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:
S = ( U )
( M L )
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

= '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*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 = '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.