The routine may be called by the names f07hdf, nagf_lapacklin_dpbtrf or its LAPACK name dpbtrf.
f07hdf forms the Cholesky factorization of a real symmetric positive definite band matrix either as if or if , where (or ) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as .
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Character(1)Input
On entry: specifies whether the upper or lower triangular part of is stored and how is to be factorized.
The upper triangular part of is stored and is factorized as , where is upper triangular.
The lower triangular part of is stored and is factorized as , where is lower triangular.
2: – IntegerInput
On entry: , the order of the matrix .
3: – IntegerInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
On entry: the symmetric positive definite band matrix .
The matrix is stored in rows to , more precisely,
if , the elements of the upper triangle of within the band must be stored with element in ;
if , the elements of the lower triangle of within the band must be stored with element in
On exit: the upper or lower triangle of is overwritten by the Cholesky factor or as specified by uplo, using the same storage format as described above.
5: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hdf is called.
6: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value.
If , dynamic memory allocation failed. See Section 9 in the Introduction to the NAG Library FL Interface for further information. An explanatory message is output, and execution of the program is terminated.
The leading minor of order is not positive definite and the factorization could not be completed. Hence itself is not positive definite. This may indicate an error in forming the matrix . There is no routine specifically designed to factorize a symmetric band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07bdf or as a full symmetric matrix, by calling f07mdf.
If , the computed factor is the exact factor of a perturbed matrix , where
is a modest linear function of , and is the machine precision.
If , a similar statement holds for the computed factor . It follows that .
8Parallelism and Performance
f07hdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately , assuming .
A call to f07hdf may be followed by calls to the routines: