Demonstration Functions - Untitled
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### B.2 Demonstration Functions

— Function File: demo ('name',n)

> Runs any examples associated with the function 'name'. Examples are stored in the script file, or in a file with the same name but no extension somewhere on your path. To keep them separate from the usual script code, all lines are prefixed by `%!`. Each example is introduced by the keyword 'demo' flush left to the prefix, with no intervening spaces. The remainder of the example can contain arbitrary octave code. For example:

>                  %!demo
>                  %! t=0:0.01:2*pi; x = sin(t);
>                  %! plot(t,x)
>                  %! %-------------------------------------------------
>                  %! % the figure window shows one cycle of a sine wave

> Note that the code is displayed before it is executed, so a simple comment at the end suffices. It is generally not necessary to use disp or printf within the demo.

> Demos are run in a function environment with no access to external variables. This means that all demos in your function must use separate initialization code. Alternatively, you can combine your demos into one huge demo, with the code:

>                  %! input("Press <enter> to continue: ","s");

> between the sections, but this is discouraged. Other techniques include using multiple plots by saying figure between each, or using subplot to put multiple plots in the same window.

> Also, since demo evaluates inside a function context, you cannot define new functions inside a demo. Instead you will have to use `eval(example('function',n))` to see them. Because eval only evaluates one line, or one statement if the statement crosses multiple lines, you must wrap your demo in "if 1 <demo stuff> endif" with the 'if' on the same line as 'demo'. For example,

>                 %!demo if 1
>                 %!  function y=f(x)
>                 %!    y=x;
>                 %!  endfunction
>                 %!  f(3)
>                 %! endif

> See also: test, example.

— Function File: example ('name',n)
— Function File: [x, idx] = example ('name',n)

> Display the code for example n associated with the function 'name', but do not run it. If n is not given, all examples are displayed.

> Called with output arguments, the examples are returned in the form of a string x, with idx indicating the ending position of the various examples.

> See `demo` for a complete explanation.

> See also: demo, test.

— Function File: speed (f, init, max_n, f2, tol)
— Function File: [order, n, T_f, T_f2] = speed (...)

> Determine the execution time of an expression for various n. The n are log-spaced from 1 to max_n. For each n, an initialization expression is computed to create whatever data are needed for the test. If a second expression is given, the execution times of the two expressions will be compared. Called without output arguments the results are presented graphically.

> f
>   The expression to evaluate.
> max_n
>   The maximum test length to run. Default value is 100. Alternatively, use `[min_n,max_n]` or for complete control, `[n1,n2,...,nk]`.
> init
>   Initialization expression for function argument values. Use k for the test number and n for the size of the test. This should compute values for all variables listed in args. Note that init will be evaluated first for k=0, so things which are constant throughout the test can be computed then. The default value is x`= randn (`n`, 1);`.
> f2
>   An alternative expression to evaluate, so the speed of the two can be compared. Default is `[]`.
> tol
>   If tol is `Inf`, then no comparison will be made between the results of expression f and expression f2. Otherwise, expression f should produce a value v and expression f2 should produce a value v2, and these shall be compared using `assert(`v`,`v2`,`tol`)`. If tol is positive, the tolerance is assumed to be absolute. If tol is negative, the tolerance is assumed to be relative. The default is `eps`.
> order
>   The time complexity of the expression `O(a n^p)`. This is a structure with fields `a` and `p`.
> n
>   The values n for which the expression was calculated and the execution time was greater than zero.
> T_f
>   The nonzero execution times recorded for the expression f in seconds.
> T_f2
>   The nonzero execution times recorded for the expression f2 in seconds. If it is needed, the mean time ratio is just `mean(T_f./T_f2)`.

> The slope of the execution time graph shows the approximate power of the asymptotic running time `O(n^p)`. This power is plotted for the region over which it is approximated (the latter half of the graph). The estimated power is not very accurate, but should be sufficient to determine the general order of your algorithm. It should indicate if for example your implementation is unexpectedly `O(n^2)` rather than `O(n)` because it extends a vector each time through the loop rather than preallocating one which is big enough. For example, in the current version of Octave, the following is not the expected `O(n)`:

>                 speed("for i=1:n,y{i}=x(i); end", "", [1000,10000])

> but it is if you preallocate the cell array `y`:

>                 speed("for i=1:n,y{i}=x(i);end", ...
>                       "x=rand(n,1);y=cell(size(x));", [1000,10000])

> An attempt is made to approximate the cost of the individual operations, but it is wildly inaccurate. You can improve the stability somewhat by doing more work for each `n`. For example:

>                 speed("airy(x)", "x=rand(n,10)", [10000,100000])

> When comparing a new and original expression, the line on the speedup ratio graph should be larger than 1 if the new expression is faster. Better algorithms have a shallow slope. Generally, vectorizing an algorithm will not change the slope of the execution time graph, but it will shift it relative to the original. For example:

>                 speed("v=sum(x)", "", [10000,100000], ...
>                       "v=0;for i=1:length(x),v+=x(i);end")

> A more complex example, if you had an original version of `xcorr` using for loops and another version using an FFT, you could compare the run speed for various lags as follows, or for a fixed lag with varying vector lengths as follows:

>                 speed("v=xcorr(x,n)", "x=rand(128,1);", 100, ...
>                       "v2=xcorr_orig(x,n)", -100*eps)
>                 speed("v=xcorr(x,15)", "x=rand(20+n,1);", 100, ...
>                       "v2=xcorr_orig(x,n)", -100*eps)

> Assuming one of the two versions is in xcorr_orig, this would compare their speed and their output values. Note that the FFT version is not exact, so we specify an acceptable tolerance on the comparison `100*eps`, and the errors should be computed relatively, as `abs((`x`-` y`)./`y`)` rather than absolutely as `abs(`x`-` y`)`.

> Type `example('speed')` to see some real examples. Note for obscure reasons, you can't run examples 1 and 2 directly using `demo('speed')`. Instead use, `eval(example('speed',1))` and `eval(example('speed',2))`.