NAG Library Routine Document

f01buf  (real_symm_posdef_fac)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

2:     $\mathbf{m1}$ – IntegerInput

On entry: $m+1$, where $m$ is the number of nonzero superdiagonals in $A$. Normally $\mathbf{m1}\ll \mathbf{n}$.

3:     $\mathbf{k}$ – IntegerInput

On entry: $k$, the change-over point in the decomposition.

Constraint: $\mathbf{m1}-1\le \mathbf{k}\le \mathbf{n}$.

4:     $\mathbf{a}\left(\mathbf{lda},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the upper triangle of the $n$ by $n$ symmetric band matrix $A$, with the diagonal of the matrix stored in the $\left(m+1\right)$th row of the array, and the $m$ superdiagonals within the band stored in the first $m$ rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if $n=6$ and $m=2$, the storage scheme is

$$\begin{array}{llllll}\text{*}& \text{*}& a_{13}& a_{24}& a_{35}& a_{46}\\ \text{*}& a_{12}& a_{23}& a_{34}& a_{45}& a_{56}\\ a_{11}& a_{22}& a_{33}& a_{44}& a_{55}& a_{66}\end{array}$$

Elements in the top left corner of the array are not used. The following code assigns the matrix elements within the band to the correct elements of the array:

    Do 20 j = 1, n
       Do 10 i = max(1,j-m1+1), j
          a(i-j+m1,j) = matrix(i,j)
 10    Continue
 20 Continue 

On exit: $A$ is overwritten by the corresponding elements of $L$, $D$ and $U$.

5:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f01buf is called.

Constraint: $\mathbf{lda}\ge \mathbf{m1}$.

6:     $\mathbf{w}\left(\mathbf{m1}\right)$ – Real (Kind=nag_wp) arrayWorkspace

7:     $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry,

$\mathbf{k}<\mathbf{m1}-1$ or $\mathbf{k}>\mathbf{n}$.

$\mathbf{ifail}=2$

$\mathbf{ifail}=3$

The matrix $A$ is not positive definite, perhaps as a result of rounding errors, giving an element of $D$ which is zero or negative. $\mathbf{ifail}=\mathbf{3}$ when the failure occurs in the leading principal sub-matrix of order k and $\mathbf{ifail}=\mathbf{2}$ when it occurs in the complement.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f01bufe.f90)

10.2

Program Data

Program Data (f01bufe.d)

10.3

Program Results

Program Results (f01bufe.r)