Functions
defined by an algorithm
Introduction
Goals
1.
to define functions from algorithms, consistent with functions already
implemented in Casyopee,
2.
to display graphs, tables and curves of these functions in the same
screens as the other functions, and allow certain calculations.
3.
to propose a simple programming structure for a first initiation.
Situations
1.
in a geometric context: trajectories
2.
in analysis : approximation of antiderivative, Euler method for
differential equations
3. ...
imagine .
.
Choices
1.
Two special variables x and y are predeclared.
2. The function is
defined
*
As a piecewise linear function f
such that f (x) = y for each
pair of values (x, y) obtained
in the iteration.
* Or
as a function f with values
in IR² f (t) = [X (t); Y (t)], such that the function X (resp. Y) is piecewise linear and, for
each integer t takes the
value of x (resp y) obtained at step t of the iteration.
3.
The other variables are identified by an alphabetical character not
already used for another objet.
4. The program
is made of two blocks
*
A declaration and initialization block.
* An
iteration block under one of four forms.
5.
Inside the body of the loop,
assigning a formula
to a declared
variable
is only
permitted.
6.
The condition for exiting (or continuating) the loop is either variable
> formula or variable e" formula or variable <formula
or variable
d" formula,
7.
The formulae
are entered like in other Casyopee
modules.
They
may include one or more program variables and other symbolic objects
defined in Casyopee (functions,
parameters).
8. There is no
input / output instruction. The
observable result of programming is the creation of a function.
9.
There is no alternative structure (if .. then .. else...).
Interface
The functions by algorithm
are created via the algebra
tab menu.
The creation window opens
The menu is
detailed below.
The
central part has two resizable panels. On the left is the program
area. One
operates in this area by way of the first 5 menu entries.
On
top of the right panel there is a test area where the successive values
of the variables involved in the iteration will be displayed. Below,
there are several choices:
(1)
the choice of the type of function
*
Choosing left corresponds to a piecewise linear function f such that f (x) = y for each pair of values (x, y) obtained in the iteration.
With
this choice, the iteration will be stopped as soon as two consecutive
values of x will be in decreasing order.
*
Choosing the right corresponds to a function f with values
in IR² f (t) = [X (t); Y (t)], such that the function X (resp. Y) is piecewise linear and, for
each integer t takes the
value of x (resp y) obtained at step t of the iteration.
With
this choice, the iteration will be continued without considering the
order of the successive values of x.
The
difference between the two options is illustrated by the last example
in the UserManual.
(2)
a field to enter a free comment to identify the function.
(3) the
maximum number of iterations. If
the program does not end with a number of iterations less than the
value chosen, it will be stopped when this number is reached.
At
the bottom of the window, there is the creation bar common to all input
windows: choice of an identifier of the function, validation or exit. Validation
is only possible after a test without syntax or execution
error.
If
you choose to exit, no function is created, but the program is not
lost: it will be available again at the next activation of the window. If
you choose OK, the function appears in the list of functions of the
algebra strand, with the comment to identify it.
Then with the context menu (right
click) you can either delete it
or change it.
The function button also
appears in the curves menu of the geometry
tab and among the buttons of the
graphical window.
See their uses in the UserManual.menu
entries
Menu entries
Declare variable
This entry creates a variable
and afffecter it an initial
value.
In the bar that opens
there is a list to choose an
identifier in the set of available characters
and a field to enter a formula.
The syntax of the formula will
be checked during a test.
After validation, a line is
created: identifier ! formula.
Note: when a line of the
declaration part is selected, pressing
the Enter key has the same effect.
Assign variable in the loop
This entry will affect a value
to a declared variable
in the body of the iteration (loop),
In the bar that opens
there is a list to choose an
identifier in the set of declared variables
and a field to enter a formula.
The syntax of the formula will
be checked during a test.
After validation, a line is
created: identifier ! formula
Note: when a line of the loop
body is selected, pressing
the Enter key has the same effect.
Modify
This entry allows you to edit a
line already entered.
Two cases:
(1) it is one of the two
lines delimiting the loop body.
Initially the structure is Loop ...
EndLoop. With this structure, the program will
stop only when the maximum number of
iterations has been reached.
In the bar that opens, three
other structures can be selected using
the list on the left.
 If the WHILE structure
is selected, two lists (one for
the variable, the other for signs
>, e", <or
d") and a formula input
field are provided for a continuation
condition.
 If the FOR structure is
selected to a list for
the loop variable and a formula
input field are provided. The
entry field is the upper value taken by the
variable.
 If theRepeat until
structure is chosen two
lists (one for the variable,
the other for one of the signs>,
e", <or d") and
a formula input field are
provided to a condition of stop.
Note: in the WHILE
(resp. Repeat until)
structure,Casyopee will check that
the variable in the continuation
(resp stop) condition
e volves in the loop.
In the structure FOR it will be verified
that there is no assignment to the
loop variable.
(2) for other lines,
an input bar lets you change the
variable and / or formula.
Note: double click on a line
has the same effect
Comment
This entry inserts a line of text that will not be evaluated.
Delete line
Removes the selected line. It is not possible to delete the declaration
lines of x and y, or those delimiting the loop body.
Test
The first click will display a check list of the program variables.
We can thus choose the variables whose values we want to examine during
the iteration.
A second click will display a
table of he sequence of values of the
selected variables.
All syntax and execution
errors are detected.
If there is no error, it is
possible to create the function by the
OK button.
Parameters
The formulae used in the program may
involve algebraic objects Casyopee,
and therefore the parameters.
During the test, the current
values of the parameters are used. It
is therefore useful to set and /
or edit the associated data (min,
max, not, etc.
This menu item will display a
parameters management window.
Reset all
Resets the program to the
initial state.
