Chapter Contents
NAG Toolbox

# Calling NAG Routines From MATLAB

Although the NAG Library is written in Fortran, you can use it from within MATLAB as if it were a collection of native MATLAB commands. The code in the toolbox will transform your MATLAB data into a form suitable for passing to Fortran, and will transform the results into MATLAB objects on successful completion of the algorithm.

## A simple example

Here is an example of how to use the NAG Library to compute the solution of a real system of linear equations, AX=B, where A is an n by n matrix and X and B are n vectors.
```a = [ 1.80,  2.88,  2.05, -0.89;
5.25, -2.95, -0.95, -3.80;
1.58, -2.69, -2.90, -1.04;
-1.11, -0.66, -0.59,  0.80];
b = [ 9.52;
24.35;
0.77;
-6.22];
[aOut, ipiv, bOut, info] = f07aa(a, b);
bOut
bOut =

1.0000
-1.0000
3.0000
-5.0000
```
Here we see that the NAG routine f07aa takes two arguments, the matrix of coefficients, A, and the vector representing the right-hand side, B (actually, since f07aa can handle multiple right-hand sides, B is really a matrix). It returns four results as follows:
aOut
the LU factorisation of A
ipiv
the pivot indices that describe the permutation matrix used to compute the result
bOut
the solution matrix
info
a diagnostic parameter that indicates whether the algorithm was successful or not
Since info=0 we know that the algorithm succeeded. Had it been non-zero a warning would have been printed (see Errors and Warnings). An example of the use of each NAG routine in MATLAB is provided in the individual routine documents.

## Optional Arguments

Many NAG routines have optional parameters. In these cases the routine will either infer a suitable value from the other inputs, or provide a default that is acceptable in most situations. A very common case is where the underlying Fortran routine requires the user to provide parameters which describe the size of a matrix, which can easily be inferred in MATLAB. For example, in the system of equations given in the previous section, it is obvious that the rank, n, of the matrix A is 4. However we can tell MATLAB that the rank is 3, in which case it will solve the system represented by the top-left 3x3 section of A, and the first three elements of B (see the section on Types for an explanation of the use of the int64 command here):
```[aOut, ipiv, bOut, info] = f07aa(a, b, 'n', int64(3));
bOut

bOut =

4.1631
-2.1249
3.9737
-6.2200
```
The last element of bOut can be ignored. Since b was a 4x1 matrix on input, it will be a 4x1 matrix on output, even though the last element is not being used. A similar outcome can be achieved by:
```[aOut, ipiv, bOut, info] = f07aa(a(1:3,1:3), b(1:3));
bOut

bOut =

4.1631
-2.1249
3.9737
```
To summarise:
1. optional parameters are provided after all compulsory parameters;
2. optional parameters are provided in pairs: a string representing the name of the parameter followed by its value;
3. optional parameters can be provided in any order.
Another common use of optional parameters is to over-ride default values. For example, g01hb computes probabilities associated with a multivariate distribution to a relative accuracy which defaults to 0.0001:
```tail = 'c';
a = [-2; -2; -2; -2];
b = [2; 2; 2; 2];
xmu = [0; 0; 0; 0];
sig = [1.0, 0.9, 0.9, 0.9;
0.9, 1.0, 0.9, 0.9;
0.9, 0.9, 1.0, 0.9;
0.9, 0.9, 0.9, 1.0];

g01hb(tail, a, b, xmu, sig)

ans =

0.9142

```
We can request more or less accuracy by varying the parameter tol:
```g01hb(tail, a, b, xmu, sig,'tol',0.1)

ans =

0.9182
```
Finally, we note that some NAG routines use a different mechanism for setting options, which involves calling an initialisation routine, a separate option setting routine, and then a computational routine. For an example of this see f12fe.

## Errors and Warnings

The NAG routines can throw a number of errors. The names of those errors and the circumstances under which they are likely to be encountered are as follows:
NAG:licenceError
A valid licence for this product could not be found.
NAG:arrayBoundError
An array provided to the routine is too small.
NAG:callBackError
An error occurred when executing an M-File passed as a parameter to the routine.
NAG:missingInputParameters
At least one compulsory input parameter is missing.
NAG:optionalParameterError
Either the optional parameters are not in name/value pairs, or the name provided does not correspond to an optional parameter. Note that this error can arise if some compulsory parameters have been omitted.
NAG:tooManyInputParameters
The user has requested too many input parameters.
NAG:tooManyOutputParameters
The user has requested too many output parameters.
NAG:typeError
A parameter provided to the routine or returned from an M-File called by the routine is of the wrong type. See the section on Types for more details.
NAG:unsetCellArrayError
A cell array has been passed without all elements being set.
NAG:valueError
An incorrect value has been provided for a parameter.
NAG:error_<s><int>
An error occurred during the execution of the routine. Details can be found by consulting the Error Indicators and Warnings section of the document for the particular routine, where <s><int> corresponds to the return value of ifail (or, in chapters f07 and f08, info) with s being 'p' for positive and 'n' for negative. See the description of nag_issue_warnings below.
In most cases the error message will give more precise details of how the error was triggered. For example a NAG:arrayBoundError might display the message:
```
??? The dimension of parameter 2 (A) should be at least 4

```
The NAG routines can also throw two kinds of warnings:
NAG:truncationWarning
A string was truncated when copying the contents of a cell array of strings into a Fortran data structure.
NAG:warning
The NAG routine returned a warning.
The latter is important, and means that on exit the value of the parameter ifail (or, in chapters f07 and f08, info) was non-zero on exit. For details about how to interpret this value the user should consult the relevant routine document. If the user does not wish to see a warning then they can disable it in the usual way, for example:
```warning('off', 'NAG:warning')
```
In this case it is vital that the user checks the value of ifail or info on exit from the routine.

### The nag_issue_warnings command

As described above, in some circumstances the NAG routine may throw an error of the form NAG:error_<s><int> or issue a warning of the form NAG:warning. Throwing errors allows the user to use try ... catch ... end blocks in their programs which is the preferred style of error handling for many people. On the other hand, when an error is thrown then none of the output parameters of the routine are available for inspection.

In previous versions of the Toolbox all NAG:error_<s><int>'s were NAG:warnings. By default, we only use warnings in cases where the output values may be of use (for example in determining the cause of the problem, or as a "warm start" in subsequent calls to the routine), or where the routine has found a solution but there are caveats, for example as to its accuracy. In all other cases we now issue an error.

Should the user prefer to receive a warning and interrogate the return value of ifail (or, in chapters f07 and f08, info) then they can use the function nag_issue_warnings to toggle the behaviour of the Toolbox as follows:

nag_issue_warnings(true)
disable the use of NAG:error_<s><int> and issue warnings
nag_issue_warnings(false)
enable the use of NAG:error_<s><int>
nag_issue_warnings()
return the current setting (true or false)
Note that this command only effects NAG:error_<s><int>'s, not the other kinds of errors documented
above.

## Types

The interfaces to NAG routines in this toolbox are quite precise about the types of their parameters. Since MATLAB assumes by default that every number is a double unless otherwise stated, this means that users need to coerce their input to the appropriate type if it is an integer, a complex number or a boolean. Similarly, in M-Files called by the NAG routine, the user must ensure that the results returned are of the appropriate type. This is to ensure the correct alignment between the MATLAB and Fortran types. Typically the user will call complex to create a complex number, and logical to create a boolean. In 64-bit versions version of the toolbox, the user will call int64 to create an integer. In other versions the user will call int32. The size of integers is chosen to be compatible with those used internally by MATLAB's implementation of LAPACK. There are numerous examples of type conversion in the routine documents. If an object of the incorrect type is provided then a NAG:typeError will be thrown:
```s01ea(0)

??? Parameter number 1 is not a complex scalar of class double.
```

## Integers

The type of integers used by the NAG Toolbox is chosen to be compatible with those used internally by MATLAB's implementation of LAPACK. In this implementation of the toolbox integers are 64 bits long, i.e. of class int64.

It should be noted that versions of MATLAB prior to 2011a do not support some operations on objects of class int64, in particular the colon operator is not defined and so they cannot be used as the start or end point of a loop. (Prior to MATLAB 2010b arithmetic operations, on int64 were not supported at all.) If using these MATLAB versions it is therefore necessary to use double variables and just cast to int64 while calling the NAG function.

If users wish to write code that is portable across versions of the NAG toolbox that use both 32 and 64 bit integers, then we provide two functions as follows:

nag_int(x)
converts x to the integer type compatible with the current version of the NAG toolbox
nag_int_name
returns the name of the integer class compatible with the current version of the NAG toolbox
These can be used as follows:
```nag_int(42);
n = zeros(10, nag_int_name);
```

## Strings

When handling string valued arguments, the underlying Fortran routines usually expect strings of a specified length (which is specified in the parameter description). If strings longer than this are supplied, they will be truncated to the specified length. This means that a typical string valued argument with minimum length 1 and a constraint that the value may be `'U'` or `'L'` may be suppled a value of `'Lower'` which would be accepted as equivalent to `'L'`. Note that you must ensure that any supplied strings are at least the minimum length by padding with spaces or other suitable characters if necessary.

## Providing M-Files as Arguments

Many NAG routines expect the user to provide an M-File to evaluate a function, which might represent an integrand, the objective function in an optimization problem etc. Here is an example showing how to solve an integral:
```d01ah(0, 1, 1e-5, 'd01ah_f', int64(0))
ans =

3.1416
```
The integrand is contained in the file 'd01ah_f.m', which looks like:
```function [result] = d01ah_f(x)
result=4.0/(1.0+x^2);
```
The user should consult the MATLAB documentation on Editing and Debugging M-Files for general advice. For every instance where a NAG routine expects an M-File to be provided, an example is given.

## Using Function Handles instead of M-Files

An alternative to providing M-Files as arguments is to use is to use function handles. These are useful in three main circumstances:
1. when the argument is an existing MATLAB command;
2. when the argument is a simple expression returning one value which can be represented as an anonymous function;
3. when the argument is a function that is local to an m-file.

An example of the first case would be integrating the sin function which can be done as follows:

```[result, abserr] = d01aj(@sin, 0, pi, 1e-5, 1e-5)

result =

2

abserr =

1.1102e-14
```

An example of the second case would be to integrate an arithmetic expression as follows:

```[result, abserr] = d01aj(@(x) 4.0/(1.0+x^2), 0, 1, 1e-5, 1e-5)

result =

3.1416

abserr =

1.7439e-14
```

An example of the third case would be a program such as solve_equations.m:

```function solve_equations

x = -ones(9,1);
[x, ~, ifail] = c05nb(@fcn, x);

if (ifail == 0)
fprintf('The root of the equation is at:\n');
disp(x);
end

function [fvec, iflag] = fcn(n,x,iflag)
fvec = zeros(n, 1);
for k = 1:double(n)
fvec(k) = (3.0-2.0*x(k))*x(k)+1.0;
if k > 1
fvec(k) = fvec(k) - x(k-1);
end
if k < n
fvec(k) = fvec(k) - 2*x(k+1);
end
end
```
Note that in this case the local function fcn is too complicated to be represented as an anonymous function.

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