The routine may be called by the names f07hef, nagf_lapacklin_dpbtrs or its LAPACK name dpbtrs.
f07hef is used to solve a real symmetric positive definite band system of linear equations , the routine must be preceded by a call to f07hdf which computes the Cholesky factorization of . The solution is computed by forward and backward substitution.
If , , where is upper triangular; the solution is computed by solving and then .
If , , where is lower triangular; the solution is computed by solving and then .
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 how has been factorized.
, where is upper triangular.
, 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: – IntegerInput
On entry: , the number of right-hand sides.
5: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab
must be at least
On entry: the Cholesky factor of , as returned by f07hdf.
6: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hef 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: the solution matrix .
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07hef 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.
For each right-hand side vector , the computed solution is the exact solution of a perturbed system of equations , where
if , ;
if , ,
is a modest linear function of , and is the machine precision.
If is the true solution, then the computed solution satisfies a forward error bound of the form
where . Note that can be much smaller than .
Forward and backward error bounds can be computed by calling f07hhf, and an estimate for () can be obtained by calling f07hgf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07hef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07hef 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 .
This routine may be followed by a call to f07hhf to refine the solution and return an error estimate.