NAG CL Interface
f07hdc (dpbtrf)

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1 Purpose

f07hdc computes the Cholesky factorization of a real symmetric positive definite band matrix.

2 Specification

#include <nag.h>
void  f07hdc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, NagError *fail)
The function may be called by the names: f07hdc, nag_lapacklin_dpbtrf or nag_dpbtrf.

3 Description

f07hdc forms the Cholesky factorization of a real symmetric positive definite band matrix A either as A=UTU if uplo=Nag_Upper or A=LLT if uplo=Nag_Lower, where U (or L) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as A.

4 References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville https://www.netlib.org/lapack/lawnspdf/lawn14.pdf
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: kd Integer Input
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
5: ab[dim] double Input/Output
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the n×n symmetric positive definite band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ab[kd+i-j+(j-1)×pdab], for j=1,,n and i=max(1,j-kd),,j;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ab[i-j+(j-1)×pdab], for j=1,,n and i=j,,min(n,j+kd);
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ab[j-i+(i-1)×pdab], for i=1,,n and j=i,,min(n,i+kd);
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ab[kd+j-i+(i-1)×pdab], for i=1,,n and j=max(1,i-kd),,i.
On exit: the upper or lower triangle of A is overwritten by the Cholesky factor U or L as specified by uplo, using the same storage format as described above.
6: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+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, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+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 leading minor of order value is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. There is no function specifically designed to factorize a symmetric band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07bdc or as a full symmetric matrix, by calling f07mdc.

7 Accuracy

If uplo=Nag_Upper, the computed factor U is the exact factor of a perturbed matrix A+E, where
|E|c(k+1)ε|UT||U| ,  
c(k+1) is a modest linear function of k+1, and ε is the machine precision.
If uplo=Nag_Lower, a similar statement holds for the computed factor L. It follows that |eij|c(k+1)εaiiajj.

8 Parallelism and Performance

f07hdc 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 f07hdc may be followed by calls to the functions:
The complex analogue of this function is f07hrc.

10 Example

This example computes the Cholesky factorization of the matrix A, where
A= ( 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 ) .  

10.1 Program Text

Program Text (f07hdce.c)

10.2 Program Data

Program Data (f07hdce.d)

10.3 Program Results

Program Results (f07hdce.r)