NAG Library Routine Document

g03baf  (rot_orthomax)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

On entry: indicates if the matrix of loadings is to be row standardized before rotation.

$\mathbf{stand}=\text{'S'}$

The loadings are row standardized.

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

The loadings are left unstandardized.

Constraint: $\mathbf{stand}=\text{'S'}$ or $\text{'U'}$.

2:     $\mathbf{g}$ – Real (Kind=nag_wp)Input

On entry: $\gamma$, the criterion constant with $\gamma =1.0$ giving varimax rotations and $\gamma =0.0$ giving quartimax rotations.

Constraint: $\mathbf{g}\ge 0.0$.

3:     $\mathbf{nvar}$ – IntegerInput

On entry: $p$, the number of original variables.

Constraint: $\mathbf{nvar}\ge \mathbf{k}$.

4:     $\mathbf{k}$ – IntegerInput

On entry: $k$, the number of derived variates or factors.

Constraint: $\mathbf{k}\ge 2$.

5:     $\mathbf{fl}\left(\mathbf{ldfl},\mathbf{k}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: $\Lambda$, the matrix of loadings. $\mathbf{fl}\left(\mathit{i},\mathit{j}\right)$ must contain the loading for the $\mathit{i}$th variable on the $\mathit{j}$th factor, for $\mathit{i}=1,2,\dots ,p$ and $\mathit{j}=1,2,\dots ,k$.

On exit: if $\mathbf{stand}=\text{'S'}$, the elements of fl are standardized so that the sum of squared elements for each row is $1.0$ and then after the computation of the rotations are rescaled; this may lead to slight differences between the input and output values of fl.

If $\mathbf{stand}=\text{'U'}$, fl will be unchanged on exit.

6:     $\mathbf{ldfl}$ – IntegerInput

On entry: the first dimension of the arrays fl and flr as declared in the (sub)program from which g03baf is called.

Constraint: $\mathbf{ldfl}\ge \mathbf{nvar}$.

7:     $\mathbf{flr}\left(\mathbf{ldfl},\mathbf{k}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the rotated matrix of loadings, ${\Lambda }^{*}$. $\mathbf{flr}\left(\mathit{i},\mathit{j}\right)$ will contain the rotated loading for the $\mathit{i}$th variable on the $\mathit{j}$th factor, for $\mathit{i}=1,2,\dots ,p$ and $\mathit{j}=1,2,\dots ,k$.

8:     $\mathbf{r}\left(\mathbf{ldr},\mathbf{k}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the matrix of rotations, $R$.

9:     $\mathbf{ldr}$ – IntegerInput

On entry: the first dimension of the array r as declared in the (sub)program from which g03baf is called.

Constraint: $\mathbf{ldr}\ge \mathbf{k}$.

10:   $\mathbf{acc}$ – Real (Kind=nag_wp)Input

On entry: indicates the accuracy required. The iterative procedure of Cooley and Lohnes (1971) will be stopped and the final refinement computed when the change in $V$ is less than $\mathbf{acc}\times \max \left(1.0,V\right)$. If acc is greater than or equal to $0.0$ but less than machine precision or if acc is greater than $1.0$, machine precision will be used instead.

Suggested value: $0.00001$.

Constraint: $\mathbf{acc}\ge 0.0$.

11:   $\mathbf{maxit}$ – IntegerInput

On entry: the maximum number of iterations.

Suggested value: $30$.

Constraint: $\mathbf{maxit}\ge 1$.

12:   $\mathbf{iter}$ – IntegerOutput

On exit: the number of iterations performed.

13:   $\mathbf{wk}\left(2\times \mathbf{nvar}+\mathbf{k}\times \mathbf{k}+5\times \left(\mathbf{k}-1\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace

14:   $\mathbf{ifail}$ – 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,	$\mathbf{k}<2$,
or	$\mathbf{nvar}<\mathbf{k}$,
or	$\mathbf{g}<0.0$,
or	$\mathbf{ldfl}<\mathbf{nvar}$,
or	$\mathbf{ldr}<\mathbf{k}$,
or	$\mathbf{acc}<0.0$,
or	$\mathbf{maxit}\le 0$,
or	$\mathbf{stand}\ne \text{'S'}$ or $\text{'U'}$.

$\mathbf{ifail}=2$

The singular value decomposition has failed to converge. This is an unlikely error exit.

$\mathbf{ifail}=3$

The algorithm to find $R$ has failed to reach the required accuracy in the given number of iterations. You should try increasing acc or increasing maxit. The returned solution should be a reasonable approximation.

$\mathbf{ifail}=-99$

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 (g03bafe.f90)

10.2

Program Data

Program Data (g03bafe.d)

10.3

Program Results

Program Results (g03bafe.r)