CPBSTF(1) LAPACK routine (version 3.2) CPBSTF(1)NAME
CPBSTF - computes a split Cholesky factorization of a complex Hermitian
positive definite band matrix A
SYNOPSIS
SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO )
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
COMPLEX AB( LDAB, * )
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:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower tri‐
angular of order n-m.
ARGUMENTS
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KD >= 0.
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 = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-
kd)<=i<=j; if UPLO = 'L', 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 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.
FURTHER DETAILS
The band storage scheme is illustrated by the following example, when N
= 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75'
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76'
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 If
UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * a31
a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * * Array
elements marked * are not used by the routine; s12' denotes conjg(s12);
the diagonal elements of S are real.
LAPACK routine (version 3.2) November 2008 CPBSTF(1)
