1CPBSTF(1)                LAPACK routine (version 3.1)                CPBSTF(1)
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NAME

6       CPBSTF - a split Cholesky factorization of a complex Hermitian positive
7       definite band matrix A
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SYNOPSIS

10       SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO )
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12           CHARACTER      UPLO
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14           INTEGER        INFO, KD, LDAB, N
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16           COMPLEX        AB( LDAB, * )
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PURPOSE

19       CPBSTF computes a split Cholesky factorization of a  complex  Hermitian
20       positive definite band matrix A.
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22       This routine is designed to be used in conjunction with CHBGST.
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24       The factorization has the form  A = S**H*S  where S is a band matrix of
25       the same bandwidth as A and the following structure:
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27         S = ( U    )
28             ( M  L )
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30       where U is upper triangular of order m = (n+kd)/2, and L is lower  tri‐
31       angular of order n-m.
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ARGUMENTS

35       UPLO    (input) CHARACTER*1
36               = 'U':  Upper triangle of A is stored;
37               = 'L':  Lower triangle of A is stored.
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39       N       (input) INTEGER
40               The order of the matrix A.  N >= 0.
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42       KD      (input) INTEGER
43               The  number of superdiagonals of the matrix A if UPLO = 'U', or
44               the number of subdiagonals if UPLO = 'L'.  KD >= 0.
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46       AB      (input/output) COMPLEX array, dimension (LDAB,N)
47               On entry, the upper or lower triangle  of  the  Hermitian  band
48               matrix A, stored in the first kd+1 rows of the array.  The j-th
49               column of A is stored in the j-th column of  the  array  AB  as
50               follows:  if  UPLO  = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
51               kd)<=i<=j;  if  UPLO  =  'L',  AB(1+i-j,j)     =   A(i,j)   for
52               j<=i<=min(n,j+kd).
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54               On exit, if INFO = 0, the factor S from the split Cholesky fac‐
55               torization A = S**H*S. See Further  Details.   LDAB     (input)
56               INTEGER The leading dimension of the array AB.  LDAB >= KD+1.
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58       INFO    (output) INTEGER
59               = 0: successful exit
60               < 0: if INFO = -i, the i-th argument had an illegal value
61               >  0:  if  INFO  = i, the factorization could not be completed,
62               because the updated element a(i,i) was negative; the  matrix  A
63               is not positive definite.
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FURTHER DETAILS

66       The band storage scheme is illustrated by the following example, when N
67       = 7, KD = 2:
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69       S = ( s11  s12  s13                     )
70           (      s22  s23  s24                )
71           (           s33  s34                )
72           (                s44                )
73           (           s53  s54  s55           )
74           (                s64  s65  s66      )
75           (                     s75  s76  s77 )
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77       If UPLO = 'U', the array AB holds:
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79       on entry:                          on exit:
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81        *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53' s64' s75'
82        *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54'  s65'  s76'
83       a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
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85       If UPLO = 'L', the array AB holds:
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87       on entry:                          on exit:
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89       a11   a22   a33   a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
90       a21  a32  a43  a54  a65  a76   *   s12' s23' s34' s54  s65  s76   * a31
91       a42  a53  a64  a64   *    *   s13' s24' s53  s64  s75   *    *
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93       Array  elements  marked  *  are  not  used by the routine; s12' denotes
94       conjg(s12); the diagonal elements of S are real.
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99 LAPACK routine (version 3.1)    November 2006                       CPBSTF(1)