  359: *
  360:             IF( UPPER ) THEN
  361:                IF( NOUNIT ) THEN
  362:                   DO 120 K = 1, N
  363:                      S = ZERO
  364:                      DO 110 I = 1, K
  365:                         S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
  366:   110                CONTINUE
  367:                      RWORK( K ) = RWORK( K ) + S
  368:   120             CONTINUE
  369:                ELSE
  370:                   DO 140 K = 1, N
  371:                      S = CABS1( X( K, J ) )
  372:                      DO 130 I = 1, K - 1
  373:                         S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
  374:   130                CONTINUE
  375:                      RWORK( K ) = RWORK( K ) + S
  376:   140             CONTINUE
  377:                END IF
  378:             ELSE
  379:                IF( NOUNIT ) THEN
  380:                   DO 160 K = 1, N
  381:                      S = ZERO
  382:                      DO 150 I = K, N
  383:                         S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
  384:   150                CONTINUE
  385:                      RWORK( K ) = RWORK( K ) + S
  386:   160             CONTINUE
  387:                ELSE
  388:                   DO 180 K = 1, N
  389:                      S = CABS1( X( K, J ) )
  390:                      DO 170 I = K + 1, N
  391:                         S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
  392:   170                CONTINUE
  393:                      RWORK( K ) = RWORK( K ) + S
  394:   180             CONTINUE
  395:                END IF
  396:             END IF
  397:          END IF
  398:          S = ZERO
  399:          DO 190 I = 1, N
  400:             IF( RWORK( I ).GT.SAFE2 ) THEN
  401:                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
  402:             ELSE
  403:                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
  404:      $             ( RWORK( I )+SAFE1 ) )
  405:             END IF
  406:   190    CONTINUE
  407:          BERR( J ) = S
  408: *
  409: *        Bound error from formula
  410: *
  411: *        norm(X - XTRUE) / norm(X) .le. FERR =
  412: *        norm( abs(inv(op(A)))*
  413: *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
  414: *
  415: *        where
  416: *          norm(Z) is the magnitude of the largest component of Z
  417: *          inv(op(A)) is the inverse of op(A)
  418: *          abs(Z) is the componentwise absolute value of the matrix or
  419: *             vector Z
  420: *          NZ is the maximum number of nonzeros in any row of A, plus 1
  421: *          EPS is machine epsilon
  422: *
  423: *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
  424: *        is incremented by SAFE1 if the i-th component of
  425: *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
  426: *
  427: *        Use ZLACN2 to estimate the infinity-norm of the matrix
  428: *           inv(op(A)) * diag(W),
  429: *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
  430: *
  431:          DO 200 I = 1, N
  432:             IF( RWORK( I ).GT.SAFE2 ) THEN
  433:                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
  434:             ELSE
  435:                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
  436:      $                      SAFE1
  437:             END IF
  438:   200    CONTINUE
  439: *
  440:          KASE = 0
  441:   210    CONTINUE
  442:          CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
  443:          IF( KASE.NE.0 ) THEN
  444:             IF( KASE.EQ.1 ) THEN
  445: *
  446: *              Multiply by diag(W)*inv(op(A)**H).
  447: *
  448:                CALL ZTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK, 1 )
  449:                DO 220 I = 1, N
  450:                   WORK( I ) = RWORK( I )*WORK( I )
  451:   220          CONTINUE
  452:             ELSE
  453: *
  454: *              Multiply by inv(op(A))*diag(W).
  455: *
  456:                DO 230 I = 1, N
  457:                   WORK( I ) = RWORK( I )*WORK( I )
  458:   230          CONTINUE
  459:                CALL ZTRSV( UPLO, TRANSN, DIAG, N, A, LDA, WORK, 1 )
  460:             END IF
  461:             GO TO 210
  462:          END IF
  463: *
  464: *        Normalize error.
  465: *
  466:          LSTRES = ZERO
  467:          DO 240 I = 1, N
  468:             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
  469:   240    CONTINUE
  470:          IF( LSTRES.NE.ZERO )
  471:      $      FERR( J ) = FERR( J ) / LSTRES
  472: *
  473:   250 CONTINUE
  474: *
  475:       RETURN
  476: *
  477: *     End of ZTRRFS
  478: *
  479:       END