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    1: *> \brief \b DLAGTF
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DLAGTF + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagtf.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagtf.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagtf.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
   22: * 
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INFO, N
   25: *       DOUBLE PRECISION   LAMBDA, TOL
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       INTEGER            IN( * )
   29: *       DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
   30: *       ..
   31: *  
   32: *
   33: *> \par Purpose:
   34: *  =============
   35: *>
   36: *> \verbatim
   37: *>
   38: *> DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
   39: *> tridiagonal matrix and lambda is a scalar, as
   40: *>
   41: *>    T - lambda*I = PLU,
   42: *>
   43: *> where P is a permutation matrix, L is a unit lower tridiagonal matrix
   44: *> with at most one non-zero sub-diagonal elements per column and U is
   45: *> an upper triangular matrix with at most two non-zero super-diagonal
   46: *> elements per column.
   47: *>
   48: *> The factorization is obtained by Gaussian elimination with partial
   49: *> pivoting and implicit row scaling.
   50: *>
   51: *> The parameter LAMBDA is included in the routine so that DLAGTF may
   52: *> be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
   53: *> inverse iteration.
   54: *> \endverbatim
   55: *
   56: *  Arguments:
   57: *  ==========
   58: *
   59: *> \param[in] N
   60: *> \verbatim
   61: *>          N is INTEGER
   62: *>          The order of the matrix T.
   63: *> \endverbatim
   64: *>
   65: *> \param[in,out] A
   66: *> \verbatim
   67: *>          A is DOUBLE PRECISION array, dimension (N)
   68: *>          On entry, A must contain the diagonal elements of T.
   69: *>
   70: *>          On exit, A is overwritten by the n diagonal elements of the
   71: *>          upper triangular matrix U of the factorization of T.
   72: *> \endverbatim
   73: *>
   74: *> \param[in] LAMBDA
   75: *> \verbatim
   76: *>          LAMBDA is DOUBLE PRECISION
   77: *>          On entry, the scalar lambda.
   78: *> \endverbatim
   79: *>
   80: *> \param[in,out] B
   81: *> \verbatim
   82: *>          B is DOUBLE PRECISION array, dimension (N-1)
   83: *>          On entry, B must contain the (n-1) super-diagonal elements of
   84: *>          T.
   85: *>
   86: *>          On exit, B is overwritten by the (n-1) super-diagonal
   87: *>          elements of the matrix U of the factorization of T.
   88: *> \endverbatim
   89: *>
   90: *> \param[in,out] C
   91: *> \verbatim
   92: *>          C is DOUBLE PRECISION array, dimension (N-1)
   93: *>          On entry, C must contain the (n-1) sub-diagonal elements of
   94: *>          T.
   95: *>
   96: *>          On exit, C is overwritten by the (n-1) sub-diagonal elements
   97: *>          of the matrix L of the factorization of T.
   98: *> \endverbatim
   99: *>
  100: *> \param[in] TOL
  101: *> \verbatim
  102: *>          TOL is DOUBLE PRECISION
  103: *>          On entry, a relative tolerance used to indicate whether or
  104: *>          not the matrix (T - lambda*I) is nearly singular. TOL should
  105: *>          normally be chose as approximately the largest relative error
  106: *>          in the elements of T. For example, if the elements of T are
  107: *>          correct to about 4 significant figures, then TOL should be
  108: *>          set to about 5*10**(-4). If TOL is supplied as less than eps,
  109: *>          where eps is the relative machine precision, then the value
  110: *>          eps is used in place of TOL.
  111: *> \endverbatim
  112: *>
  113: *> \param[out] D
  114: *> \verbatim
  115: *>          D is DOUBLE PRECISION array, dimension (N-2)
  116: *>          On exit, D is overwritten by the (n-2) second super-diagonal
  117: *>          elements of the matrix U of the factorization of T.
  118: *> \endverbatim
  119: *>
  120: *> \param[out] IN
  121: *> \verbatim
  122: *>          IN is INTEGER array, dimension (N)
  123: *>          On exit, IN contains details of the permutation matrix P. If
  124: *>          an interchange occurred at the kth step of the elimination,
  125: *>          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
  126: *>          returns the smallest positive integer j such that
  127: *>
  128: *>             abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL,
  129: *>
  130: *>          where norm( A(j) ) denotes the sum of the absolute values of
  131: *>          the jth row of the matrix A. If no such j exists then IN(n)
  132: *>          is returned as zero. If IN(n) is returned as positive, then a
  133: *>          diagonal element of U is small, indicating that
  134: *>          (T - lambda*I) is singular or nearly singular,
  135: *> \endverbatim
  136: *>
  137: *> \param[out] INFO
  138: *> \verbatim
  139: *>          INFO is INTEGER
  140: *>          = 0   : successful exit
  141: *>          .lt. 0: if INFO = -k, the kth argument had an illegal value
  142: *> \endverbatim
  143: *
  144: *  Authors:
  145: *  ========
  146: *
  147: *> \author Univ. of Tennessee 
  148: *> \author Univ. of California Berkeley 
  149: *> \author Univ. of Colorado Denver 
  150: *> \author NAG Ltd. 
  151: *
  152: *> \date November 2011
  153: *
  154: *> \ingroup auxOTHERcomputational
  155: *
  156: *  =====================================================================
  157:       SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
  158: *
  159: *  -- LAPACK computational routine (version 3.4.0) --
  160: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  161: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  162: *     November 2011
  163: *
  164: *     .. Scalar Arguments ..
  165:       INTEGER            INFO, N
  166:       DOUBLE PRECISION   LAMBDA, TOL
  167: *     ..
  168: *     .. Array Arguments ..
  169:       INTEGER            IN( * )
  170:       DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
  171: *     ..
  172: *
  173: * =====================================================================
  174: *
  175: *     .. Parameters ..
  176:       DOUBLE PRECISION   ZERO
  177:       PARAMETER          ( ZERO = 0.0D+0 )
  178: *     ..
  179: *     .. Local Scalars ..
  180:       INTEGER            K
  181:       DOUBLE PRECISION   EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
  182: *     ..
  183: *     .. Intrinsic Functions ..
  184:       INTRINSIC          ABS, MAX
  185: *     ..
  186: *     .. External Functions ..
  187:       DOUBLE PRECISION   DLAMCH
  188:       EXTERNAL           DLAMCH
  189: *     ..
  190: *     .. External Subroutines ..
  191:       EXTERNAL           XERBLA
  192: *     ..
  193: *     .. Executable Statements ..
  194: *
  195:       INFO = 0
  196:       IF( N.LT.0 ) THEN
  197:          INFO = -1
  198:          CALL XERBLA( 'DLAGTF', -INFO )
  199:          RETURN
  200:       END IF
  201: *
  202:       IF( N.EQ.0 )
  203:      $   RETURN
  204: *
  205:       A( 1 ) = A( 1 ) - LAMBDA
  206:       IN( N ) = 0
  207:       IF( N.EQ.1 ) THEN
  208:          IF( A( 1 ).EQ.ZERO )
  209:      $      IN( 1 ) = 1
  210:          RETURN
  211:       END IF
  212: *
  213:       EPS = DLAMCH( 'Epsilon' )
  214: *
  215:       TL = MAX( TOL, EPS )
  216:       SCALE1 = ABS( A( 1 ) ) + ABS( B( 1 ) )
  217:       DO 10 K = 1, N - 1
  218:          A( K+1 ) = A( K+1 ) - LAMBDA
  219:          SCALE2 = ABS( C( K ) ) + ABS( A( K+1 ) )
  220:          IF( K.LT.( N-1 ) )
  221:      $      SCALE2 = SCALE2 + ABS( B( K+1 ) )
  222:          IF( A( K ).EQ.ZERO ) THEN
  223:             PIV1 = ZERO
  224:          ELSE
  225:             PIV1 = ABS( A( K ) ) / SCALE1
  226:          END IF
  227:          IF( C( K ).EQ.ZERO ) THEN
  228:             IN( K ) = 0
  229:             PIV2 = ZERO
  230:             SCALE1 = SCALE2
  231:             IF( K.LT.( N-1 ) )
  232:      $         D( K ) = ZERO
  233:          ELSE
  234:             PIV2 = ABS( C( K ) ) / SCALE2
  235:             IF( PIV2.LE.PIV1 ) THEN
  236:                IN( K ) = 0
  237:                SCALE1 = SCALE2
  238:                C( K ) = C( K ) / A( K )
  239:                A( K+1 ) = A( K+1 ) - C( K )*B( K )
  240:                IF( K.LT.( N-1 ) )
  241:      $            D( K ) = ZERO
  242:             ELSE
  243:                IN( K ) = 1
  244:                MULT = A( K ) / C( K )
  245:                A( K ) = C( K )
  246:                TEMP = A( K+1 )
  247:                A( K+1 ) = B( K ) - MULT*TEMP
  248:                IF( K.LT.( N-1 ) ) THEN
  249:                   D( K ) = B( K+1 )
  250:                   B( K+1 ) = -MULT*D( K )
  251:                END IF
  252:                B( K ) = TEMP
  253:                C( K ) = MULT
  254:             END IF
  255:          END IF
  256:          IF( ( MAX( PIV1, PIV2 ).LE.TL ) .AND. ( IN( N ).EQ.0 ) )
  257:      $      IN( N ) = K
  258:    10 CONTINUE
  259:       IF( ( ABS( A( N ) ).LE.SCALE1*TL ) .AND. ( IN( N ).EQ.0 ) )
  260:      $   IN( N ) = N
  261: *
  262:       RETURN
  263: *
  264: *     End of DLAGTF
  265: *
  266:       END