NAG Library Routine Document
F08UFF (DPBSTF) computes a split Cholesky factorization of a real symmetric positive definite band matrix.
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The routine may be called by its
F08UFF (DPBSTF) computes a split Cholesky factorization of a real symmetric positive definite band matrix
. It is designed to be used in conjunction with F08UEF (DSBGST)
The factorization has the form
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
- 1: – CHARACTER(1)Input
: indicates whether the upper or lower triangular part of
- 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
, the number of superdiagonals,
, of the matrix
If , the number of subdiagonals, , of the matrix .
- 4: – REAL (KIND=nag_wp) arrayInput/Output
the second dimension of the array BB
must be at least
symmetric positive definite band matrix
The matrix is stored in rows
, 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
: the first dimension of the array BB
as declared in the (sub)program from which F08UFF (DPBSTF) is called.
- 6: – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , 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
is a modest linear function of
is the machine precision
. It follows that
8 Parallelism and Performance
F08UFF (DPBSTF) 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 (DPBSTF) may be followed by a call to F08UEF (DSBGST)
to solve the generalized eigenproblem
are banded and
is positive definite.
The complex analogue of this routine is F08UTF (ZPBSTF)
See Section 10
in F08UEF (DSBGST).