F08UFF (DPBSTF) (PDF version)
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NAG Library Manual

NAG Library Routine Document

F08UFF (DPBSTF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1  Purpose

F08UFF (DPBSTF) computes a split Cholesky factorization of a real symmetric positive definite band matrix.

2  Specification

SUBROUTINE F08UFF ( UPLO, N, KB, BB, LDBB, INFO)
INTEGER  N, KB, LDBB, INFO
REAL (KIND=nag_wp)  BB(LDBB,*)
CHARACTER(1)  UPLO
The routine may be called by its LAPACK name dpbstf.

3  Description

F08UFF (DPBSTF) computes a split Cholesky factorization of a real symmetric positive definite band matrix B. It is designed to be used in conjunction with F08UEF (DSBGST).
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 – INTEGERInput
On entry: n, the order of the matrix B.
Constraint: N0.
3:     KB – INTEGERInput
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:     BBLDBB* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array BB must be at least max1,N.
On entry: the n by 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 BBkb+1+i-jj​ for ​max1,j-kbij;
  • if UPLO='L', the elements of the lower triangle of B within the band must be stored with element Bij in BB1+i-jj​ for ​jiminn,j+kb.
On exit: B is overwritten by the elements of its split Cholesky factor S.
5:     LDBB – INTEGERInput
On entry: the first dimension of the array BB as declared in the (sub)program from which F08UFF (DPBSTF) is called.
Constraint: LDBBKB+1.
6:     INFO – INTEGEROutput
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
If INFO=i, the factorization could not be completed, because the updated element bi,i 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
Eck+1εSTS,  
ck+1 is a modest linear function of k+1, and ε is the machine precision. It follows that eijck+1εbiibjj.

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.

9  Further Comments

The total number of floating-point operations is approximately n k+1 2, assuming nk.
A call to F08UFF (DPBSTF) may be followed by a call to F08UEF (DSBGST) 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 (ZPBSTF).

10  Example

See Section 10 in F08UEF (DSBGST).

F08UFF (DPBSTF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016