The routine may be called by the names f07haf, nagf_lapacklin_dpbsv or its LAPACK name dpbsv.
f07haf uses the Cholesky decomposition to factor as if or if , where is an upper triangular band matrix, and is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as . The factored form of is then used to solve the system of equations .
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Character(1)Input
On entry: if , the upper triangle of is stored.
If , the lower triangle of is stored.
2: – IntegerInput
On entry: , the number of linear equations, i.e., the order of the matrix .
3: – IntegerInput
On entry: , the number of superdiagonals of the matrix if , or the number of subdiagonals if .
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
5: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
On entry: the upper or lower triangle of the symmetric 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: if , the triangular factor or from the Cholesky factorization or of the band matrix , in the same storage format as .
6: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07haf is called.
7: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
On entry: the right-hand side matrix .
On exit: if , the solution matrix .
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07haf is called.
9: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The leading minor of order of is not positive definite, so the factorization could not be completed, and the solution has not been computed.
The computed solution for a single right-hand side, , satisfies an equation of the form
and is the machine precision. An approximate error bound for the computed solution is given by
where , the condition number of with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
f07hbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bff solves and returns a forward error bound and condition estimate. f04bff calls f07haf to solve the equations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07haf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07haf 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.
When , the total number of floating-point operations is approximately , where is the number of superdiagonals and is the number of right-hand sides.