- 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, * )
PURPOSEZPBSV computes the solution to a complex system of linear equations
A * X = 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
A = U**H * U, if UPLO = 'U', or
A = L * L**H, if UPLO = 'L',
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.
- UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': 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 = 'U', or the number of subdiagonals if UPLO = 'L'. 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 = '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). 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 DETAILSThe band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U':
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 = 'L' 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.