Getting started with Yacas

Introduction

Yacas (Yet Another Computer Algebra System) is a small and highly flexible general-purpose computer algebra system and programming language. The language has a familiar, C-like infix-operator syntax. The distribution contains a small library of mathematical functions, but its real strength is in the language in which you can easily write your own symbolic manipulation algorithms. The core engine supports arbitrary precision arithmetic (for faster calculations, it can also optionally be linked with the GNU arbitrary precision math library libgmp) and is able to execute symbolic manipulations on various mathematical objects by following user-defined rules.

Currently, the Yacas programming language is stable and seems powerful enough for all computer algebra applications. External libraries providing additional functionality may be dynamically loaded into Yacas via the "plugin" mechanism.

Installing Yacas

Read the file INSTALL for instructions on how to compile Yacas. Yacas is portable across most Unix-ish platforms and requires only a standard C++ compiler such as g++.

The base Yacas application accepts text as input and returns text as output. This makes it rather platform-independent. Apart from Unix-like systems, Yacas has been compiled on Windows and on EPOC32, aka Psion (which doesn't come with a standard C++ library!). The source code to compile Yacas for Windows can be found at the Sourceforge repository .

For Unix, compilation basically amounts to the standard sequence 

./configure
make
make install
This will install the binaries to /usr/local/bin and the library files to /usr/local/share/yacas/.

The arbitrary precision math in Yacas will be generally faster if you compile Yacas with the libgmp library (the option --enable-gmp for the configure script). Precompiled Red Hat (RPM) and Debian (DEB) packages are also available.

Additionally, LaTeX-formatted documentation in PostScript and PDF formats can be produced by the command 

make texdocs

Using the console mode

You can run Yacas in the console mode simply by typing yacas. The Yacas command prompt looks like this: 
In>
and Yacas's answers appear after the prompt 
Out>

A Yacas session may be terminated by typing Exit() or quit. Pressing ^C will also quit Yacas; however, pressing ^C while Yacas is busy with a calculation will stop just that calculation. A session can be restarted (forgetting all previous definitions and results) by typing 
restart

Typically, you would enter one statement per line, for example 

In> Sin(Pi/2);
Out> 1;

Statements should end with a semicolon (;) although this is not required (Yacas will append a semicolon at end of line to finish the statement).

All documentation is accessible from the Yacas prompt. If you type 

In> ??
you should be able to read all available manuals; Yacas will run lynx or another browser to show you the HTML documentation. You can also get help on individual functions: to read about the function Sum(), type 
In> ?Sum

Type Example(); to get some random examples of Yacas calculations.

The command line has a history list, so it should be easy to browse through the expressions you entered previously using the Up and Down arrow keys. Typing the first few characters of a previous expression and then hitting the TAB key makes Yacas recall the last expression in the history list that matches these first characters.

Commands spanning multiple lines can (and actually have to) be entered by using a trailing backslash \ at end of each continued line. For example: 

In> a:=2+3+
Error on line 1 in file [CommandLine]
Line error occurred on:
>>>
Error parsing expression


In> a:=2+3+ \
In> 1
Out> 6;
The error after our first attempt occurred because Yacas has appended a semicolon at end of the first line and 2+3+; is not a valid Yacas expression.

Incidentally, any text Yacas prints without a prompt is either messages printed by functions as their side-effect, or error messages. Resulting values of expressions are always printed after an Out> prompt.

Yacas as a symbolic calculator

We are ready to try some calculations. Yacas uses a C-like infix syntax and is case-sensitive. Here are some exact manipulations with fractions for a start: 

In> 1/14+5/21*(30-(1+1/2)*5^2);
Out> -12/7;

The standard scripts already contain a simple math library for symbolic simplification of basic algebraic functions. Any names such as x are treated as independent, symbolic variables and are not evaluated by default. 

In> 0+x;
Out> x;
In> x+1*y;
Out> x+y;
In> Sin(ArcSin(alpha))+ArcCos(Cos(beta));
Out> alpha+beta;
In> (x+y)^3-(x-y)^3
Out> (x+y)^3-(x-y)^3;
In> Simplify(%)
Out> 6*x^2*y+2*y^3;

The special operator % automatically recalls the result from the previous line. The function Simplify attempts to reduce an expression to a simpler form. Note that standard function names in Yacas are typically capitalized. Multiple capitalization such as ArcSin is sometimes used. The underscore character _ is a reserved operator symbol and cannot be part of variable or function names.

Yacas can deal with arbitrary precision numbers: 

In> 20!;
Out> 2432902008176640000;

When dealing with floating point numbers, the command Precision(n); can be used to specify that all floating point numbers should have a fixed precision of n digits: 

In> Precision(30);
Out> True;
In> N(1/243);
Out> 0.004115226337448559670781893004;
Note that we need to enter N() to force the approximate calculation, otherwise the fraction would have been left unevaluated. The value True is a boolean constant.

Analytic derivatives of functions can be evaluated: 

In> D(x) Sin(x);
Out> Cos(x);
In> D(x) D(x) Sin(x);
Out> -Sin(x);

Rational numbers will stay rational as long as the numerator and denominator are integers, so 55/10 will evaluate to 11/2. You can override this behaviour by using the numerical evaluation function N(). For example, N(55/10) will evaluate to 5.5 . This behaviour holds for most math functions. Yacas will try to maintain an exact answer (in terms of integers or fractions) instead of using floating point numbers, unless N() is used. Where the value for the constant pi is needed, use the built-in variable Pi. It will be replaced by the (approximate) numerical value when N(Pi) is called. Yacas knows some simplification rules using Pi (especially with trigonometric functions). The imaginary unit i is denoted I and complex numbers can be entered as either expressions involving I or explicitly Complex(a,b) for a+ib.

Some simple equation solving algorithms are in place: 

In> Solve(a+x*y==z,x);
Out> (z-a)/y;
In> Solve({11*x+3*y==1,2*x+y==0},{x,y})
Out> {{1/5,-2/5}};
(Note the use of the == operator, which does not evaluate to anything, to denote an "equation" object.) Currently Solve is rather limited and only deals with equations where the variable to be solved for only occurs once in the equation. In the future there will be more sophisticated algorithms.

Taylor series are supported, for example: 

In> Taylor(x,0,3) Exp(x)
Out> 1+x+(1/2)*x^2+(1/6)*x^3;
As this form of the answer may be a little bit hard to read, you might then type 
In> PrettyForm(%);
        / 1 \    2   / 1 \    3
1 + x + | - | * x  + | - | * x
        \ 2 /        \ 6 /

Out> True;

The function PrettyForm() tries to render the formula in a better format for reading, using ASCII text. You can also export an expression to TeX by typing TeXForm(...).

Variables

Yacas supports variables: 

In> Set(a,Cos(0));
Out> True;
In> a:=a+1;
Out> 2;
The variable a has now been globally set to 2. The function Set() and the operator := can both be used to assign values to global variables. (Variables local to procedures can also be defined; see below the chapters on programming.) To clear a variable binding, execute Clear(a); "a" will now evaluate to just a. This is one of the properties of the evaluation scheme of Yacas: when some object can not be evaluated or transformed any further, it is returned as the final result.

Currently there is no difference between assigning variables using Set() or using the operator :=. The latter can however also assign lists and define functions.

Functions

The := operator can be used to define functions: 

f(x):=2*x*x
will define a new function, f, that accepts one argument and returns twice the square of that argument.

One and the same function name such as f may be used by different functions if they take different numbers of arguments (but not if they merely take different types of arguments, since Yacas does not have a strict type system): 

In> f(x):=x^2;
Out> True;
In> f(x,y):=x*y;
Out> True;
In> f(3)+f(3,2);
Out> 15;
Functions may return values of any type, or may even return values of different types at different times.

Yacas predefines True and False as boolean values. Functions returning boolean values are called predicates. For example, IsNumber() and IsInteger() are predicates defined in the standard library: 

In> IsNumber(2+x);
Out> False;
In> IsInteger(15/5);
Out> True;

When assigning variables, the right hand side is evaluated before it is assigned. Thus 

a:=2*2
will set a to 4. This is however not the case for functions. When entering f(x):=x+x the right hand side, x+x, is not evaluated before being assigned. This can be forced by using Eval(): 

f(x):=Eval(x+x)
will first evaluate x+x to 2*x before assigning it to the user function f. This specific example is not a very useful one but it will come in handy when the operation being performed on the right hand side is expensive. For example, if we evaluate a Taylor series expansion before assigning it to the user-defined function, the engine doesn't need to create the Taylor series expansion each time that user-defined function is called.

Strings and lists

In addition to numbers and variables, Yacas supports strings and lists. Strings are simply sequences of characters enclosed by double quotes, for example: 
"this is a string with \"quotes\" in it"

Lists are ordered groups of items, as usual. Yacas represents lists by putting the objects between braces and separating them with commas. The list consisting of objects a, b, and c could be entered by typing {a,b,c}. In Yacas, vectors are represented as lists and matrices as lists of lists. In fact, any Yacas expression can be converted to a list (see below).

Items in a list can be accessed through the [ ] operator. Examples: when you enter 

uu:={a,b,c,d,e,f};
then 

uu[2];
evaluates to b, and 

uu[2 .. 4];
evaluates to {b,c,d}. The "range" expression 

2 .. 4
evaluates to {2,3,4}. Note that spaces around the .. operator are necessary, or else the parser will not be able to distinguish it from a part of a number.

Another use of lists is the associative list, sometimes called a hash table, which is implemented in Yacas simply as a list of key-value pairs. Keys must be strings and values may be any objects. Associative lists can also work as mini-databases. As an example, first enter 

u:={};
and then 

u["name"]:="Isaia";
u["occupation"]:="prophet";
u["is alive"]:=False;

Now, u["name"] would return "Isaia". The list u now contains three sublists, as we can see: 

In> u;
Out> { {"is alive", False}, {"occupation",
  "prophet"}, {"name", "Isaia"} };

Lists evaluate their arguments, and return a list with results of evaluating each element. So, typing {1+2,3}; would evaluate to {3,3}.

Assignment of multiple variables is also possible using lists. For instance, {x,y}:={2!,3!} will result in 2 being assigned to x and 6 to y.

The idea of using lists to represent expressions dates back to the language LISP developed in the 1970's. From a small set of operations on lists, very powerful symbolic manipulation algorithms can be built. Lists can also be used as function arguments when a variable number of arguments are expected.

Let's try some list operations now: 

In> m:={a,b,c};
Out> True;


In> Length(m);
Out> 3;


In> Reverse(m);
Out> {c,b,a};


In> Concat(m,m);
Out> {a,b,c,a,b,c};


In> m[1]:="blah blah";
Out> True;
In> m;
Out> {"blah blah",b,c};


In> Nth(m,2);
Out> b;

Many more list operations are described in the reference manual.

Linear Algebra

Vectors of fixed dimension are represented as lists of their components. The list {1,2,3} would be a three-dimensional vector with components 1, 2 and 3. Matrices are represented as a vector of vectors.

Vector components can be assigned values just like list items, since they are in fact list items: 
In> l:=ZeroVector(3);
Out> True;
In> l;
Out> {0,0,0};
In> l[ 2 ]:=2;
Out> True;
In> l;
Out> {0,2,0};

Yacas can perform multiplication of matrices, vectors and numbers as usual in linear algebra: 

In> v:={1,0,0,0}
Out> {1,0,0,0};
In> E4:={ {0,u1,0,0},{d0,0,u2,0},
  {0,d1,0,0},{0,0,d2,0}}
Out> {{0,u1,0,0},{d0,0,u2,0},
  {0,d1,0,0},{0,0,d2,0}};
In> CharacteristicEquation(E4,x)
Out> x^4-x*u2*d1*x-u1*d0*x^2;
In> Expand(%,x)
Out> x^4-(u2*d1+u1*d0)*x^2;
In> v+E4*v+E4*E4*v+E4*E4*E4*v
Out> {1+u1*d0,d0+(d0*u1+u2*d1)*d0,
  d1*d0,d2*d1*d0};

The standard Yacas script library also includes taking the determinant and inverse of a matrix, finding eigenvectors and eigenvalues (in simple cases) and solving linear sets of equations, such as A*x=b where A is a matrix, and x and b are vectors. There are several more matrix operations supported. See the reference manual for more details.

Control flow: conditionals, loops, blocks

The Yacas language includes some constructs and functions for control flow. Looping can be done with either a ForEach() or a While() function call. The function ForEach(x, list) body executes its body for each element of the list and assigns the variable x to that element each time. The function call While(predicate) body repeats the "body" until the "predicate" returns False.

Conditional execution is implemented by the If(predicate, body1, body2) function call, which works like the C language construct (predicate) ? body1 : body2. If the condition is true, "body1" is evaluated, otherwise "body2" is evaluated, and the corresponding value is returned. For example, the absolute value of a number can be computed with: 

absx := If( x>=0, x, -x );
(The library function Abs() does this already.)

If several operations need to be executed in sequence to obtain a result, you can use a Prog() function call or equivalently the [ ] construct.

To illustrate these features, let us create a list of all even integers from 2 to 20 and compute the product of all those integers except those divisible by 3. (What follows is not necessarily the most economical way to do it in Yacas.) 

In> L := {};
Out> {};
In> i := 2;
Out> 2;
In> While(i<=20) [ L:= Append(L, i); \
  i := i+2; ]
Out> True;
In> L;
Out> {2,4,6,8,10,12,14,16,18,20};
In> answer := 1;
Out> 1;
In> ForEach(i, L) If (Mod(i, 3)!=0, \
  answer := answer * i);
Out> True;
In> answer;
Out> 2867200;

We used a shorter form of If(predicate, body) with only one body which is executed when the condition holds. If the condition does not hold, this function call returns False.