CPBSTF(l) ) CPBSTF(l)
CPBSTF - compute a split Cholesky factorization of a complex Hermitian positive definite
band matrix A
SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO )
INTEGER INFO, KD, LDAB, N
COMPLEX AB( LDAB, * )
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 triangular of order n-m.
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 sub-
diagonals 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 col-
umn 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.
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 version 3.0 15 June 2000 CPBSTF(l)