# zpbsv (l) - Linux Manuals

## NAME

ZPBSV - computes the solution to a complex system of linear equations A * X = B,

## SYNOPSIS

SUBROUTINE ZPBSV(
UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )

CHARACTER UPLO

INTEGER INFO, KD, LDAB, LDB, N, NRHS

COMPLEX*16 AB( LDAB, * ), B( LDB, * )

## PURPOSE

ZPBSV computes the solution to a complex system of linear equations
B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices.
The Cholesky decomposition is used to factor A as

U**H U,  if UPLO aqUaq, or

L**H,  if UPLO aqLaq,
where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

## ARGUMENTS

UPLO (input) CHARACTER*1
= aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., 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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 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). See below for further details. On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

## FURTHER DETAILS

The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = aqUaq:
On entry: On exit:

a13  a24  a35  a46           u13  u24  u35  u46
a12  a23  a34  a45  a56        u12  u23  u34  u45  u56
a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66 Similarly, if UPLO = aqLaq the format of A is as follows:
On entry: On exit:

a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
a21  a32  a43  a54  a65        l21  l32  l43  l54  l65   *
a31  a42  a53  a64           l31  l42  l53  l64      * Array elements marked * are not used by the routine.