NAG CL Interface
f08ufc (dpbstf)

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1 Purpose

f08ufc computes a split Cholesky factorization of a real symmetric positive definite band matrix.

2 Specification

#include <nag.h>
void  f08ufc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kb, double bb[], Integer pdbb, NagError *fail)
The function may be called by the names: f08ufc, nag_lapackeig_dpbstf or nag_dpbstf.

3 Description

f08ufc computes a split Cholesky factorization of a real symmetric positive definite band matrix B. It is designed to be used in conjunction with f08uec.
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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: indicates whether the upper or lower triangular part of B is stored.
uplo=Nag_Upper
The upper triangular part of B is stored.
uplo=Nag_Lower
The lower triangular part of B is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix B.
Constraint: n0.
4: kb Integer Input
On entry: if uplo=Nag_Upper, the number of superdiagonals, kb, of the matrix B.
If uplo=Nag_Lower, the number of subdiagonals, kb, of the matrix B.
Constraint: kb0.
5: bb[dim] double Input/Output
Note: the dimension, dim, of the array bb must be at least max(1,pdbb×n).
On entry: the n×n symmetric positive definite band matrix B.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Bij, depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Bij is stored in bb[kb+i-j+(j-1)×pdbb], for j=1,,n and i=max(1,j-kb),,j;
if order=Nag_ColMajor and uplo=Nag_Lower,
Bij is stored in bb[i-j+(j-1)×pdbb], for j=1,,n and i=j,,min(n,j+kb);
if order=Nag_RowMajor and uplo=Nag_Upper,
Bij is stored in bb[j-i+(i-1)×pdbb], for i=1,,n and j=i,,min(n,i+kb);
if order=Nag_RowMajor and uplo=Nag_Lower,
Bij is stored in bb[kb+j-i+(i-1)×pdbb], for i=1,,n and j=max(1,i-kb),,i.
On exit: B is overwritten by the elements of its split Cholesky factor S.
6: pdbb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix B in the array bb.
Constraint: pdbbkb+1.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kb=value.
Constraint: kb0.
On entry, n=value.
Constraint: n0.
On entry, pdbb=value.
Constraint: pdbb>0.
NE_INT_2
On entry, pdbb=value and kb=value.
Constraint: pdbbkb+1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_POS_DEF
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

f08ufc 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 function. 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 f08ufc may be followed by a call to f08uec to solve the generalized eigenproblem Az=λBz, where A and B are banded and B is positive definite.
The complex analogue of this function is f08utc.

10 Example

See f08uec.