# 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 `--with-numlib=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

or, alternatively, by passing `--enable-ps-doc` or `--enable-pdf-doc` to `./configure` when building `Yacas`. In the latter case, the documentation will be automatically rebuilt every time the documentation changes (which is useful when maintaining the documentation).

In addition, there is also a Java version of the lower-level interpreter. The code for this Java version can be found in the directory "JavaYacas", and can be compiled with the make file "makefile.yacas", by typing in:

    make -f makefile.yacas

The interpreter can then be invoked from the command line with:

    java -jar yacas.jar

or alternatively it can be invoked as an applet, by opening yacasconsole.html.

The binary files that comprise the entire binary release for the Java version are:

- yacas.jar - the Java class files in one jar file.
- scripts.zip - all the scripts compressed in to one zip file.
- yacasconsole.html - the file that launches the applet.
- yacasconsole.html.hints - the hints that are shown in the applet (the grey box with commands that match what you are typing in at that moment).
- yacas.gif - the logo shown in the applet version.

To get copy-pasting to work in the applet version, you need to tell the Java virtual machine that you trust this applet to get or set data on the clipboard of your OS.

On Unix-like systems this means creating a file ~/.java.policy if it does not already exist, and then adding the following lines to it:

    grant codeBase "http://yacas.sourceforge.net/*" {
    permission java.awt.AWTPermission "accessClipboard";
    };

This grants all applets residing at http://www.xs4all.nl/~apinkus/* access to the clipboard.

This can be done in a similar way on Windows, but the directory where you need to place the .java.policy file depends on the version of Windows that is running.

The Java version has almost all the features the C++ version has. In fact, there is no reason the Java version should not have all the same features. At the time of writing (version 1.0.58), plugins are not available yet, not all command line arguments are available yet, and the command line prompt does not have the history yet.

##

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 in interactive sessions (`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.

When a few characters have been typed, the command line will use the characters before the cursor as a filter into the history, and allow you to browse through all the commands in the history that start with these characters quickly, instead of browsing through the entire history.

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))+Tan(ArcTan(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.

The `N` function has an optional second argument, the required precision:

    In> N(1/234,10)
    Out> 0.0042735042;
    In> N(1/234,20)
    Out> 0.0042735042735042735;
    In> N(1/234,30)
    Out> 0.004273504273504273504273504273;

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);

The `D` function also accepts an argument specifying how often the derivative has to be taken. In that case, the above expressions can also be written as:

    In> D(x,1)Sin(x)
    Out> Cos(x);
    In> D(x,2)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 behavior by using the numerical evaluation function `N()`. For example, `N(55/10)` will evaluate to `5.5` . This behavior 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(x/(1+x) == a, x);
    Out> {x==a/(1-a)};
    In> Solve(x^2+x == 0, x);
    Out> {x==0,x==(-1)};

(Note the use of the `==` operator, which does not evaluate to anything, to denote an "equation" object.) Currently `Solve` is rather limited, but 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+x, 3*Sin(p)}` would be a three-dimensional vector with components 1, 2+x and 3*Sin(p). 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`.

The above example is not the shortest possible way to write out the algorithm. A more 'functional' approach would go like this:

First construct a list with all even numbers from `2` to `20`. For this we use the `..` operator to set up all numbers from one to ten, and then multiply that with two.

    In> 2*(1 .. 10)
    Out> {2,4,6,8,10,12,14,16,18,20};

Now we want an expression that returns all the even numbers up to `20` which are not divisible by `3`. For this we can use `Select`, which takes as first argument a predicate that should return `True` if the list item is to be accepted, and false otherwise, and as second argument the list in question:

    In> Select({{n},Mod(n,3)!=0},2*(1 .. 10))
    Out> {2,4,8,10,14,16,20};

The numbers `6`, `12` and `18` have been correctly filtered out.

All that remains is to factor the items in this list. For this we can use `Factorize`, which accepts a list as argument and returns the product of all the items in that list:

    In> Factorize(Select({{n},Mod(n,3)!=0},2*(1 .. 10)))
    Out> 2867200;

A more flexible function one can use is UnFlatten:

    In> UnFlatten({a,b,c},"*",1)
    Out> a*b*c;
    In> UnFlatten({a,b,c},"+",0)
    Out> a+b+c;

Using `UnFlatten`, the result becomes:

    In> UnFlatten(Select({{n},Mod(n,3)!=0},2*(1 .. 10)),"*",1)
    Out> 2867200;