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@node Test and Demo Functions
@appendix Test and Demo Functions
@cindex test functions

Octave includes a number of functions to allow the integration of testing
and demonstration code in the source code of the functions themselves.

@menu
* Test Functions::
* Demonstration Functions::
@end menu

@node Test Functions
@section Test Functions

@c test scripts/testfun/test.m
@anchor{XREFtest}
@deftypefn  {} {} test @var{name}
@deftypefnx {} {} test @var{name} quiet|normal|verbose
@deftypefnx {} {} test ("@var{name}", "quiet|normal|verbose", @var{fid})
@deftypefnx {} {} test ("@var{name}", "quiet|normal|verbose", @var{fname})
@deftypefnx {} {@var{success} =} test (@dots{})
@deftypefnx {} {[@var{n}, @var{nmax}, @var{nxfail}, @var{nbug}, @var{nskip}, @var{nrtskip}, @var{nregression}] =} test (@dots{})
@deftypefnx {} {[@var{code}, @var{idx}] =} test ("@var{name}", "grabdemo")
@deftypefnx {} {} test ([], "explain", @var{fid})
@deftypefnx {} {} test ([], "explain", @var{fname})

Perform built-in self-tests from the first file in the loadpath matching
@var{name}.

@code{test} can be called in either command or functional form.  The exact
operation of test is determined by a combination of mode (interactive or
batch), reporting level (@qcode{"quiet"}, @qcode{"normal"},
@qcode{"verbose"}), and whether a logfile or summary output variable is
used.

The default mode when @code{test} is called from the command line is
interactive.  In this mode, tests will be run until the first error is
encountered, or all tests complete successfully.  In batch mode, all tests
are run regardless of any failures, and the results are collected for
reporting.  Tests which require user interaction, i.e., demo blocks,
are never run in batch mode.

Batch mode is enabled by either 1) specifying a logfile using the third
argument @var{fname} or @var{fid}, or 2) requesting an output argument
such as @var{success}, @var{n}, etc.

The optional second argument determines the amount of output to generate and
which types of tests to run.  The default value is @qcode{"normal"}.
Requesting an output argument will suppress printing the final summary
message and any intermediate warnings, unless verbose reporting is
enabled.

@table @asis
@item @qcode{"quiet"}
Print a summary message when all tests pass, or print an error with the
results of the first bad test when a failure occurs.  Don't run tests which
require user interaction.

@item @qcode{"normal"}
Display warning messages about skipped tests or failing xtests during test
execution.
Print a summary message when all tests pass, or print an error with the
results of the first bad test when a failure occurs.  Don't run tests which
require user interaction.

@item @qcode{"verbose"}
Display tests before execution.  Print all warning messages.  In interactive
mode, run all tests including those which require user interaction.
@end table

The optional third input argument specifies a logfile where results of the
tests should be written.  The logfile may be a character string
(@var{fname}) or an open file descriptor ID (@var{fid}).  To enable batch
processing, but still print the results to the screen, use @code{stdout} for
@var{fid}.

When called with just a single output argument @var{success}, @code{test}
returns true if all of the tests were successful.  If called with more
than one output argument then the number of successful tests (@var{n}),
the total number of tests in the file (@var{nmax}), the number of xtest
failures (@var{nxfail}), the number of tests failed due known bugs
(@var{nbug}), the number of tests skipped due to missing features
(@var{nskip}), the number of tests skipped due to run-time
conditions (@var{nrtskip}), and the number of regressions
(@var{nregression}) are returned.

Example

@example
@group
test sind
@result{}
PASSES 5 out of 5 tests

[n, nmax] = test ("sind")
@result{}
n =  5
nmax =  5
@end group
@end example

Additional Calling Syntaxes

If the second argument is the string @qcode{"grabdemo"}, the contents of
any built-in demo blocks are extracted but not executed.  The text for all
code blocks is concatenated and returned as @var{code} with @var{idx} being
a vector of positions of the ends of each demo block.  For an easier way to
extract demo blocks from files, @xref{XREFexample,,example}.

If the second argument is @qcode{"explain"} then @var{name} is ignored and
an explanation of the line markers used in @code{test} output reports is
written to the file specified by @var{fname} or @var{fid}.

@seealso{@ref{XREFassert,,assert}, @ref{XREFfail,,fail}, @ref{XREFdemo,,demo}, @ref{XREFexample,,example}, @ref{XREFerror,,error}}
@end deftypefn


@code{test} scans the named script file looking for lines which start
with the identifier @samp{%!}.  The prefix is stripped off and the rest
of the line is processed through the Octave interpreter.  If the code
generates an error, then the test is said to fail.

Since @code{eval()} will stop at the first error it encounters, you must
divide your tests up into blocks, with anything in a separate
block evaluated separately.  Blocks are introduced by valid keywords like
@code{test}, @code{function}, or @code{assert} immediately following @samp{%!}.
A block is defined by indentation as in Python.  Lines beginning with
@samp{%!<whitespace>} are part of the preceeding block.

For example:

@example
@group
%!test error ("this test fails!")
%!test "test doesn't fail.  it doesn't generate an error"
@end group
@end example

When a test fails, you will see something like:

@example
@group
  ***** test error ("this test fails!")
!!!!! test failed
this test fails!
@end group
@end example

Generally, to test if something works, you want to assert that it
produces a correct value.  A real test might look something like

@example
@group
%!test
%! @var{a} = [1, 2, 3; 4, 5, 6]; B = [1; 2];
%! expect = [ @var{a} ; 2*@var{a} ];
%! get = kron (@var{b}, @var{a});
%! if (any (size (expect) != size (get)))
%!   error ("wrong size: expected %d,%d but got %d,%d",
%!          size (expect), size (get));
%! elseif (any (any (expect != get)))
%!   error ("didn't get what was expected.");
%! endif
@end group
@end example

To make the process easier, use the @code{assert} function.  For example,
with @code{assert} the previous test is reduced to:

@example
@group
%!test
%! @var{a} = [1, 2, 3; 4, 5, 6]; @var{b} = [1; 2];
%! assert (kron (@var{b}, @var{a}), [ @var{a}; 2*@var{a} ]);
@end group
@end example

@code{assert} can accept a tolerance so that you can compare results
absolutely or relatively.  For example, the following all succeed:

@example
@group
%!test assert (1+eps, 1, 2*eps)           # absolute error
%!test assert (100+100*eps, 100, -2*eps)  # relative error
@end group
@end example

You can also do the comparison yourself, but still have assert
generate the error:

@example
@group
%!test assert (isempty ([]))
%!test assert ([1, 2; 3, 4] > 0)
@end group
@end example

Because @code{assert} is so frequently used alone in a test block, there
is a shorthand form:

@example
%!assert (@dots{})
@end example

@noindent
which is equivalent to:

@example
%!test assert (@dots{})
@end example

Occasionally a block of tests will depend on having optional
functionality in Octave.  Before testing such blocks the availability of
the required functionality must be checked.  A @code{%!testif HAVE_XXX}
block will only be run if Octave was compiled with functionality
@samp{HAVE_XXX}.  For example, the sparse single value decomposition,
@code{svds()}, depends on having the @sc{arpack} library.  All of the tests
for @code{svds} begin with

@example
%!testif HAVE_ARPACK
@end example

@noindent
Review @file{config.h} or @code{__octave_config_info__ ("build_features")}
to see some of the possible values to check.

Sometimes during development there is a test that should work but is
known to fail.  You still want to leave the test in because when the
final code is ready the test should pass, but you may not be able to
fix it immediately.  To avoid unnecessary bug reports for these known
failures, mark the block with @code{xtest} rather than @code{test}:

@example
@group
%!xtest assert (1==0)
%!xtest fail ("success=1", "error")
@end group
@end example

@noindent
In this case, the test will run and any failure will be reported.
However, testing is not aborted and subsequent test blocks will be
processed normally.  Another use of @code{xtest} is for statistical
tests which should pass most of the time but are known to fail
occasionally.

Each block is evaluated in its own function environment, which means
that variables defined in one block are not automatically shared
with other blocks.  If you do want to share variables, then you
must declare them as @code{shared} before you use them.  For example, the
following declares the variable @var{a}, gives it an initial value (default
is empty), and then uses it in several subsequent tests.

@example
@group
%!shared @var{a}
%! @var{a} = [1, 2, 3; 4, 5, 6];
%!assert (kron ([1; 2], @var{a}), [ @var{a}; 2*@var{a} ])
%!assert (kron ([1, 2], @var{a}), [ @var{a}, 2*@var{a} ])
%!assert (kron ([1,2; 3,4], @var{a}), [ @var{a},2*@var{a}; 3*@var{a},4*@var{a} ])
@end group
@end example

You can share several variables at the same time:

@example
%!shared @var{a}, @var{b}
@end example

You can also share test functions:

@example
@group
%!function @var{a} = fn (@var{b})
%!  @var{a} = 2*@var{b};
%!endfunction
%!assert (fn(2), 4)
@end group
@end example

Note that all previous variables and values are lost when a new
shared block is declared.

Remember that @code{%!function} begins a new block and that
@code{%!endfunction} ends this block.  Be aware that until a new block
is started, lines starting with @samp{%!<space>} will be discarded as comments.
The following is nearly identical to the example above, but does nothing.

@example
@group
%!function @var{a} = fn (@var{b})
%!  @var{a} = 2*@var{b};
%!endfunction
%! assert (fn(2), 4)
@end group
@end example

@noindent
Because there is a space after @samp{%!} the @code{assert} statement does
not begin a new block and this line is treated as a comment.

Error and warning blocks are like test blocks, but they only succeed
if the code generates an error.  You can check the text of the error
is correct using an optional regular expression @code{<pattern>}.
For example:

@example
%!error <passes!> error ("this test passes!")
@end example

If the code doesn't generate an error, the test fails.  For example:

@example
%!error "this is an error because it succeeds."
@end example

@noindent
produces

@example
@group
  ***** error "this is an error because it succeeds."
!!!!! test failed: no error
@end group
@end example

It is important to automate the tests as much as possible, however
some tests require user interaction.  These can be isolated into
demo blocks, which if you are in batch mode, are only run when
called with @code{demo} or the @code{verbose} option to @code{test}.
The code is displayed before it is executed.  For example,

@example
@group
%!demo
%! @var{t} = [0:0.01:2*pi]; @var{x} = sin (@var{t});
%! plot (@var{t}, @var{x});
%! # you should now see a sine wave in your figure window
@end group
@end example

@noindent
produces

@example
@group
funcname example 1:
 @var{t} = [0:0.01:2*pi]; @var{x} = sin (@var{t});
 plot (@var{t}, @var{x});
 # you should now see a sine wave in your figure window

Press <enter> to continue:
@end group
@end example

Note that demo blocks cannot use any shared variables.  This is so
that they can be executed by themselves, ignoring all other tests.

If you want to temporarily disable a test block, put @code{#} in place
of the block type.  This creates a comment block which is echoed
in the log file but not executed.  For example:

@example
@group
%!#demo
%! @var{t} = [0:0.01:2*pi]; @var{x} = sin (@var{t});
%! plot (@var{t}, @var{x});
%! # you should now see a sine wave in your figure window
@end group
@end example

@noindent
The following trivial code snippet provides examples for the use of
fail, assert, error, and xtest:

@example
@group
function @var{output} = must_be_zero (@var{input})
  if (@var{input} != 0)
    error ("Nonzero input!")
  endif
  @var{output} = @var{input};
endfunction

%!fail ("must_be_zero (1)")
%!assert (must_be_zero (0), 0)
%!error <Nonzero> must_be_zero (1)
%!xtest error ("This code generates an error")
@end group
@end example

@noindent
When putting this in a file @file{must_be_zero.m}, and running the test, we see

@example
@group
test must_be_zero verbose

@result{}
>>>>> /path/to/must_be_zero.m
***** fail ("must_be_zero (1)")
***** assert (must_be_zero (0), 0)
***** error <Nonzero> must_be_zero (1)
***** xtest error ("This code generates an error")
!!!!! known failure
This code generates an error
PASSES 3 out of 4 tests (1 expected failure)
@end group
@end example

@subsubheading Block type summary:

@table @code
@item  %!test
@itemx %!test <MESSAGE>
Check that entire block is correct.  If @code{<MESSAGE>} is present, the
test block is interpreted as for @code{xtest}.

@item  %!testif HAVE_XXX
@itemx %!testif HAVE_XXX, HAVE_YYY, @dots{}
@itemx %!testif HAVE_XXX, HAVE_YYY @dots{}; RUNTIME_COND
@itemx %!testif @dots{} <MESSAGE>
Check block only if Octave was compiled with feature @w{@code{HAVE_XXX}}.
@w{@code{RUNTIME_COND}} is an optional expression to evaluate to check
whether some condition is met when the test is executed.  If
@w{@code{RUNTIME_COND}} is false, the test is skipped.  If @code{<MESSAGE>}
is present, the test block is interpreted as for @code{xtest}.

@item  %!xtest
@itemx %!xtest <MESSAGE>
Check block, report a test failure but do not abort testing.  If
@code{<MESSAGE>} is present, then the text of the message is displayed
if the test fails, like this:

@example
!!!!! Known bug:  MESSAGE
@end example

@noindent
If the message is an integer, it is interpreted as a bug ID for the
Octave bug tracker and reported as

@example
!!!!! Known bug: https://octave.org/testfailure/?BUG-ID
@end example

@noindent
in which BUG-ID is the integer bug number.  The intent is to allow
clearer documentation of known problems.

@item  %!error
@itemx %!error <MESSAGE>
@itemx %!warning
@itemx %!warning <MESSAGE>
Check for correct error or warning message.  If @code{<MESSAGE>} is
supplied it is interpreted as a regular expression pattern that is
expected to match the error or warning message.

@item %!demo
Demo only executes in interactive mode.

@item %!#
Comment.  Ignore everything within the block

@item %!shared x,y,z
Declare variables for use in multiple tests.

@item %!function
Define a function for use in multiple tests.

@item %!endfunction
Close a function definition.

@item %!assert (x, y, tol)

@item %!assert <MESSAGE> (x, y, tol)

@item %!fail (CODE, PATTERN)

@item %!fail <MESSAGE> (CODE, PATTERN)
Shorthand for @code{%!test assert (x, y, tol)} or
@code{%!test fail (CODE, PATTERN)}.  If @code{<MESSAGE>} is present, the
test block is interpreted as for @code{xtest}.

@end table

@anchor{test-message-anchor}

When coding tests the Octave convention is that lines that begin with a block
type do not have a semicolon at the end.  Any code that is within a block,
however, is normal Octave code and usually will have a trailing semicolon.
For example,

@example
@group
## bare block instantiation
%!assert (sin (0), 0)
@end group
@end example

@noindent
but

@example
@group
## test block with normal Octave code
%!test
%! assert (sin (0), 0);
@end group
@end example

You can also create test scripts for built-in functions and your own C++
functions.  To do so, put a file with the bare function name (no .m
extension) in a directory in the load path and it will be discovered by
the @code{test} function.  Alternatively, you can embed tests directly in your
C++ code:

@example
@group
/*
%!test disp ("this is a test")
*/
@end group
@end example

@noindent
or

@example
@group
#if 0
%!test disp ("this is a test")
#endif
@end group
@end example

@noindent
However, in this case the raw source code will need to be on the load
path and the user will have to remember to type
@code{test ("funcname.cc")}.

@c assert scripts/testfun/assert.m
@anchor{XREFassert}
@deftypefn  {} {} assert (@var{cond})
@deftypefnx {} {} assert (@var{cond}, @var{errmsg})
@deftypefnx {} {} assert (@var{cond}, @var{errmsg}, @dots{})
@deftypefnx {} {} assert (@var{cond}, @var{msg_id}, @var{errmsg}, @dots{})
@deftypefnx {} {} assert (@var{observed}, @var{expected})
@deftypefnx {} {} assert (@var{observed}, @var{expected}, @var{tol})

Produce an error if the specified condition is not met.

@code{assert} can be called in three different ways.

@table @code
@item  assert (@var{cond})
@itemx assert (@var{cond}, @var{errmsg})
@itemx assert (@var{cond}, @var{errmsg}, @dots{})
@itemx assert (@var{cond}, @var{msg_id}, @var{errmsg}, @dots{})
Called with a single argument @var{cond}, @code{assert} produces an error if
@var{cond} is false (numeric zero).

Any additional arguments are passed to the @code{error} function for
processing.

@item assert (@var{observed}, @var{expected})
Produce an error if observed is not the same as expected.

Note that @var{observed} and @var{expected} can be scalars, vectors,
matrices, strings, cell arrays, or structures.

@item assert (@var{observed}, @var{expected}, @var{tol})
Produce an error if observed is not the same as expected but equality
comparison for numeric data uses a tolerance @var{tol}.

If @var{tol} is positive then it is an absolute tolerance which will produce
an error if @code{abs (@var{observed} - @var{expected}) > abs (@var{tol})}.

If @var{tol} is negative then it is a relative tolerance which will produce
an error if @code{abs (@var{observed} - @var{expected}) >
abs (@var{tol} * @var{expected})}.

If @var{expected} is zero @var{tol} will always be interpreted as an
absolute tolerance.

If @var{tol} is not scalar its dimensions must agree with those of
@var{observed} and @var{expected} and tests are performed on an
element-by-element basis.
@end table
@seealso{@ref{XREFfail,,fail}, @ref{XREFtest,,test}, @ref{XREFerror,,error}, @ref{XREFisequal,,isequal}}
@end deftypefn


@c fail scripts/testfun/fail.m
@anchor{XREFfail}
@deftypefn  {} {} fail (@var{code})
@deftypefnx {} {} fail (@var{code}, @var{pattern})
@deftypefnx {} {} fail (@var{code}, "warning")
@deftypefnx {} {} fail (@var{code}, "warning", @var{pattern})

Return true if @var{code} fails with an error message matching
@var{pattern}, otherwise produce an error.

@var{code} must be in the form of a string that is passed to the Octave
interpreter via the @code{evalin} function, i.e., a (quoted) string constant
or a string variable.

Note that if @var{code} runs successfully, rather than failing, the error
printed is:

@example
          expected error <.> but got none
@end example

If called with two arguments, the return value will be true only if
@var{code} fails with an error message containing @var{pattern}
(case sensitive).  If the code fails with a different error than the one
specified in @var{pattern} then the message produced is:

@example
@group
          expected <@var{pattern}>
          but got <text of actual error>
@end group
@end example

The angle brackets are not part of the output.

When called with the @qcode{"warning"} option @code{fail} will produce an
error if executing the code produces no warning.
@seealso{@ref{XREFassert,,assert}, @ref{XREFerror,,error}}
@end deftypefn


@node Demonstration Functions
@section Demonstration Functions

@c demo scripts/testfun/demo.m
@anchor{XREFdemo}
@deftypefn  {} {} demo @var{name}
@deftypefnx {} {} demo @var{name} @var{n}
@deftypefnx {} {} demo ("@var{name}")
@deftypefnx {} {} demo ("@var{name}", @var{n})

Run example code block @var{n} associated with the function @var{name}.

If @var{n} is not specified, all examples are run.

The preferred location for example code blocks is embedded within the script
m-file immediately following the code that it exercises.  Alternatively,
the examples may be stored in a file with the same name but no extension
located on Octave's load path.  To separate examples from regular script
code all lines are prefixed by @code{%!}.  Each example must also be
introduced by the keyword @qcode{"demo"} flush left to the prefix with no
intervening spaces.  The remainder of the example can contain arbitrary
Octave code.  For example:

@example
@group
%!demo
%! t = 0:0.01:2*pi;
%! x = sin (t);
%! plot (t, x);
%! title ("one cycle of a sine wave");
%! #-------------------------------------------------
%! # the figure window shows one cycle of a sine wave
@end group
@end example

Note that the code is displayed before it is executed so that a simple
comment at the end suffices for labeling what is being shown.  For plots,
labeling can also be done with @code{title} or @code{text}.  It is generally
@strong{not} necessary to use @code{disp} or @code{printf} within the demo.

Demos are run in a stand-alone function environment with no access to
external variables.  This means that every demo must have separate
initialization code.  Alternatively, all demos can be combined into a single
large demo with the code

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

@noindent
between the sections, but this usage is discouraged.  Other techniques to
avoid multiple initialization blocks include using multiple plots with a new
@code{figure} command between each plot, or using @code{subplot} to put
multiple plots in the same window.

Finally, because @code{demo} evaluates within a function context it is not
possible to define new functions within the code.  Anonymous functions make
a good substitute in most instances.  If function blocks @strong{must} be
used then the code @code{eval (example ("function", n))} will allow Octave
to see them.  This has its own problems, however, as @code{eval} only
evaluates one line or statement at a time.  In this case the function
declaration must be wrapped with @qcode{"if 1 <demo stuff> endif"} where
@qcode{"if"} is on the same line as @qcode{"demo"}.  For example:

@example
@group
%!demo if 1
%!  function y = f(x)
%!    y = x;
%!  endfunction
%!  f(3)
%! endif
@end group
@end example

@seealso{@ref{XREFrundemos,,rundemos}, @ref{XREFexample,,example}, @ref{XREFtest,,test}}
@end deftypefn


@c example scripts/testfun/example.m
@anchor{XREFexample}
@deftypefn  {} {} example @var{name}
@deftypefnx {} {} example @var{name} @var{n}
@deftypefnx {} {} example ("@var{name}")
@deftypefnx {} {} example ("@var{name}", @var{n})
@deftypefnx {} {[@var{s}, @var{idx}] =} example (@dots{})

Display the code for example @var{n} associated with the function
@var{name}, but do not run it.

If @var{n} is not specified, all examples are displayed.

When called with output arguments, the examples are returned in the form of
a string @var{s}, with @var{idx} indicating the ending position of the
various examples.

See @code{demo} for a complete explanation.
@seealso{@ref{XREFdemo,,demo}, @ref{XREFtest,,test}}
@end deftypefn


@c rundemos scripts/testfun/rundemos.m
@anchor{XREFrundemos}
@deftypefn  {} {} rundemos ()
@deftypefnx {} {} rundemos (@var{directory})
Execute built-in demos for all m-files in the specified @var{directory}.

Demo blocks in any C++ source files (@file{*.cc}) will also be executed
for use with dynamically linked oct-file functions.

If no directory is specified, operate on all directories in Octave's search
path for functions.
@seealso{@ref{XREFdemo,,demo}, @ref{XREFruntests,,runtests}, @ref{XREFpath,,path}}
@end deftypefn


@c runtests scripts/testfun/runtests.m
@anchor{XREFruntests}
@deftypefn  {} {} runtests ()
@deftypefnx {} {} runtests (@var{directory})
Execute built-in tests for all m-files in the specified @var{directory}.

Test blocks in any C++ source files (@file{*.cc}) will also be executed
for use with dynamically linked oct-file functions.

If no directory is specified, operate on all directories in Octave's search
path for functions.
@seealso{@ref{XREFrundemos,,rundemos}, @ref{XREFtest,,test}, @ref{XREFpath,,path}}
@end deftypefn


@c speed scripts/testfun/speed.m
@anchor{XREFspeed}
@deftypefn  {} {} speed (@var{f}, @var{init}, @var{max_n}, @var{f2}, @var{tol})
@deftypefnx {} {[@var{order}, @var{n}, @var{T_f}, @var{T_f2}] =} speed (@dots{})

Determine the execution time of an expression (@var{f}) for various input
values (@var{n}).

The @var{n} are log-spaced from 1 to @var{max_n}.  For each @var{n}, an
initialization expression (@var{init}) is computed to create any data needed
for the test.  If a second expression (@var{f2}) is given then the
execution times of the two expressions are compared.  When called without
output arguments the results are printed to stdout and displayed
graphically.

@table @code
@item @var{f}
The code expression to evaluate.

@item @var{max_n}
The maximum test length to run.  The default value is 100.  Alternatively,
use @code{[min_n, max_n]} or specify the @var{n} exactly with
@code{[n1, n2, @dots{}, nk]}.

@item @var{init}
Initialization expression for function argument values.  Use @var{k} for
the test number and @var{n} for the size of the test.  This should compute
values for all variables used by @var{f}.  Note that @var{init} will be
evaluated first for @math{k = 0}, so things which are constant throughout
the test series can be computed once.  The default value is
@code{@var{x} = randn (@var{n}, 1)}.

@item @var{f2}
An alternative expression to evaluate, so that the speed of two
expressions can be directly compared.  The default is @code{[]}.

@item @var{tol}
Tolerance used to compare the results of expression @var{f} and expression
@var{f2}.  If @var{tol} is positive, the tolerance is an absolute one.
If @var{tol} is negative, the tolerance is a relative one.  The default is
@code{eps}.  If @var{tol} is @code{Inf}, then no comparison will be made.

@item @var{order}
The time complexity of the expression @math{O(a*n^p)}.  This is a
structure with fields @code{a} and @code{p}.

@item @var{n}
The values @var{n} for which the expression was calculated @strong{AND}
the execution time was greater than zero.

@item @var{T_f}
The nonzero execution times recorded for the expression @var{f} in seconds.

@item @var{T_f2}
The nonzero execution times recorded for the expression @var{f2} in seconds.
If required, the mean time ratio is simply @code{mean (T_f ./ T_f2)}.

@end table

The slope of the execution time graph shows the approximate power of the
asymptotic running time @math{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 an algorithm.  It should indicate if, for
example, the implementation is unexpectedly @math{O(n^2)} rather than
@math{O(n)} because it extends a vector each time through the loop rather
than pre-allocating storage.  In the current version of Octave, the
following is not the expected @math{O(n)}.

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

@noindent
But it is if you preallocate the cell array @code{y}:

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

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

@example
speed ("airy(x)", "x = rand (n, 10)", [10000, 100000])
@end example

When comparing two different expressions (@var{f}, @var{f2}), the slope of
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 will shift it relative to the original.  For example:

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

The following is a more complex example.  If there was an original version
of @code{xcorr} using for loops and a second version using an FFT, then
one could compare the run speed for various lags as follows, or for a fixed
lag with varying vector lengths as follows:

@example
@group
speed ("xcorr (x, n)", "x = rand (128, 1);", 100,
       "xcorr_orig (x, n)", -100*eps)
speed ("xcorr (x, 15)", "x = rand (20+n, 1);", 100,
       "xcorr_orig (x, n)", -100*eps)
@end group
@end example

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
one must specify an acceptable tolerance on the comparison @code{100*eps}.
In this case, the comparison should be computed relatively, as
@code{abs ((@var{x} - @var{y}) ./ @var{y})} rather than absolutely as
@code{abs (@var{x} - @var{y})}.

Type @kbd{example ("speed")} to see some real examples or
@kbd{demo ("speed")} to run them.
@end deftypefn

