NAG Library Routine Document

F07HFF (DPBEQU)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     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:     N – INTEGERInput

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

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

3:     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:     AB(LDAB,$*$) – 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:     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:     S(N) – REAL (KIND=nag_wp) arrayOutput

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

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

$\mathbf{INFO}>0$

If $\mathbf{INFO}=i$, the $i$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07hffe.f90)

9.2  Program Data

Program Data (f07hffe.d)

9.3  Program Results

Program Results (f07hffe.r)