NAG FL Interface
f08uff (dpbstf)

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1 Purpose

f08uff computes a split Cholesky factorization of a real symmetric positive definite band matrix.

2 Specification

Fortran Interface
Subroutine f08uff ( uplo, n, kb, bb, ldbb, info)
Integer, Intent (In) :: n, kb, ldbb
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: bb(ldbb,*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f08uff_ (const char *uplo, const Integer *n, const Integer *kb, double bb[], const Integer *ldbb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f08uff, nagf_lapackeig_dpbstf or its LAPACK name dpbstf.

3 Description

f08uff computes a split Cholesky factorization of a real symmetric positive definite band matrix B. It is designed to be used in conjunction with f08uef.
The factorization has the form B=STS, where S is a band matrix of the same bandwidth as B and the following structure: S is upper triangular in the first (n+k)/2 rows, and transposed — hence, lower triangular — in the remaining rows. For example, if n=9 and k=2, then
S = ( s11 s12 s13 s22 s23 s24 s33 s34 s35 s44 s45 s55 s64 s65 s66 s75 s76 s77 s86 s87 s88 s97 s98 s99 ) .  

4 References

None.

5 Arguments

1: uplo Character(1) Input
On entry: indicates whether the upper or lower triangular part of B is stored.
uplo='U'
The upper triangular part of B is stored.
uplo='L'
The lower triangular part of B is stored.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix B.
Constraint: n0.
3: kb Integer Input
On entry: if uplo='U', the number of superdiagonals, kb, of the matrix B.
If uplo='L', the number of subdiagonals, kb, of the matrix B.
Constraint: kb0.
4: bb(ldbb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array bb must be at least max(1,n).
On entry: the n×n symmetric positive definite band matrix B.
The matrix is stored in rows 1 to kb+1, more precisely,
  • if uplo='U', the elements of the upper triangle of B within the band must be stored with element Bij in bb(kb+1+i-j,j)​ for ​max(1,j-kb)ij;
  • if uplo='L', the elements of the lower triangle of B within the band must be stored with element Bij in bb(1+i-j,j)​ for ​jimin(n,j+kb).
On exit: B is overwritten by the elements of its split Cholesky factor S.
5: ldbb Integer Input
On entry: the first dimension of the array bb as declared in the (sub)program from which f08uff is called.
Constraint: ldbbkb+1.
6: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
The factorization could not be completed, because the updated element b(value,value) would be the square root of a negative number. Hence B is not positive definite. This may indicate an error in forming the matrix B.

7 Accuracy

The computed factor S is the exact factor of a perturbed matrix (B+E), where
|E|c(k+1)ε|ST||S|,  
c(k+1) is a modest linear function of k+1, and ε is the machine precision. It follows that |eij|c(k+1)ε(biibjj).

8 Parallelism 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.

9 Further Comments

The total number of floating-point operations is approximately n (k+1) 2, assuming nk.
A call to f08uff may be followed by a call to f08uef to solve the generalized eigenproblem Az=λBz, where A and B are banded and B is positive definite.
The complex analogue of this routine is f08utf.

10 Example

See f08uef.