- 1 Installation
- 2 First steps
- 3 Common CAS commands
- 4 Probabilities and statistics
- 5 Graphics
- 6 Programs
- 7 Keyboard shortcuts.
- 8 Remarks
- 9 Copyright and Thanks to.
- 10 Developer infos.
- A Command list.
Abstract: This document explains how to run efficiently CAS on Casio CG50 calculators. CAS is an adapted version of the Giac/Xcas computer algebra system (CAS) for this calculator.
CAS are not allowed during exams in some countries, it is the user responsability to check the rules before running CAS in an exam. The authors shall not be held responsible for misuse of CAS in exam conditions.
This document is interactive, you can modify and run commands by clicking in the ok button or by hitting Enter.
To install or update CAS, get on your computer the file khicasen.g3a
Connect the USB cable of the calculator, type F1 for USB key connection
and copy the file
khicasen.g3a on the calculator-“key”
then disconnect the key from your computer.
If you test from the emulator,
from the main menu of the calculator (MENU), go to Memory
then F3 (Import/Export), then F1 (Import files),
select the file
khicasen.g3a, type F1 to save to
the calculator root directory, confirm with F1 if you upgrade.
Be patient, the transfert will take several minutes (about 5).
Once the transfert is finished, you should see the snowflake icon of Xcas in the main menu.
2 First steps
From the main menu (MENU), move the cursor to the Xcas icon and hit EXE. This opens a “shell” where you can write most Xcas commands.
For example, type
then EXE, you should see the result
You can copy a command from the history of commands by hitting the
up and down arrow keys (once or more) and EXE. Then you can
modify the command and run it with EXE. For example, up arrow,
1/3 and hit EXE.
The last result is stored in
ans(), hit the
calculator key to get it (keys: shift then
It is recommended to store the result in a variable if you want
to reuse a result later. There are two ways to store a value in a
=>using the key, for example
2=>Astores 2 in variable A. Now, every time you write
Ain a computation, it will be replaced by 2.
- left-store with
:=, type shift then F1. for example
A:=2does the same as
If you begin a commandline by hitting ,
the system will automatically insert
it is therefore easy to store a result in a variable.
The most popular Xcas commands are available from the Catalog, where
they are shortly explained with an example.
Hit F4 (CATALOG), choose a submenu,
Algebra, hit EXE,
move the selection to a command, for example
Now F6 will display a short help with an example. Hit F2
to copy the example in the commandline.
You can run it as is (EXE) or modify it and run it (EXE) if
you want to factor another polynomial.
When a command returns an expression, it is displayed in 2d mode. You can move with the pad if the expression is larger than the display. Type + and - to modify the fontsize. Type EXIT to go back to the shell.
Now, try to type the command
Hint: type F4 (CATALOG), then select
When a command returns a graph, it will be displayed in a 2d frame.
You can modify the displayed area with
(zoom in or out,
(-) does a partial zoomout along ),
the cursor keys,
/ (orthonormalisation of the frame),
* (autoscale), VAR or OPTN is a switch to display or hide axis.
Type EXIT to go back to the shell.
The KhiCAS menu (F6) has an item
5 Clear that will erase
the display and optionally erase the variables (in order to start
a new exercice with a fresh CAS). You will see the
/* DEL EXE to */ restart?. Type ON
to erase the display only (variables are not erased)
or DEL then EXE to erase variables.
Hit MENU to leave CAS. If you launch another application, the variables and history will be saved, they will be restored if you come back to CAS (except if you disable auto-save from KhiCAS F6 menu). On the emulator, auto-save is sometimes slow (10 to 20 seconds), it’s faster on the calculator. This is also true for other computations, the calculator is faster, up to 4 times faster than my computer.
It is recommended to hit MENU before OFF, otherwise when you power on the calculator, a history command may be added to the commandline (press AC/ON or/and EXE to restore normal behavior).
3 Common CAS commands
3.1 Expand and factor
From catalog, select
factor: factorization. Shortcut
=>*( key then *), for example
cfactorto factor over .
partfrac: expands a polynomial or performs partial fraction expansion over a fraction. Shortcut
=>+( then + key), for example
simplify: tries to simplify an expression. Shortcut
=>/( key then /), for example
ratnormal: rewrite as an irreducible fraction.
From catalog, select
diff: derivative. Shortcut
'(F3) for derivative with respect to , example
are equivalent. For th-derivative, add , for example 3rd derivative
integrate: antiderivative (1 or 2 arguments) for example
Defined integration with 4 arguments, for example
computes . For an approximate computation, enter one boundary as an approx number, for example
Shortcut: shift F3.
limit: limit of an expression. Example
tabvar: table of variations of an expression. for example
one can check with the graph
series: Taylor expansion or asymptotic serie expansion, for example
polynomialif you do not want to have the remainder term.
sum: discrete summation, for example
computes the sum and rewrites it factored.
Shortcut: ALPHA F3.
From catalog, select
solvesolves an equation exactly. Takes the variable to solve for as second argument, unless it is
x, for example
If exact solving fails, run
fsolvefor approx solving, either with an iterative method starting with a guess
, or by dichotomy
For complex solutions, run
It is possible to restrict solutions using assumptions on the variable, for example
solvecan also solve (simple) polynomial systems, enter a list of equations as 1st argument and a list of variables as 2nd argument, for example intersection of a circle and a line:
linsolveto solve linear systems. enter a list of equations as 1st argument and a list of variables as 2nd argument, example: linsolve([x+2y=3,x-y=7],[x,y])
desolveto solve exactly a differential equation. for example, to solve , type
Another example with an initial condition:
odesolvefor approx solving or
plotodefor a graphic representation of the approx. solution.
rsolvesolves some recurrence relations , for example to solve the arithmetico-geometric recurrence , type:
When required, the distinction between integer arithmetic
and polynomial arithmetic is done by a prefix
integer commands. For example
ifactor for integer factorization
factor for polynomial factorization
cfactor for polynomial factorization over ).
Some commands work for integers and polynomials, like
From catalog, select
irem(a,b)quotient and remainder of euclidean division of two integers.
isprime(n)checks whether is prime. This is a probabilisitic test for large values of .
ifactor(n)factorizes an integer (not too large, since algorithms used are trial division and Pollard-, there is no space left in memory for quadratic sieve), for example
Shortcut then * (
lcm(a,b)GCD and LCM of two integers or polynomials.
iegcd(a,b)returns 3 integers such that where is the GCD of et , and .
ichinrem([a,m],[b,n])returns (if possible) such that and (if are coprime, exists).
powmod(a,n,m)returns computed by the fast modular powering algorithm
ascconverts a string to a list of ASCII code,
charconverts back a list to a string. These commands may be used to easily write cryptographic algorithms with string messages.
From catalog, select
The default variable is , otherwise you can specify it as last
optional argument. For example
degree(x^2*y,x) return 2,
degree(x^2*y,y) returns 1.
coeff(P,n)coefficient of in ,
lcoeff(P)leading coefficient of , for example
degre(P)degree of polynomial
rem(P,Q)quotient and remainder of euclidean division of
proot(P): approx. roots of (all roots, real and complex)
interp(X,Y): for two lists of the same size, returns the interpolating polynomial such that .
resultant(P,Q): resultant of polynomials and
hermite(x,n): -th Hermite polynomial (orthogonal for the density on )
laguerre(x,n,a): -th Laguerre polynomial
legendre(x,n): -th Legendre polynomial (orthogonal for the density on )
tchebyshev2(n)Tchebyshev polynomials of 1st and 2nd kind defined by :
3.5 Linear algebra, vectors, matrices
Xcas does not make distinction between vectors and lists. For example,
defines 2 vectors and , then
dot will compute
the scalar product of and :
A matrix is a list of lists of the same size. You can enter a matrix element by element using the matrix editor (shift-MATR or F6 4). Enter a new variable name to create a new matrix or the name of an existing variable to edit a matrix. For small matrices, it is also convenient to enter them directly in the commandline, for example to define type
It is recommended to store matrices in variables!
If a matrix is defined by a formula, then it’s better to use the
matrix command, for example:
returns the matrix where coefficient line and column is (beware, indices begin at 0).
idn(n) to get the identity matrix of order
get a matrix with random coefficients with dimensions .
For basic arithmetic on matrices, use keyboard operators
+ - *, inverse). Otherwise, open catalog and
eigenvalues and eigenvectors of matrix .
finds the Jordan normal form of matrix , returns matrices and such that , with upper triangular (diagonal if is diagonalizable)
computes matrix to the -th power, where is symbolic.
rref: row reduction to echelon form
lu: factorization of matrix , returns a permutation and two matrices (lower) and (upper) such that . The result of
may be passed as an argument to the command
to solve a system by solving two triangular systems (in instead of ).
qrfactorization of matrix , is orthogonal and upper triangular, .
svd(A)singular value decomposition of matrix returns orthogonal, vector of singular values, orthogonal such that
The ratio of the largest and the smallest singular value of is the condition number of relative to the Euclidean norm.
4 Probabilities and statistics
4.1 Random numbers
From catalog, select
(real in ) or
(integer between 1 and ). Other commands with prefix
rand are available, followed
by the name of the law, for example
returns a random integer according to binomial law of parameters .
For a random vector or matrix, run
Alglin, Matrice submenu),
for example for a vector with 10 random
reals according to normal law (mean 0, stddev 1), type
From catalog, select
There you will find a few distribution laws:
uniformd. Other distribution must be keyed in:
To get the cumulated distribution function, enter the law name then
_cdf suffix (shortcut: select
cdf in the catalog at the end and press F1).
Inverse cumulated distribution function follows the same principle
_icdf suffix (shortcut:
cdf in the catalog and press F2).
Example : find the centered interval for the normal law of
mean 5000, standard deviation 200, such that the probability
to be outside is 5%
4.3 1-d statistics
The statistic functions are taking lists as arguments, for example
From catalog, select
: arithmetic mean of a list
: standard deviation of a list.
to get an unbiaised estimate of the standard deviation of a population from a sample
returns respectivly the median, first and third quartile of a list.
For 1-d statistics with frequencies, replace
by two lists of the same length, the first list being the
values of the serie, the second list the frequencies.
For graphic representations, open catalog,
4.4 2-d statistics
From catalog, select
correlation(X,Y): correlation of two lists and of the same length.
covariance(X,Y):: covariance of two lists and of the same length.
- regression computations:
run commands with suffix
_regression(X,Y), for example
linear_regression(X,Y)returns coefficients of the linear regression line .
linear_regression_plot(X,Y)and all commands of suffix
_regression_plotwill display the line (or curve) of the regression. These commands will also print the coefficient that give information on the quality of adjustment ( near 1 is good).
From catalog, select
Graphics (shortcut 7).
plot(f(x),x=a..b)plot expression for . Discretization option:
xstep=, for example
Default is 384 evaluations per plot (one per horizontal pixel).
plotseq(f(x),x=[u0,a,b])webplot for a recurrent sequence of first term , for example if , with a plot on
plotparam([x(t),y(t)],t=tm..tM)parametric plot for . Discretization option:
plotpolar(r(theta),theta=a..b)polar plot of for , for example
plotlist(l): plot a list
l, i.e. draws a polygonal line with vertices (index starts at 0).
plotlist([X1,Y1],[X2,Y2],...)polygonal line with vertices the points of coordinates
polygonscatterplot(X,Y)for two lists
X,Yof the same size, draws the points or a polygonal line of vertices
histogram(l,class_min,class_size)plots the histogram of data in
l, class size
class_size, first class starts at
class_min. Example: check the random generator quality
plotcontour(f(x,y),[x=xmin..xmax,y=ymin..ymax],[l0,l1,...])plot implicit curves .
plotfield(f(t,y),[t=tmin..tmax,y=ymin..ymax])plot the field of tangents for the differential equation . Add the optional last parameter
,plotode=[t0,y0]to plot simultaneously the solution with initial condition . Example for and
plotodemay be used outside of
For simultaneous plots, write commands separated by
For display options, press the OPTN key:
display=colorcolor option: select a color then press F2, for example
display=line_width_8: change segments width (including inside polygonal line used to plot a curve). Simultaneous display options should be added with
+. For example
- Circles and rectangles with edges parallel to the coordinate
axis may be filled with
display=filled(this attribute might be added to other attributes)
- If you want to define the display window (overwriting
the autoscale computation), select
gl_yand add an or interval, for example
gl_commands must preced the plotting command.
- If you want to remove axes, select
axesand press F2 (
axes=0must preced the plotting command. Axes can be removed interactively when the graph screen is displayed by pressing VARS.
You can program either with Xcas-like syntax (English or French) or with Python-like syntax.
Example : function defined by an algebraic expression
for example simple confidence interval for a frequency in a sample
Second example : more precise confidence interval for a frequency
in a sample of size :
To avoid computing twice the same quantity, one can insert a local variable. Run shift-PRGM to enter function, local, return and ffunction)
The commandline is not well adapted to write these kinds of functions. For non algebraic functions, it is best to run the program editor. Press F6, select New program, enter a filename like conf, then F6, type the program above with the help of the shift-PRGM shortcut for programming structures. Type F6 EXE to check the syntax. Once the program is correct, save it (F6 2), then type EXIT. Now you can call your program from the commandline like this
Third example : a loop printing integer squares
from 1 to in Python syntax.
Check that Python syntax is enabled in the F6 menu, if it is not checked,
check it (7).
Type F6 New program, enter
test for the program name then F6.
This should create a file named
test.py and open the program
editor, you should see the following template for a function definition
(press F5 to lock the keyboard in alpha lowercase), move to the end of
the line and press EXE to input a newline.
Type Shift-PRGM then
3 for, then F5 J space alpha, then
6 in range(a,b). Type
then F1 (
:). Type Shift-EXE to insert a newline
then Alpha SPACE,
F4 (CATALOG), EXE (
1 All), P, select
j,j^2) then EXE.
def f(n): for j in range(1,n+1): print(j,j^2)
^ means power,
** is also accepted like in
Now, type F6, select
1. Check syntax. If syntax is correct,
you will see
Success in the status line. Otherwise, the
first error line number and token will be displayed and cursor will
be positionned at the line where the error was detected. Note that
the error may be before this line but it was only detected later. Note
also that if you are using Python syntax compatibility, programming
structures are translated into Xcas, errors are displayed after
translation, therefore you might see token errors like
that were added by the translator.
If the program is correct, you can save it with the F6 menu (save or save as).
You can run it from the commandline by pressing EXIT then for
f(10) should display all squares from 1 to 10.
The turtle is a nice way to learn programming. The turtle is a small
robot that you can move, it handles a pen that marks its path.
Type F6, New program, enter
logo as filename, then select F1 Turtle.
You should see
efface which mean clear the screen. You can
access to the turtle commands using shift-MENU (move the cursor to
a command and press F6 for help). For example try
(forward). Checking the syntax (F6 1 or ON)
will display the turtle window moves.
You can enter several moves in your script, and organize them in functions.
function square(n) repete(4,avance n,tourne_gauche); ffunction:; efface; for n from 1 to 10 do square(10*n); od;
Another example of non algebraic function: the euclidean algorithm
to compute the GCD of two integers.
Press shift-EXE to insert a newline.
! is in the
Programmation_cmds (11, shortcut )
function pgcd(a,b) while b!=0 do a,b:=b,irem(a,b); od; return a; ffunction
def pgcd(a,b): while b!=0: a,b=b,a % b return a
If your program has runtime errors or if you want
to see it run step by step, run
debug on it,
Unlike adaptations of Micro-Python by calculator manufacturers (including
Casio), the Python syntax in Xcas is fully integrated.
You can therefore use all Xcas commands and data types in your programs.
This corresponds approximatively to importing Python modules
There is also a small
pixelised graphic commands set
set_pixel() to synchronize
c parameter is optional,
draw_arc(x,y,rx,ry,t1,t2,c) draws an ellipsis arc).
And you can somewhat replace
with graphic commands of CAS
histogram and all
commands). Plus you have natural access to data types
like rationnals or expressions, and you can run CAS commands on them.
The complete list of commands available on the calculator
is given in appendix. For documentation on commands not listed
in the catalog categories, please refer to Xcas documentation.
7 Keyboard shortcuts.
- F1 to F3 : depends on mode (Python/Xcas) and shift/alpha state, see labels
- F4: catalog.
- F5: uppercase to lowercase switch. If alpha mode is not active, locks the keyboard in alpha lowercase.
- F6: File and config menu
(-): returns the
- shift PRGM: programming commands
- OPTN: all options
- shift-QUIT: turtle commands
- shift-List: create or edit a list
- shift-Mat: create or edit a matrix
- angle key:
- yellow shifted fraction:
- red r key:
- red key:
- SD key:
- yellow shifted SD key:
In programming editor
- shift-cursor key: move to begin/end of line or file
- shift CLIP: begin selection. Move the cursor to the selection end, type DEL to remove the selection (it will be copied to clipboard) or again shift-CLIP to copy selection to clipboard without removal. Type AC/ON to cancel selection.
- EXE: if a search/replace is currently active (F6 6) find next word occurence. Otherwise add a newline.
- shift EXE: add a newline.
- DEL: remove selection or previous character if no selection active
- shift PASTE: copy clipboard
- Shift-INS (touche DEL): remove current line and copy to clipboard
- AC/ON: cancel selection or cancel search/replace or check syntax (like F6 1)
- EXIT: leave editor to the commandline. Type EXIT again to come back to the editor.
This adaptation is not a full version of Xcas because the maximal size for a Casio add-in is too small (2 Mo). There are more complete adaptations of Xcas for calculators, for more informations, refer to Giac/Xcas homepage.
9 Copyright and Thanks to.
- Giac and CAS, computing kernel (c) B. Parisse et R. De Graeve, 2018.
- CAS interface adapted by B. Parisse from Eigenmath source code by Gabrial Maia and from Xcas source code.
- CAS license GPL2. See details in
LICENSE.GPL2file, inside khicasio.zip or GPL2 on the Free Software Foundation website. The source code of CAS and of the required libraries libtommath and USTL are available in the Casio section of my webpage (see section 10).
- Thanks to the active members of tiplanet and Planete Casio for answering questions and testing during the time I developed CAS. Thanks to all contributors of the Prizm programming portal. Thanks to Pavel Demin for compilation tricks that spared about 135K.
- Thanks to Casio France for sending me a calculator and an emulator license.
10 Developer infos.
To build this add-in, I have installed the gcc cross-compiler
for sh3eb CPU following this
I have configured gcc like this
../gcc-5.3.0/configure --target=sh3eb-elf --prefix="$HOME/opt/sh3eb-elf" --disable-nls --disable-shared --disable-multilib --enable-languages=c,c++ --without-headers
Unfortunately, there is no support for sh3eb in the
(C librairy) of gcc, nor for
libfxcg.tar.gz with a few modification
(corrections of small bugs, added missing functions like qsort, ...),
it is available in
this folder (unarchive and compile with make).
In this folder you will also find
tommath.tgz (big integer
ustl.tar.gz (standard template library)
that I also had to modify to make it work with sh3eb-elf-g++,
with partial success, i.e. enough support to build Giac
(vector/string/map supported, I/O on files are not supported,
there is a custom iostream file for cin/cout minimal support).
Unarchive and compile with make.
The file sh3eb-elf.tar.gz is a binary version of gcc/libraries for GNU/Linux debian 9.
The file khicas.tgz is the source of the port of Giac.
mkxcas script to compile Giac.
A Command list.
You can test a command in the commandline at the bottom of the document.
To get short help on a command, click on
+ to increase
the document console size, then type
by the commandname.