.TH ZPBSTF 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME ZPBSTF - a split Cholesky factorization of a complex Hermitian positive definite band matrix A .SH SYNOPSIS .TP 19 SUBROUTINE ZPBSTF( 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*16 AB( LDAB, * ) .SH 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: .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; .br = \(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*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 = \(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: .br S = ( s11 s12 s13 ) .br ( s22 s23 s24 ) .br ( s33 s34 ) .br ( s44 ) .br ( s53 s54 s55 ) .br ( s64 s65 s66 ) .br ( s75 s76 s77 ) .br If UPLO = \(aqU\(aq, the array AB holds: .br 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: .br on entry: on exit: .br 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