NAG Library Routine Document

F03ACF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     A(LDA,M1) – REAL (KIND=nag_wp) arrayInput

On entry: the lower triangle of the $n$ by $n$ positive definite symmetric band matrix $A$, with the diagonal of the matrix stored in the $\left(m+1\right)$th column of the array, and the $m$ subdiagonals within the band stored in the first $m$ columns of the array. Each row of the matrix is stored in the corresponding row of the array. For example, if $n=5$ and $m=2$, the storage scheme is

$$\left(\begin{array}{lll}\text{*}& \text{*}& a_{11}\\ \text{*}& a_{21}& a_{22}\\ a_{31}& a_{32}& a_{33}\\ a_{42}& a_{43}& a_{44}\\ a_{53}& a_{54}& a_{55}\end{array}\right)\text{.}$$

The elements in the top left corner of the array are not used. The following code may be used to assign elements within the band of the lower triangle of the matrix to the correct elements of the array:

 DO 20 I = 1, N DO 10 J = MAX(1,I-M), I A(I,J-I+M+1) = matrix(I,J) 10 CONTINUE 20 CONTINUE 

2:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F03ACF is called.

Constraint: $\mathbf{LDA}\ge \mathbf{N}$.

3:     N – INTEGERInput

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

4:     M – INTEGERInput

On entry: $m$, the number of subdiagonals within the band of $A$.

5:     DET – REAL (KIND=nag_wp)Output

On exit: the determinant of $A$.

6:     RL(LDRL,M1) – REAL (KIND=nag_wp) arrayOutput

On exit: the lower triangular matrix $L$, stored in the same way as $A$, except that in place of the diagonal elements, their reciprocals are stored.

7:     LDRL – INTEGERInput

On entry: the first dimension of the array RL as declared in the (sub)program from which F03ACF is called.

Constraint: $\mathbf{LDRL}\ge \mathbf{N}$.

8:     M1 – INTEGERInput

On entry: the value $m+1$.

9:     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

$\mathbf{IFAIL}=1$

The matrix $A$ is not positive definite, possibly due to rounding errors.

$\mathbf{IFAIL}=2$

Overflow. The value of the determinant is too large to be held in the computer.

$\mathbf{IFAIL}=3$

Underflow. The value of the determinant is too small to be held in the computer.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f03acfe.f90)

9.2  Program Data

Program Data (f03acfe.d)

9.3  Program Results

Program Results (f03acfe.r)