1DPBEQU(1)                LAPACK routine (version 3.2)                DPBEQU(1)
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NAME

6       DPBEQU  -  computes  row  and column scalings intended to equilibrate a
7       symmetric positive definite band matrix A and reduce its condition num‐
8       ber (with respect to the two-norm)
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SYNOPSIS

11       SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
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13           CHARACTER      UPLO
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15           INTEGER        INFO, KD, LDAB, N
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17           DOUBLE         PRECISION AMAX, SCOND
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19           DOUBLE         PRECISION AB( LDAB, * ), S( * )
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PURPOSE

22       DPBEQU  computes row and column scalings intended to equilibrate a sym‐
23       metric positive definite band matrix A and reduce its condition  number
24       (with  respect  to the two-norm).  S contains the scale factors, S(i) =
25       1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)
26       = S(i)*A(i,j)*S(j) has ones on the diagonal.  This choice of S puts the
27       condition number of B within a factor N of the smallest possible condi‐
28       tion number over all possible diagonal scalings.
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ARGUMENTS

31       UPLO    (input) CHARACTER*1
32               = 'U':  Upper triangular of A is stored;
33               = 'L':  Lower triangular of A is stored.
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35       N       (input) INTEGER
36               The order of the matrix A.  N >= 0.
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38       KD      (input) INTEGER
39               The  number of superdiagonals of the matrix A if UPLO = 'U', or
40               the number of subdiagonals if UPLO = 'L'.  KD >= 0.
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42       AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
43               The upper or lower triangle of the  symmetric  band  matrix  A,
44               stored in the first KD+1 rows of the array.  The j-th column of
45               A is stored in the j-th column of the array AB as  follows:  if
46               UPLO  =  'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if
47               UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
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49       LDAB     (input) INTEGER
50                The leading dimension of the array A.  LDAB >= KD+1.
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52       S       (output) DOUBLE PRECISION array, dimension (N)
53               If INFO = 0, S contains the scale factors for A.
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55       SCOND   (output) DOUBLE PRECISION
56               If INFO = 0, S contains the ratio of the smallest S(i)  to  the
57               largest  S(i).   If  SCOND >= 0.1 and AMAX is neither too large
58               nor too small, it is not worth scaling by S.
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60       AMAX    (output) DOUBLE PRECISION
61               Absolute value of largest matrix  element.   If  AMAX  is  very
62               close to overflow or very close to underflow, the matrix should
63               be scaled.
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65       INFO    (output) INTEGER
66               = 0:  successful exit
67               < 0:  if INFO = -i, the i-th argument had an illegal value.
68               > 0:  if INFO = i, the i-th diagonal element is nonpositive.
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72 LAPACK routine (version 3.2)    November 2008                       DPBEQU(1)
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