.TH CPBSTF 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
CPBSTF - a split Cholesky factorization of a complex Hermitian positive definite band matrix A
.SH SYNOPSIS
.TP 19
SUBROUTINE CPBSTF(
UPLO, N, KD, AB, LDAB, INFO )
.TP 19
.ti +4
CHARACTER
UPLO
.TP 19
.ti +4
INTEGER
INFO, KD, LDAB, N
.TP 19
.ti +4
COMPLEX
AB( LDAB, * )
.SH PURPOSE
CPBSTF computes a split Cholesky factorization of a complex
Hermitian positive definite band matrix A.
This routine is designed to be used in conjunction with CHBGST.
The factorization has the form A = S**H*S where S is a band matrix
of the same bandwidth as A and the following structure:
.br
S = ( U )
.br
( M L )
.br
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
.br
.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
= \(aqU\(aq: Upper triangle of A is stored;
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= \(aqL\(aq: Lower triangle of A is stored.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = \(aqU\(aq,
or the number of subdiagonals if UPLO = \(aqL\(aq. KD >= 0.
.TP 8
AB (input/output) COMPLEX array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first kd+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = \(aqU\(aq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = \(aqL\(aq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the factor S from the split Cholesky
factorization A = S**H*S. See Further Details.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 0: if INFO = i, the factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
.SH FURTHER DETAILS
The band storage scheme is illustrated by the following example, when
N = 7, KD = 2:
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S = ( s11 s12 s13 )
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( s22 s23 s24 )
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( s33 s34 )
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( s44 )
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( s53 s54 s55 )
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( s64 s65 s66 )
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( s75 s76 s77 )
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If UPLO = \(aqU\(aq, the array AB holds:
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on entry: on exit:
.br
* * a13 a24 a35 a46 a57 * * s13 s24 s53\(aq s64\(aq s75\(aq
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54\(aq s65\(aq s76\(aq
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = \(aqL\(aq, the array AB holds:
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on entry: on exit:
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a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12\(aq s23\(aq s34\(aq s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13\(aq s24\(aq s53 s64 s75 * *
Array elements marked * are not used by the routine; s12\(aq denotes
conjg(s12); the diagonal elements of S are real.
.br