NAG Library Routine Document

F01BUF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     N – INTEGERInput

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

2:     M1 – INTEGERInput

On entry: $m+1$, where $m$ is the number of nonzero superdiagonals in $A$. Normally $\mathbf{M1}\ll \mathbf{N}$.

3:     K – INTEGERInput

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

Constraint: $\mathbf{M1}-1\le \mathbf{K}\le \mathbf{N}$.

4:     A(LDA,N) – 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:     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:     W(M1) – REAL (KIND=nag_wp) arrayWorkspace

7:     IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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,

$\mathbf{K}<\mathbf{M1}$ 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.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f01bufe.f90)

9.2  Program Data

Program Data (f01bufe.d)

9.3  Program Results

Program Results (f01bufe.r)