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    1: *> \brief \b DTPRFS
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DTPRFS + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtprfs.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtprfs.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtprfs.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
   22: *                          FERR, BERR, WORK, IWORK, INFO )
   23: * 
   24: *       .. Scalar Arguments ..
   25: *       CHARACTER          DIAG, TRANS, UPLO
   26: *       INTEGER            INFO, LDB, LDX, N, NRHS
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       INTEGER            IWORK( * )
   30: *       DOUBLE PRECISION   AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
   31: *      $                   WORK( * ), X( LDX, * )
   32: *       ..
   33: *  
   34: *
   35: *> \par Purpose:
   36: *  =============
   37: *>
   38: *> \verbatim
   39: *>
   40: *> DTPRFS provides error bounds and backward error estimates for the
   41: *> solution to a system of linear equations with a triangular packed
   42: *> coefficient matrix.
   43: *>
   44: *> The solution matrix X must be computed by DTPTRS or some other
   45: *> means before entering this routine.  DTPRFS 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 = 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] AP
   89: *> \verbatim
   90: *>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
   91: *>          The upper or lower triangular matrix A, packed columnwise in
   92: *>          a linear array.  The j-th column of A is stored in the array
   93: *>          AP as follows:
   94: *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   95: *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
   96: *>          If DIAG = 'U', the diagonal elements of A are not referenced
   97: *>          and are assumed to be 1.
   98: *> \endverbatim
   99: *>
  100: *> \param[in] B
  101: *> \verbatim
  102: *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
  103: *>          The right hand side matrix B.
  104: *> \endverbatim
  105: *>
  106: *> \param[in] LDB
  107: *> \verbatim
  108: *>          LDB is INTEGER
  109: *>          The leading dimension of the array B.  LDB >= max(1,N).
  110: *> \endverbatim
  111: *>
  112: *> \param[in] X
  113: *> \verbatim
  114: *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
  115: *>          The solution matrix X.
  116: *> \endverbatim
  117: *>
  118: *> \param[in] LDX
  119: *> \verbatim
  120: *>          LDX is INTEGER
  121: *>          The leading dimension of the array X.  LDX >= max(1,N).
  122: *> \endverbatim
  123: *>
  124: *> \param[out] FERR
  125: *> \verbatim
  126: *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
  127: *>          The estimated forward error bound for each solution vector
  128: *>          X(j) (the j-th column of the solution matrix X).
  129: *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
  130: *>          is an estimated upper bound for the magnitude of the largest
  131: *>          element in (X(j) - XTRUE) divided by the magnitude of the
  132: *>          largest element in X(j).  The estimate is as reliable as
  133: *>          the estimate for RCOND, and is almost always a slight
  134: *>          overestimate of the true error.
  135: *> \endverbatim
  136: *>
  137: *> \param[out] BERR
  138: *> \verbatim
  139: *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
  140: *>          The componentwise relative backward error of each solution
  141: *>          vector X(j) (i.e., the smallest relative change in
  142: *>          any element of A or B that makes X(j) an exact solution).
  143: *> \endverbatim
  144: *>
  145: *> \param[out] WORK
  146: *> \verbatim
  147: *>          WORK is DOUBLE PRECISION array, dimension (3*N)
  148: *> \endverbatim
  149: *>
  150: *> \param[out] IWORK
  151: *> \verbatim
  152: *>          IWORK is INTEGER array, dimension (N)
  153: *> \endverbatim
  154: *>
  155: *> \param[out] INFO
  156: *> \verbatim
  157: *>          INFO is INTEGER
  158: *>          = 0:  successful exit
  159: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  160: *> \endverbatim
  161: *
  162: *  Authors:
  163: *  ========
  164: *
  165: *> \author Univ. of Tennessee 
  166: *> \author Univ. of California Berkeley 
  167: *> \author Univ. of Colorado Denver 
  168: *> \author NAG Ltd. 
  169: *
  170: *> \date November 2011
  171: *
  172: *> \ingroup doubleOTHERcomputational
  173: *
  174: *  =====================================================================
  175:       SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
  176:      $                   FERR, BERR, WORK, IWORK, INFO )
  177: *
  178: *  -- LAPACK computational routine (version 3.4.0) --
  179: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  180: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  181: *     November 2011
  182: *
  183: *     .. Scalar Arguments ..
  184:       CHARACTER          DIAG, TRANS, UPLO
  185:       INTEGER            INFO, LDB, LDX, N, NRHS
  186: *     ..
  187: *     .. Array Arguments ..
  188:       INTEGER            IWORK( * )
  189:       DOUBLE PRECISION   AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
  190:      $                   WORK( * ), X( LDX, * )
  191: *     ..
  192: *
  193: *  =====================================================================
  194: *
  195: *     .. Parameters ..
  196:       DOUBLE PRECISION   ZERO
  197:       PARAMETER          ( ZERO = 0.0D+0 )
  198:       DOUBLE PRECISION   ONE
  199:       PARAMETER          ( ONE = 1.0D+0 )
  200: *     ..
  201: *     .. Local Scalars ..
  202:       LOGICAL            NOTRAN, NOUNIT, UPPER
  203:       CHARACTER          TRANST
  204:       INTEGER            I, J, K, KASE, KC, NZ
  205:       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
  206: *     ..
  207: *     .. Local Arrays ..
  208:       INTEGER            ISAVE( 3 )
  209: *     ..
  210: *     .. External Subroutines ..
  211:       EXTERNAL           DAXPY, DCOPY, DLACN2, DTPMV, DTPSV, XERBLA
  212: *     ..
  213: *     .. Intrinsic Functions ..
  214:       INTRINSIC          ABS, MAX
  215: *     ..
  216: *     .. External Functions ..
  217:       LOGICAL            LSAME
  218:       DOUBLE PRECISION   DLAMCH
  219:       EXTERNAL           LSAME, DLAMCH
  220: *     ..
  221: *     .. Executable Statements ..
  222: *
  223: *     Test the input parameters.
  224: *
  225:       INFO = 0
  226:       UPPER = LSAME( UPLO, 'U' )
  227:       NOTRAN = LSAME( TRANS, 'N' )
  228:       NOUNIT = LSAME( DIAG, 'N' )
  229: *
  230:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  231:          INFO = -1
  232:       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
  233:      $         LSAME( TRANS, 'C' ) ) THEN
  234:          INFO = -2
  235:       ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
  236:          INFO = -3
  237:       ELSE IF( N.LT.0 ) THEN
  238:          INFO = -4
  239:       ELSE IF( NRHS.LT.0 ) THEN
  240:          INFO = -5
  241:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  242:          INFO = -8
  243:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
  244:          INFO = -10
  245:       END IF
  246:       IF( INFO.NE.0 ) THEN
  247:          CALL XERBLA( 'DTPRFS', -INFO )
  248:          RETURN
  249:       END IF
  250: *
  251: *     Quick return if possible
  252: *
  253:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
  254:          DO 10 J = 1, NRHS
  255:             FERR( J ) = ZERO
  256:             BERR( J ) = ZERO
  257:    10    CONTINUE
  258:          RETURN
  259:       END IF
  260: *
  261:       IF( NOTRAN ) THEN
  262:          TRANST = 'T'
  263:       ELSE
  264:          TRANST = 'N'
  265:       END IF
  266: *
  267: *     NZ = maximum number of nonzero elements in each row of A, plus 1
  268: *
  269:       NZ = N + 1
  270:       EPS = DLAMCH( 'Epsilon' )
  271:       SAFMIN = DLAMCH( 'Safe minimum' )
  272:       SAFE1 = NZ*SAFMIN
  273:       SAFE2 = SAFE1 / EPS
  274: *
  275: *     Do for each right hand side
  276: *
  277:       DO 250 J = 1, NRHS
  278: *
  279: *        Compute residual R = B - op(A) * X,
  280: *        where op(A) = A or A**T, depending on TRANS.
  281: *
  282:          CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
  283:          CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
  284:          CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
  285: *
  286: *        Compute componentwise relative backward error from formula
  287: *
  288: *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
  289: *
  290: *        where abs(Z) is the componentwise absolute value of the matrix
  291: *        or vector Z.  If the i-th component of the denominator is less
  292: *        than SAFE2, then SAFE1 is added to the i-th components of the
  293: *        numerator and denominator before dividing.
  294: *
  295:          DO 20 I = 1, N
  296:             WORK( I ) = ABS( B( I, J ) )
  297:    20    CONTINUE
  298: *
  299:          IF( NOTRAN ) THEN
  300: *
  301: *           Compute abs(A)*abs(X) + abs(B).
  302: *
  303:             IF( UPPER ) THEN
  304:                KC = 1
  305:                IF( NOUNIT ) THEN
  306:                   DO 40 K = 1, N
  307:                      XK = ABS( X( K, J ) )
  308:                      DO 30 I = 1, K
  309:                         WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
  310:    30                CONTINUE
  311:                      KC = KC + K
  312:    40             CONTINUE
  313:                ELSE
  314:                   DO 60 K = 1, N
  315:                      XK = ABS( X( K, J ) )
  316:                      DO 50 I = 1, K - 1
  317:                         WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
  318:    50                CONTINUE
  319:                      WORK( K ) = WORK( K ) + XK
  320:                      KC = KC + K
  321:    60             CONTINUE
  322:                END IF
  323:             ELSE
  324:                KC = 1
  325:                IF( NOUNIT ) THEN
  326:                   DO 80 K = 1, N
  327:                      XK = ABS( X( K, J ) )
  328:                      DO 70 I = K, N
  329:                         WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
  330:    70                CONTINUE
  331:                      KC = KC + N - K + 1
  332:    80             CONTINUE
  333:                ELSE
  334:                   DO 100 K = 1, N
  335:                      XK = ABS( X( K, J ) )
  336:                      DO 90 I = K + 1, N
  337:                         WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
  338:    90                CONTINUE
  339:                      WORK( K ) = WORK( K ) + XK
  340:                      KC = KC + N - K + 1
  341:   100             CONTINUE
  342:                END IF
  343:             END IF
  344:          ELSE
  345: *
  346: *           Compute abs(A**T)*abs(X) + abs(B).
  347: *
  348:             IF( UPPER ) THEN
  349:                KC = 1
  350:                IF( NOUNIT ) THEN
  351:                   DO 120 K = 1, N
  352:                      S = ZERO
  353:                      DO 110 I = 1, K
  354:                         S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
  355:   110                CONTINUE
  356:                      WORK( K ) = WORK( K ) + S
  357:                      KC = KC + K
  358:   120             CONTINUE
  359:                ELSE
  360:                   DO 140 K = 1, N
  361:                      S = ABS( X( K, J ) )
  362:                      DO 130 I = 1, K - 1
  363:                         S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
  364:   130                CONTINUE
  365:                      WORK( K ) = WORK( K ) + S
  366:                      KC = KC + K
  367:   140             CONTINUE
  368:                END IF
  369:             ELSE
  370:                KC = 1
  371:                IF( NOUNIT ) THEN
  372:                   DO 160 K = 1, N
  373:                      S = ZERO
  374:                      DO 150 I = K, N
  375:                         S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
  376:   150                CONTINUE
  377:                      WORK( K ) = WORK( K ) + S
  378:                      KC = KC + N - K + 1
  379:   160             CONTINUE
  380:                ELSE
  381:                   DO 180 K = 1, N
  382:                      S = ABS( X( K, J ) )
  383:                      DO 170 I = K + 1, N
  384:                         S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
  385:   170                CONTINUE
  386:                      WORK( K ) = WORK( K ) + S
  387:                      KC = KC + N - K + 1
  388:   180             CONTINUE
  389:                END IF
  390:             END IF
  391:          END IF
  392:          S = ZERO
  393:          DO 190 I = 1, N
  394:             IF( WORK( I ).GT.SAFE2 ) THEN
  395:                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
  396:             ELSE
  397:                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
  398:      $             ( WORK( I )+SAFE1 ) )
  399:             END IF
  400:   190    CONTINUE
  401:          BERR( J ) = S
  402: *
  403: *        Bound error from formula
  404: *
  405: *        norm(X - XTRUE) / norm(X) .le. FERR =
  406: *        norm( abs(inv(op(A)))*
  407: *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
  408: *
  409: *        where
  410: *          norm(Z) is the magnitude of the largest component of Z
  411: *          inv(op(A)) is the inverse of op(A)
  412: *          abs(Z) is the componentwise absolute value of the matrix or
  413: *             vector Z
  414: *          NZ is the maximum number of nonzeros in any row of A, plus 1
  415: *          EPS is machine epsilon
  416: *
  417: *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
  418: *        is incremented by SAFE1 if the i-th component of
  419: *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
  420: *
  421: *        Use DLACN2 to estimate the infinity-norm of the matrix
  422: *           inv(op(A)) * diag(W),
  423: *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
  424: *
  425:          DO 200 I = 1, N
  426:             IF( WORK( I ).GT.SAFE2 ) THEN
  427:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
  428:             ELSE
  429:                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
  430:             END IF
  431:   200    CONTINUE
  432: *
  433:          KASE = 0
  434:   210    CONTINUE
  435:          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
  436:      $                KASE, ISAVE )
  437:          IF( KASE.NE.0 ) THEN
  438:             IF( KASE.EQ.1 ) THEN
  439: *
  440: *              Multiply by diag(W)*inv(op(A)**T).
  441: *
  442:                CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 )
  443:                DO 220 I = 1, N
  444:                   WORK( N+I ) = WORK( I )*WORK( N+I )
  445:   220          CONTINUE
  446:             ELSE
  447: *
  448: *              Multiply by inv(op(A))*diag(W).
  449: *
  450:                DO 230 I = 1, N
  451:                   WORK( N+I ) = WORK( I )*WORK( N+I )
  452:   230          CONTINUE
  453:                CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
  454:             END IF
  455:             GO TO 210
  456:          END IF
  457: *
  458: *        Normalize error.
  459: *
  460:          LSTRES = ZERO
  461:          DO 240 I = 1, N
  462:             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
  463:   240    CONTINUE
  464:          IF( LSTRES.NE.ZERO )
  465:      $      FERR( J ) = FERR( J ) / LSTRES
  466: *
  467:   250 CONTINUE
  468: *
  469:       RETURN
  470: *
  471: *     End of DTPRFS
  472: *
  473:       END