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    1:       SUBROUTINE DPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK,
    2:      $                   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          UPLO
   13:       INTEGER            INFO, KD, LDAB, N
   14:       DOUBLE PRECISION   ANORM, RCOND
   15: *     ..
   16: *     .. Array Arguments ..
   17:       INTEGER            IWORK( * )
   18:       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
   19: *     ..
   20: *
   21: *  Purpose
   22: *  =======
   23: *
   24: *  DPBCON estimates the reciprocal of the condition number (in the
   25: *  1-norm) of a real symmetric positive definite band matrix using the
   26: *  Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.
   27: *
   28: *  An estimate is obtained for norm(inv(A)), and the reciprocal of the
   29: *  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
   30: *
   31: *  Arguments
   32: *  =========
   33: *
   34: *  UPLO    (input) CHARACTER*1
   35: *          = 'U':  Upper triangular factor stored in AB;
   36: *          = 'L':  Lower triangular factor stored in AB.
   37: *
   38: *  N       (input) INTEGER
   39: *          The order of the matrix A.  N >= 0.
   40: *
   41: *  KD      (input) INTEGER
   42: *          The number of superdiagonals of the matrix A if UPLO = 'U',
   43: *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
   44: *
   45: *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
   46: *          The triangular factor U or L from the Cholesky factorization
   47: *          A = U**T*U or A = L*L**T of the band matrix A, stored in the
   48: *          first KD+1 rows of the array.  The j-th column of U or L is
   49: *          stored in the j-th column of the array AB as follows:
   50: *          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
   51: *          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).
   52: *
   53: *  LDAB    (input) INTEGER
   54: *          The leading dimension of the array AB.  LDAB >= KD+1.
   55: *
   56: *  ANORM   (input) DOUBLE PRECISION
   57: *          The 1-norm (or infinity-norm) of the symmetric band matrix A.
   58: *
   59: *  RCOND   (output) DOUBLE PRECISION
   60: *          The reciprocal of the condition number of the matrix A,
   61: *          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
   62: *          estimate of the 1-norm of inv(A) computed in this routine.
   63: *
   64: *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
   65: *
   66: *  IWORK   (workspace) INTEGER array, dimension (N)
   67: *
   68: *  INFO    (output) INTEGER
   69: *          = 0:  successful exit
   70: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   71: *
   72: *  =====================================================================
   73: *
   74: *     .. Parameters ..
   75:       DOUBLE PRECISION   ONE, ZERO
   76:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
   77: *     ..
   78: *     .. Local Scalars ..
   79:       LOGICAL            UPPER
   80:       CHARACTER          NORMIN
   81:       INTEGER            IX, KASE
   82:       DOUBLE PRECISION   AINVNM, SCALE, SCALEL, SCALEU, SMLNUM
   83: *     ..
   84: *     .. Local Arrays ..
   85:       INTEGER            ISAVE( 3 )
   86: *     ..
   87: *     .. External Functions ..
   88:       LOGICAL            LSAME
   89:       INTEGER            IDAMAX
   90:       DOUBLE PRECISION   DLAMCH
   91:       EXTERNAL           LSAME, IDAMAX, DLAMCH
   92: *     ..
   93: *     .. External Subroutines ..
   94:       EXTERNAL           DLACN2, DLATBS, DRSCL, XERBLA
   95: *     ..
   96: *     .. Intrinsic Functions ..
   97:       INTRINSIC          ABS
   98: *     ..
   99: *     .. Executable Statements ..
  100: *
  101: *     Test the input parameters.
  102: *
  103:       INFO = 0
  104:       UPPER = LSAME( UPLO, 'U' )
  105:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  106:          INFO = -1
  107:       ELSE IF( N.LT.0 ) THEN
  108:          INFO = -2
  109:       ELSE IF( KD.LT.0 ) THEN
  110:          INFO = -3
  111:       ELSE IF( LDAB.LT.KD+1 ) THEN
  112:          INFO = -5
  113:       ELSE IF( ANORM.LT.ZERO ) THEN
  114:          INFO = -6
  115:       END IF
  116:       IF( INFO.NE.0 ) THEN
  117:          CALL XERBLA( 'DPBCON', -INFO )
  118:          RETURN
  119:       END IF
  120: *
  121: *     Quick return if possible
  122: *
  123:       RCOND = ZERO
  124:       IF( N.EQ.0 ) THEN
  125:          RCOND = ONE
  126:          RETURN
  127:       ELSE IF( ANORM.EQ.ZERO ) THEN
  128:          RETURN
  129:       END IF
  130: *
  131:       SMLNUM = DLAMCH( 'Safe minimum' )
  132: *
  133: *     Estimate the 1-norm of the inverse.
  134: *
  135:       KASE = 0
  136:       NORMIN = 'N'
  137:    10 CONTINUE
  138:       CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
  139:       IF( KASE.NE.0 ) THEN
  140:          IF( UPPER ) THEN
  141: *
  142: *           Multiply by inv(U').
  143: *
  144:             CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
  145:      $                   KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
  146:      $                   INFO )
  147:             NORMIN = 'Y'
  148: *
  149: *           Multiply by inv(U).
  150: *
  151:             CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N,
  152:      $                   KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
  153:      $                   INFO )
  154:          ELSE
  155: *
  156: *           Multiply by inv(L).
  157: *
  158:             CALL DLATBS( 'Lower', 'No transpose', 'Non-unit', NORMIN, N,
  159:      $                   KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ),
  160:      $                   INFO )
  161:             NORMIN = 'Y'
  162: *
  163: *           Multiply by inv(L').
  164: *
  165:             CALL DLATBS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N,
  166:      $                   KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ),
  167:      $                   INFO )
  168:          END IF
  169: *
  170: *        Multiply by 1/SCALE if doing so will not cause overflow.
  171: *
  172:          SCALE = SCALEL*SCALEU
  173:          IF( SCALE.NE.ONE ) THEN
  174:             IX = IDAMAX( N, WORK, 1 )
  175:             IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
  176:      $         GO TO 20
  177:             CALL DRSCL( N, SCALE, WORK, 1 )
  178:          END IF
  179:          GO TO 10
  180:       END IF
  181: *
  182: *     Compute the estimate of the reciprocal condition number.
  183: *
  184:       IF( AINVNM.NE.ZERO )
  185:      $   RCOND = ( ONE / AINVNM ) / ANORM
  186: *
  187:    20 CONTINUE
  188: *
  189:       RETURN
  190: *
  191: *     End of DPBCON
  192: *
  193:       END