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    1:       SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       CHARACTER          UPLO
   10:       INTEGER            INFO, KD, LDAB, N
   11:       DOUBLE PRECISION   AMAX, SCOND
   12: *     ..
   13: *     .. Array Arguments ..
   14:       DOUBLE PRECISION   AB( LDAB, * ), S( * )
   15: *     ..
   16: *
   17: *  Purpose
   18: *  =======
   19: *
   20: *  DPBEQU computes row and column scalings intended to equilibrate a
   21: *  symmetric positive definite band matrix A and reduce its condition
   22: *  number (with respect to the two-norm).  S contains the scale factors,
   23: *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
   24: *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
   25: *  choice of S puts the condition number of B within a factor N of the
   26: *  smallest possible condition number over all possible diagonal
   27: *  scalings.
   28: *
   29: *  Arguments
   30: *  =========
   31: *
   32: *  UPLO    (input) CHARACTER*1
   33: *          = 'U':  Upper triangular of A is stored;
   34: *          = 'L':  Lower triangular of A is stored.
   35: *
   36: *  N       (input) INTEGER
   37: *          The order of the matrix A.  N >= 0.
   38: *
   39: *  KD      (input) INTEGER
   40: *          The number of superdiagonals of the matrix A if UPLO = 'U',
   41: *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
   42: *
   43: *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
   44: *          The upper or lower triangle of the symmetric band matrix A,
   45: *          stored in the first KD+1 rows of the array.  The j-th column
   46: *          of A is stored in the j-th column of the array AB as follows:
   47: *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
   48: *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
   49: *
   50: *  LDAB     (input) INTEGER
   51: *          The leading dimension of the array A.  LDAB >= KD+1.
   52: *
   53: *  S       (output) DOUBLE PRECISION array, dimension (N)
   54: *          If INFO = 0, S contains the scale factors for A.
   55: *
   56: *  SCOND   (output) DOUBLE PRECISION
   57: *          If INFO = 0, S contains the ratio of the smallest S(i) to
   58: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   59: *          large nor too small, it is not worth scaling by S.
   60: *
   61: *  AMAX    (output) DOUBLE PRECISION
   62: *          Absolute value of largest matrix element.  If AMAX is very
   63: *          close to overflow or very close to underflow, the matrix
   64: *          should be scaled.
   65: *
   66: *  INFO    (output) INTEGER
   67: *          = 0:  successful exit
   68: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
   69: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   70: *
   71: *  =====================================================================
   72: *
   73: *     .. Parameters ..
   74:       DOUBLE PRECISION   ZERO, ONE
   75:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
   76: *     ..
   77: *     .. Local Scalars ..
   78:       LOGICAL            UPPER
   79:       INTEGER            I, J
   80:       DOUBLE PRECISION   SMIN
   81: *     ..
   82: *     .. External Functions ..
   83:       LOGICAL            LSAME
   84:       EXTERNAL           LSAME
   85: *     ..
   86: *     .. External Subroutines ..
   87:       EXTERNAL           XERBLA
   88: *     ..
   89: *     .. Intrinsic Functions ..
   90:       INTRINSIC          MAX, MIN, SQRT
   91: *     ..
   92: *     .. Executable Statements ..
   93: *
   94: *     Test the input parameters.
   95: *
   96:       INFO = 0
   97:       UPPER = LSAME( UPLO, 'U' )
   98:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
   99:          INFO = -1
  100:       ELSE IF( N.LT.0 ) THEN
  101:          INFO = -2
  102:       ELSE IF( KD.LT.0 ) THEN
  103:          INFO = -3
  104:       ELSE IF( LDAB.LT.KD+1 ) THEN
  105:          INFO = -5
  106:       END IF
  107:       IF( INFO.NE.0 ) THEN
  108:          CALL XERBLA( 'DPBEQU', -INFO )
  109:          RETURN
  110:       END IF
  111: *
  112: *     Quick return if possible
  113: *
  114:       IF( N.EQ.0 ) THEN
  115:          SCOND = ONE
  116:          AMAX = ZERO
  117:          RETURN
  118:       END IF
  119: *
  120:       IF( UPPER ) THEN
  121:          J = KD + 1
  122:       ELSE
  123:          J = 1
  124:       END IF
  125: *
  126: *     Initialize SMIN and AMAX.
  127: *
  128:       S( 1 ) = AB( J, 1 )
  129:       SMIN = S( 1 )
  130:       AMAX = S( 1 )
  131: *
  132: *     Find the minimum and maximum diagonal elements.
  133: *
  134:       DO 10 I = 2, N
  135:          S( I ) = AB( J, I )
  136:          SMIN = MIN( SMIN, S( I ) )
  137:          AMAX = MAX( AMAX, S( I ) )
  138:    10 CONTINUE
  139: *
  140:       IF( SMIN.LE.ZERO ) THEN
  141: *
  142: *        Find the first non-positive diagonal element and return.
  143: *
  144:          DO 20 I = 1, N
  145:             IF( S( I ).LE.ZERO ) THEN
  146:                INFO = I
  147:                RETURN
  148:             END IF
  149:    20    CONTINUE
  150:       ELSE
  151: *
  152: *        Set the scale factors to the reciprocals
  153: *        of the diagonal elements.
  154: *
  155:          DO 30 I = 1, N
  156:             S( I ) = ONE / SQRT( S( I ) )
  157:    30    CONTINUE
  158: *
  159: *        Compute SCOND = min(S(I)) / max(S(I))
  160: *
  161:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  162:       END IF
  163:       RETURN
  164: *
  165: *     End of DPBEQU
  166: *
  167:       END