The routine may be called by the names f08uff, nagf_lapackeig_dpbstf or its LAPACK name dpbstf.
f08uff computes a split Cholesky factorization of a real symmetric positive definite band matrix . It is designed to be used in conjunction with f08uef.
The factorization has the form , where is a band matrix of the same bandwidth as and the following structure: is upper triangular in the first rows, and transposed — hence, lower triangular — in the remaining rows. For example, if and , then
1: – Character(1)Input
On entry: indicates whether the upper or lower triangular part of is stored.
The upper triangular part of is stored.
The lower triangular part of is stored.
2: – IntegerInput
On entry: , the order of the matrix .
3: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array bb
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: is overwritten by the elements of its split Cholesky factor .
5: – IntegerInput
On entry: the first dimension of the array bb as declared in the (sub)program from which f08uff 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. An explanatory message is output, and execution of the program is terminated.
The factorization could not be completed, because the updated element would be the square root of a negative number. Hence is not positive definite. This may indicate an error in forming the matrix .
The computed factor is the exact factor of a perturbed matrix , where
is a modest linear function of , and is the machine precision. It follows that .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08uff 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 f08uff may be followed by a call to f08uef to solve the generalized eigenproblem , where and are banded and is positive definite.