# zpbstf (l) - Linux Man Pages

## NAME

ZPBSTF - computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A

## SYNOPSIS

SUBROUTINE ZPBSTF(
UPLO, N, KD, AB, LDAB, INFO )

CHARACTER UPLO

INTEGER INFO, KD, LDAB, N

COMPLEX*16 AB( LDAB, * )

## PURPOSE

ZPBSTF computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A. This routine is designed to be used in conjunction with ZHBGST. 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:

)

)
where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

## ARGUMENTS

UPLO (input) CHARACTER*1
= aqUaq: Upper triangle of A is stored;
= aqLaq: 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 = aqUaq, or the number of subdiagonals if UPLO = aqLaq. KD >= 0.
AB (input/output) COMPLEX*16 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 = aqUaq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = aqLaq, 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 = aqUaq, the array AB holds:
on entry: on exit:

a13  a24  a35  a46  a57        s13  s24  s53aq s64aq s75aq
a12  a23  a34  a45  a56  a67     s12  s23  s34  s54aq s65aq s76aq a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 If UPLO = aqLaq, 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 * s12aq s23aq s34aq s54 s65 s76 * a31 a42 a53 a64 a64 * * s13aq s24aq s53 s64 s75 * * Array elements marked * are not used by the routine; s12aq denotes conjg(s12); the diagonal elements of S are real.