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    1: *> \brief \b ZLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download ZLAQHB + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqhb.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqhb.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqhb.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       CHARACTER          EQUED, UPLO
   25: *       INTEGER            KD, LDAB, N
   26: *       DOUBLE PRECISION   AMAX, SCOND
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       DOUBLE PRECISION   S( * )
   30: *       COMPLEX*16         AB( LDAB, * )
   31: *       ..
   32: *
   33: *
   34: *> \par Purpose:
   35: *  =============
   36: *>
   37: *> \verbatim
   38: *>
   39: *> ZLAQHB equilibrates a Hermitian band matrix A
   40: *> using the scaling factors in the vector S.
   41: *> \endverbatim
   42: *
   43: *  Arguments:
   44: *  ==========
   45: *
   46: *> \param[in] UPLO
   47: *> \verbatim
   48: *>          UPLO is CHARACTER*1
   49: *>          Specifies whether the upper or lower triangular part of the
   50: *>          symmetric matrix A is stored.
   51: *>          = 'U':  Upper triangular
   52: *>          = 'L':  Lower triangular
   53: *> \endverbatim
   54: *>
   55: *> \param[in] N
   56: *> \verbatim
   57: *>          N is INTEGER
   58: *>          The order of the matrix A.  N >= 0.
   59: *> \endverbatim
   60: *>
   61: *> \param[in] KD
   62: *> \verbatim
   63: *>          KD is INTEGER
   64: *>          The number of super-diagonals of the matrix A if UPLO = 'U',
   65: *>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
   66: *> \endverbatim
   67: *>
   68: *> \param[in,out] AB
   69: *> \verbatim
   70: *>          AB is COMPLEX*16 array, dimension (LDAB,N)
   71: *>          On entry, the upper or lower triangle of the symmetric band
   72: *>          matrix A, stored in the first KD+1 rows of the array.  The
   73: *>          j-th column of A is stored in the j-th column of the array AB
   74: *>          as follows:
   75: *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
   76: *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
   77: *>
   78: *>          On exit, if INFO = 0, the triangular factor U or L from the
   79: *>          Cholesky factorization A = U**H *U or A = L*L**H of the band
   80: *>          matrix A, in the same storage format as A.
   81: *> \endverbatim
   82: *>
   83: *> \param[in] LDAB
   84: *> \verbatim
   85: *>          LDAB is INTEGER
   86: *>          The leading dimension of the array AB.  LDAB >= KD+1.
   87: *> \endverbatim
   88: *>
   89: *> \param[out] S
   90: *> \verbatim
   91: *>          S is DOUBLE PRECISION array, dimension (N)
   92: *>          The scale factors for A.
   93: *> \endverbatim
   94: *>
   95: *> \param[in] SCOND
   96: *> \verbatim
   97: *>          SCOND is DOUBLE PRECISION
   98: *>          Ratio of the smallest S(i) to the largest S(i).
   99: *> \endverbatim
  100: *>
  101: *> \param[in] AMAX
  102: *> \verbatim
  103: *>          AMAX is DOUBLE PRECISION
  104: *>          Absolute value of largest matrix entry.
  105: *> \endverbatim
  106: *>
  107: *> \param[out] EQUED
  108: *> \verbatim
  109: *>          EQUED is CHARACTER*1
  110: *>          Specifies whether or not equilibration was done.
  111: *>          = 'N':  No equilibration.
  112: *>          = 'Y':  Equilibration was done, i.e., A has been replaced by
  113: *>                  diag(S) * A * diag(S).
  114: *> \endverbatim
  115: *
  116: *> \par Internal Parameters:
  117: *  =========================
  118: *>
  119: *> \verbatim
  120: *>  THRESH is a threshold value used to decide if scaling should be done
  121: *>  based on the ratio of the scaling factors.  If SCOND < THRESH,
  122: *>  scaling is done.
  123: *>
  124: *>  LARGE and SMALL are threshold values used to decide if scaling should
  125: *>  be done based on the absolute size of the largest matrix element.
  126: *>  If AMAX > LARGE or AMAX < SMALL, scaling is done.
  127: *> \endverbatim
  128: *
  129: *  Authors:
  130: *  ========
  131: *
  132: *> \author Univ. of Tennessee
  133: *> \author Univ. of California Berkeley
  134: *> \author Univ. of Colorado Denver
  135: *> \author NAG Ltd.
  136: *
  137: *> \ingroup complex16OTHERauxiliary
  138: *
  139: *  =====================================================================
  140:       SUBROUTINE ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
  141: *
  142: *  -- LAPACK auxiliary routine --
  143: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  144: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  145: *
  146: *     .. Scalar Arguments ..
  147:       CHARACTER          EQUED, UPLO
  148:       INTEGER            KD, LDAB, N
  149:       DOUBLE PRECISION   AMAX, SCOND
  150: *     ..
  151: *     .. Array Arguments ..
  152:       DOUBLE PRECISION   S( * )
  153:       COMPLEX*16         AB( LDAB, * )
  154: *     ..
  155: *
  156: *  =====================================================================
  157: *
  158: *     .. Parameters ..
  159:       DOUBLE PRECISION   ONE, THRESH
  160:       PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
  161: *     ..
  162: *     .. Local Scalars ..
  163:       INTEGER            I, J
  164:       DOUBLE PRECISION   CJ, LARGE, SMALL
  165: *     ..
  166: *     .. External Functions ..
  167:       LOGICAL            LSAME
  168:       DOUBLE PRECISION   DLAMCH
  169:       EXTERNAL           LSAME, DLAMCH
  170: *     ..
  171: *     .. Intrinsic Functions ..
  172:       INTRINSIC          DBLE, MAX, MIN
  173: *     ..
  174: *     .. Executable Statements ..
  175: *
  176: *     Quick return if possible
  177: *
  178:       IF( N.LE.0 ) THEN
  179:          EQUED = 'N'
  180:          RETURN
  181:       END IF
  182: *
  183: *     Initialize LARGE and SMALL.
  184: *
  185:       SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
  186:       LARGE = ONE / SMALL
  187: *
  188:       IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
  189: *
  190: *        No equilibration
  191: *
  192:          EQUED = 'N'
  193:       ELSE
  194: *
  195: *        Replace A by diag(S) * A * diag(S).
  196: *
  197:          IF( LSAME( UPLO, 'U' ) ) THEN
  198: *
  199: *           Upper triangle of A is stored in band format.
  200: *
  201:             DO 20 J = 1, N
  202:                CJ = S( J )
  203:                DO 10 I = MAX( 1, J-KD ), J - 1
  204:                   AB( KD+1+I-J, J ) = CJ*S( I )*AB( KD+1+I-J, J )
  205:    10          CONTINUE
  206:                AB( KD+1, J ) = CJ*CJ*DBLE( AB( KD+1, J ) )
  207:    20       CONTINUE
  208:          ELSE
  209: *
  210: *           Lower triangle of A is stored.
  211: *
  212:             DO 40 J = 1, N
  213:                CJ = S( J )
  214:                AB( 1, J ) = CJ*CJ*DBLE( AB( 1, J ) )
  215:                DO 30 I = J + 1, MIN( N, J+KD )
  216:                   AB( 1+I-J, J ) = CJ*S( I )*AB( 1+I-J, J )
  217:    30          CONTINUE
  218:    40       CONTINUE
  219:          END IF
  220:          EQUED = 'Y'
  221:       END IF
  222: *
  223:       RETURN
  224: *
  225: *     End of ZLAQHB
  226: *
  227:       END