NAG Library Routine Document

f07hff (dpbequ)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{uplo}$ – Character(1)Input

On entry: indicates whether the upper or lower triangular part of $A$ is stored in the array ab, as follows:

$\mathbf{uplo}=\text{'U'}$

The upper triangle of $A$ is stored.

$\mathbf{uplo}=\text{'L'}$

The lower triangle of $A$ is stored.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathbf{n}$ – IntegerInput

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

Constraint: $\mathbf{n}\ge 0$.

3:     $\mathbf{kd}$ – IntegerInput

On entry: $k_d$, the number of superdiagonals of the matrix $A$ if $\mathbf{uplo}=\text{'U'}$, or the number of subdiagonals if $\mathbf{uplo}=\text{'L'}$.

Constraint: $\mathbf{kd}\ge 0$.

4:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array ab must be at least $\max \left(1,\mathbf{n}\right)$.

On entry: the upper or lower triangle of the symmetric positive definite band matrix $A$ whose scaling factors are to be computed.

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

Only the elements of the array ab corresponding to the diagonal elements of $A$ are referenced. (Row $\left(k_d+1\right)$ of ab when $\mathbf{uplo}=\text{'U'}$, row $1$ of ab when $\mathbf{uplo}=\text{'L'}$.)

5:     $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f07hff (dpbequ) is called.

Constraint: $\mathbf{ldab}\ge \mathbf{kd}+1$.

6:     $\mathbf{s}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: if $\mathbf{info}=\mathbf{0}$, s contains the diagonal elements of the scaling matrix $S$.

7:     $\mathbf{scond}$ – Real (Kind=nag_wp)Output

On exit: if $\mathbf{info}=\mathbf{0}$, scond contains the ratio of the smallest value of s to the largest value of s. If $\mathbf{scond}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by $S$.

8:     $\mathbf{amax}$ – Real (Kind=nag_wp)Output

On exit: $\max \left|a_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.

9:     $\mathbf{info}$ – IntegerOutput

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$\mathbf{info}>0$

The $〈\mathit{\text{value}}〉$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07hffe.f90)

10.2

Program Data

Program Data (f07hffe.d)

10.3

Program Results

Program Results (f07hffe.r)