Other Alias

Symbolic Algebra in Pure Perl., See user manual ``NAME''.

## NAME

Math::Algebra::Symbols - Symbolic Algebra in Pure Perl.

User guide.

## SYNOPSIS

Example symbols.pl

#!perl -w -I..
#______________________________________________________________________
# Symbolic algebra.
# [email protected], 2004.
#______________________________________________________________________

use Math::Algebra::Symbols hyper=>1;
use Test::Simple tests=>5;

(\$n, \$x, \$y) = symbols(qw(n x y));

\$a += (\$x**8 - 1)/(\$x-1);
\$b += sin(\$x)**2 + cos(\$x)**2;
\$c += (sin(\$n*\$x) + cos(\$n*\$x))->d->d->d->d / (sin(\$n*\$x)+cos(\$n*\$x));
\$d = tanh(\$x+\$y) == (tanh(\$x)+tanh(\$y))/(1+tanh(\$x)*tanh(\$y));
(\$e,\$f) = @{(\$x**2 eq 5*\$x-6) > \$x};

print "\$a\n\$b\n\$c\n\$d\n\$e,\$f\n";

ok("\$a" eq '\$x+\$x**2+\$x**3+\$x**4+\$x**5+\$x**6+\$x**7+1');
ok("\$b" eq '1');
ok("\$c" eq '\$n**4');
ok("\$d" eq '1');
ok("\$e,\$f" eq '2,3');

## DESCRIPTION

This package supplies a set of functions and operators to manipulate operator expressions algebraically using the familiar Perl syntax.

These expressions are constructed from ``Symbols'', ``Operators'', and ``Functions'', and processed via ``Methods''. For examples, see: ``Examples''.

Symbols

Symbols are created with the exported symbols() constructor routine:

Example t/constants.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: constants.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>1;

my (\$x, \$y, \$i, \$o, \$pi) = symbols(qw(x y i 1 pi));

ok( "\$x \$y \$i \$o \$pi"   eq   '\$x \$y i 1 \$pi'  );
```

The symbols() routine constructs references to symbolic variables and symbolic constants from a list of names and integer constants.

The special symbol i is recognized as the square root of -1.

The special symbol pi is recognized as the smallest positive real that satisfies:

Example t/ipi.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: constants.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$i, \$pi) = symbols(qw(i pi));

ok(  exp(\$i*\$pi)  ==   -1  );
ok(  exp(\$i*\$pi) <=>  '-1' );
```

Constructor Routine Name

If you wish to use a different name for the constructor routine, say S:

Example t/ipi2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: constants.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols symbols=>'S';
use Test::Simple tests=>2;

my (\$i, \$pi) = S(qw(i pi));

ok(  exp(\$i*\$pi)  ==   -1  );
ok(  exp(\$i*\$pi) <=>  '-1' );
```

Big Integers

Symbols automatically uses big integers if needed.

Example t/bigInt.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: bigInt.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>1;

my \$z = symbols('1234567890987654321/1234567890987654321');

ok( eval \$z eq '1');
```

Operators

``Symbols'' can be combined with ``Operators'' to create symbolic expressions:

Arithmetic operators

Arithmetic Operators: + - * / **

Example t/x2y2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplification.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$y) = symbols(qw(x y));

ok(  (\$x**2-\$y**2)/(\$x-\$y)  ==  \$x+\$y  );
ok(  (\$x**2-\$y**2)/(\$x-\$y)  !=  \$x-\$y  );
ok(  (\$x**2-\$y**2)/(\$x-\$y) <=> '\$x+\$y' );
```

The operators: += -= *= /= are overloaded to work symbolically rather than numerically. If you need numeric results, you can always eval() the resulting symbolic expression.

Square root Operator: sqrt

Example t/ix.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: sqrt(-1).
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$x, \$i) = symbols(qw(x i));

ok(  sqrt(-\$x**2)  ==  \$i*\$x  );
ok(  sqrt(-\$x**2)  <=> 'i*\$x' );
```

The square root is represented by the symbol i, which allows complex expressions to be processed by Math::Complex.

Exponential Operator: exp

Example t/expd.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: exp.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$x, \$i) = symbols(qw(x i));

ok(   exp(\$x)->d(\$x)  ==   exp(\$x)  );
ok(   exp(\$x)->d(\$x) <=>  'exp(\$x)' );
```

The exponential operator.

Logarithm Operator: log

Example t/logExp.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: log: need better example.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>1;

my (\$x) = symbols(qw(x));

ok(   log(\$x) <=>  'log(\$x)' );
```

Logarithm to base e.

Note: the above result is only true for x > 0. Symbols does not include domain and range specifications of the functions it uses.

Sine and Cosine Operators: sin and cos

Example t/sinCos.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplification.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x) = symbols(qw(x));

ok(  sin(\$x)**2 + cos(\$x)**2  ==  1  );
ok(  sin(\$x)**2 + cos(\$x)**2  !=  0  );
ok(  sin(\$x)**2 + cos(\$x)**2 <=> '1' );
```

This famous trigonometric identity is not preprogrammed into Symbols as it is in commercial products.

Instead: an expression for sin() is constructed using the complex exponential: ``exp'', said expression is algebraically multiplied out to prove the identity. The proof steps involve large intermediate expressions in each step, as yet I have not provided a means to neatly lay out these intermediate steps and thus provide a more compelling demonstration of the ability of Symbols to verify such statements from first principles.

Relational operators

Relational operators: ==, !=

Example t/x2y2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplification.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$y) = symbols(qw(x y));

ok(  (\$x**2-\$y**2)/(\$x-\$y)  ==  \$x+\$y  );
ok(  (\$x**2-\$y**2)/(\$x-\$y)  !=  \$x-\$y  );
ok(  (\$x**2-\$y**2)/(\$x-\$y) <=> '\$x+\$y' );
```

The relational equality operator == compares two symbolic expressions and returns TRUE(1) or FALSE(0) accordingly. != produces the opposite result.

Relational operator: eq

Example t/eq.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: solving.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$v, \$t) = symbols(qw(x v t));

ok(  (\$v eq \$x / \$t)->solve(qw(x in terms of v t))  ==  \$v*\$t  );
ok(  (\$v eq \$x / \$t)->solve(qw(x in terms of v t))  !=  \$v+\$t  );
ok(  (\$v eq \$x / \$t)->solve(qw(x in terms of v t)) <=> '\$v*\$t' );
```

The relational operator eq is a synonym for the minus - operator, with the expectation that later on the solve() function will be used to simplify and rearrange the equation. You may prefer to use eq instead of - to enhance readability, there is no functional difference.

Complex operators

Complex operators: the dot operator: ^

Example t/dot.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: dot operator.  Note the low priority
# of the ^ operator.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$a, \$b, \$i) = symbols(qw(a b i));

ok(  ((\$a+\$i*\$b)^(\$a-\$i*\$b))  ==  \$a**2-\$b**2  );
ok(  ((\$a+\$i*\$b)^(\$a-\$i*\$b))  !=  \$a**2+\$b**2  );
ok(  ((\$a+\$i*\$b)^(\$a-\$i*\$b)) <=> '\$a**2-\$b**2' );
```

Note the use of brackets: The ^ operator has low priority.

The ^ operator treats its left hand and right hand arguments as complex numbers, which in turn are regarded as two dimensional vectors to which the vector dot product is applied.

Complex operators: the cross operator: x

Example t/cross.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: cross operator.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$i) = symbols(qw(x i));

ok(  \$i*\$x x \$x  ==  \$x**2  );
ok(  \$i*\$x x \$x  !=  \$x**3  );
ok(  \$i*\$x x \$x <=> '\$x**2' );
```

The x operator treats its left hand and right hand arguments as complex numbers, which in turn are regarded as two dimensional vectors defining the sides of a parallelogram. The x operator returns the area of this parallelogram.

Note the space before the x, otherwise Perl is unable to disambiguate the expression correctly.

Complex operators: the conjugate operator: ~

Example t/conjugate.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: dot operator.  Note the low priority
# of the ^ operator.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$y, \$i) = symbols(qw(x y i));

ok(  ~(\$x+\$i*\$y)  ==  \$x-\$i*\$y  );
ok(  ~(\$x-\$i*\$y)  ==  \$x+\$i*\$y  );
ok(  ((\$x+\$i*\$y)^(\$x-\$i*\$y)) <=> '\$x**2-\$y**2' );
```

The ~ operator returns the complex conjugate of its right hand side.

Complex operators: the modulus operator: abs

Example t/abs.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: dot operator.  Note the low priority
# of the ^ operator.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$i) = symbols(qw(x i));

ok(  abs(\$x+\$i*\$x)  ==  sqrt(2*\$x**2)  );
ok(  abs(\$x+\$i*\$x)  !=  sqrt(2*\$x**3)  );
ok(  abs(\$x+\$i*\$x) <=> 'sqrt(2*\$x**2)' );
```

The abs operator returns the modulus (length) of its right hand side.

Complex operators: the unit operator: !

Example t/unit.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: unit operator.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>4;

my (\$i) = symbols(qw(i));

ok(  !\$i      == \$i                         );
ok(  !\$i     <=> 'i'                        );
ok(  !(\$i+1) <=>  '1/(sqrt(2))+i/(sqrt(2))' );
ok(  !(\$i-1) <=> '-1/(sqrt(2))+i/(sqrt(2))' );
```

The ! operator returns a complex number of unit length pointing in the same direction as its right hand side.

Equation Manipulation Operators

Equation Manipulation Operators: Simplify operator: +=

Example t/simplify.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplify.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$x) = symbols(qw(x));

ok(  (\$x**8 - 1)/(\$x-1)  ==  \$x+\$x**2+\$x**3+\$x**4+\$x**5+\$x**6+\$x**7+1  );
ok(  (\$x**8 - 1)/(\$x-1) <=> '\$x+\$x**2+\$x**3+\$x**4+\$x**5+\$x**6+\$x**7+1' );
```

The simplify operator += is a synonym for the simplify() method, if and only if, the target on the left hand side initially has a value of undef.

Admittedly this is very strange behavior: it arises due to the shortage of over-rideable operators in Perl: in particular it arises due to the shortage of over-rideable unary operators in Perl. Never-the-less: this operator is useful as can be seen in the Synopsis, and the desired pre-condition can always achieved by using my.

Equation Manipulation Operators: Solve operator: >

Example t/solve2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplify.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$t) = symbols(qw(t));

my \$rabbit  = 10 + 5 * \$t;
my \$fox     = 7 * \$t * \$t;
my (\$a, \$b) = @{(\$rabbit eq \$fox) > \$t};

ok( "\$a" eq  '1/14*sqrt(305)+5/14'  );
ok( "\$b" eq '-1/14*sqrt(305)+5/14'  );
```

The solve operator > is a synonym for the solve() method.

The priority of > is higher than that of eq, so the brackets around the equation to be solved are necessary until Perl provides a mechanism for adjusting operator priority (cf. Algol 68).

If the equation is in a single variable, the single variable may be named after the > operator without the use of [...]:

``` use Math::Algebra::Symbols;
my \$rabbit  = 10 + 5 * \$t;
my \$fox     = 7 * \$t * \$t;
my (\$a, \$b) = @{(\$rabbit eq \$fox) > \$t};
print "\$a\n";
# 1/14*sqrt(305)+5/14
```

If there are multiple solutions, (as in the case of polynomials), > returns an array of symbolic expressions containing the solutions.

This example was provided by Mike Schilli [email protected].

Functions

Perl operator overloading is very useful for producing compact representations of algebraic expressions. Unfortunately there are only a small number of operators that Perl allows to be overloaded. The following functions are used to provide capabilities not easily expressed via Perl operator overloading.

These functions may either be called as methods from symbols constructed by the ``Symbols'' construction routine, or they may be exported into the user's namespace as described in ``EXPORT''.

Trigonometric and Hyperbolic functions

Trigonometric functions

Example t/sinCos2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>1;

my (\$x, \$y) = symbols(qw(x y));

ok( (sin(\$x)**2 == (1-cos(2*\$x))/2) );
```

The trigonometric functions cos, sin, tan, sec, csc, cot are available, either as exports to the caller's name space, or as methods.

Hyperbolic functions

Example t/tanh.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols hyper=>1;
use Test::Simple tests=>1;

my (\$x, \$y) = symbols(qw(x y));

ok( tanh(\$x+\$y)==(tanh(\$x)+tanh(\$y))/(1+tanh(\$x)*tanh(\$y)));
```

The hyperbolic functions cosh, sinh, tanh, sech, csch, coth are available, either as exports to the caller's name space, or as methods.

Complex functions

Complex functions: re and im

``` use Math::Algebra::Symbols complex=>1;
```

Example t/reIm.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$x, \$i) = symbols(qw(x i));

ok( (\$i*\$x)->re   <=>  0    );
ok( (\$i*\$x)->im   <=>  '\$x' );
```

The re and im functions return an expression which represents the real and imaginary parts of the expression, assuming that symbolic variables represent real numbers.

Complex functions: dot and cross

Example t/dotCross.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my \$i = symbols(qw(i));

ok( (\$i+1)->cross(\$i-1)   <=>  2 );
ok( (\$i+1)->dot  (\$i-1)   <=>  0 );
```

The dot and cross operators are available as functions, either as exports to the caller's name space, or as methods.

Complex functions: conjugate, modulus and unit

Example t/conjugate2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my \$i = symbols(qw(i));

ok( (\$i+1)->unit      <=>  '1/(sqrt(2))+i/(sqrt(2))' );
ok( (\$i+1)->modulus   <=>  'sqrt(2)'                 );
ok( (\$i+1)->conjugate <=>  '1-i'                     );
```

The conjugate, abs and unit operators are available as functions: conjugate, modulus and unit, either as exports to the caller's name space, or as methods. The confusion over the naming of: the abs operator being the same as the modulus complex function; arises over the limited set of Perl operator names available for overloading.

Methods

Methods for manipulating Equations

Simplifying equations: simplify()

Example t/simplify2.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplify.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$x) = symbols(qw(x));

my \$y  = ((\$x**8 - 1)/(\$x-1))->simplify();  # Simplify method
my \$z +=  (\$x**8 - 1)/(\$x-1);               # Simplify via +=

ok( "\$y" eq '\$x+\$x**2+\$x**3+\$x**4+\$x**5+\$x**6+\$x**7+1' );
ok( "\$z" eq '\$x+\$x**2+\$x**3+\$x**4+\$x**5+\$x**6+\$x**7+1' );
```

Simplify() attempts to simplify an expression. There is no general simplification algorithm: consequently simplifications are carried out on ad hoc basis. You may not even agree that the proposed simplification for a given expressions is indeed any simpler than the original. It is for these reasons that simplification has to be explicitly requested rather than being performed automagically.

At the moment, simplifications consist of polynomial division: when the expression consists, in essence, of one polynomial divided by another, an attempt is made to perform polynomial division, the result is returned if there is no remainder.

The += operator may be used to simplify and assign an expression to a Perl variable. Perl operator overloading precludes the use of = in this manner.

Substituting into equations: sub()

Example t/sub.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: expression substitution for a variable.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>2;

my (\$x, \$y) = symbols(qw(x y));

my \$e  = 1+\$x+\$x**2/2+\$x**3/6+\$x**4/24+\$x**5/120;

ok(  \$e->sub(x=>\$y**2, z=>2)  <=> '\$y**2+1/2*\$y**4+1/6*\$y**6+1/24*\$y**8+1/120*\$y**10+1'  );
ok(  \$e->sub(x=>1)            <=>  '163/60');
```

The sub() function example on line #1 demonstrates replacing variables with expressions. The replacement specified for z has no effect as z is not present in this equation.

Line #2 demonstrates the resulting rational fraction that arises when all the variables have been replaced by constants. This package does not convert fractions to decimal expressions in case there is a loss of accuracy, however:

``` my \$e2 = \$e->sub(x=>1);
\$result = eval "\$e2";
```

or similar will produce approximate results.

At the moment only variables can be replaced by expressions. Mike Schilli, [email protected], has proposed that substitutions for expressions should also be allowed, as in:

``` \$x/\$y => \$z
```

Solving equations: solve()

Example t/solve1.t

``` #!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplify.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests=>3;

my (\$x, \$v, \$t) = symbols(qw(x v t));

ok(   (\$v eq \$x / \$t)->solve(qw(x in terms of v t))  ==  \$v*\$t  );
ok(   (\$v eq \$x / \$t)->solve(qw(x in terms of v t))  !=  \$v/\$t  );
ok(   (\$v eq \$x / \$t)->solve(qw(x in terms of v t)) <=> '\$v*\$t' );
```

solve() assumes that the equation on the left hand side is equal to zero, applies various simplifications, then attempts to rearrange the equation to obtain an equation for the first variable in the parameter list assuming that the other terms mentioned in the parameter list are known constants. There may of course be other unknown free variables in the equation to be solved: the proposed solution is automatically tested against the original equation to check that the proposed solution removes these variables, an error is reported via die() if it does not.

Example t/solve.t

``` #!perl -w -I..
#______________________________________________________________________
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::Simple tests => 2;

my (\$x) = symbols(qw(x));

my  \$p = \$x**2-5*\$x+6;        # Quadratic polynomial
my (\$a, \$b) = @{(\$p > \$x )};  # Solve for x

print "x=\$a,\$b\n";            # Roots

ok(\$a == 2);
ok(\$b == 3);
```

If there are multiple solutions, (as in the case of polynomials), solve() returns an array of symbolic expressions containing the solutions.

Methods for performing Calculus

Differentiation: d()

Example t/differentiation.t

``` #!perl -w -I..
#______________________________________________________________________
# Symbolic algebra.
# [email protected], 2004, Perl License.
#______________________________________________________________________

use Math::Algebra::Symbols;
use Test::More tests => 5;

\$x = symbols(qw(x));

ok(  sin(\$x)    ==  sin(\$x)->d->d->d->d);
ok(  cos(\$x)    ==  cos(\$x)->d->d->d->d);
ok(  exp(\$x)    ==  exp(\$x)->d(\$x)->d('x')->d->d);
ok( (1/\$x)->d   == -1/\$x**2);
ok(  exp(\$x)->d->d->d->d <=> 'exp(\$x)' );
```

d() differentiates the equation on the left hand side by the named variable.

The variable to be differentiated by may be explicitly specified, either as a string or as single symbol; or it may be heuristically guessed as follows:

If the equation to be differentiated refers to only one symbol, then that symbol is used. If several symbols are present in the equation, but only one of t, x, y, z is present, then that variable is used in honor of Newton, Leibnitz, Cauchy.

Example of Equation Solving: the focii of a hyperbola:

``` use Math::Algebra::Symbols;
my (\$a, \$b, \$x, \$y, \$i, \$o) = symbols(qw(a b x y i 1));
print
"Hyperbola: Constant difference between distances from focii to locus of y=1/x",
"\n  Assume by symmetry the focii are on ",
"\n    the line y=x:                     ",  \$f1 = \$x + \$i * \$x,
"\n  and equidistant from the origin:    ",  \$f2 = -\$f1,
"\n  Choose a convenient point on y=1/x: ",  \$a = \$o+\$i,
"\n        and a general point on y=1/x: ",  \$b = \$y+\$i/\$y,
"\n  Difference in distances from focii",
"\n    From convenient point:            ",  \$A = abs(\$a - \$f2) - abs(\$a - \$f1),
"\n    From general point:               ",  \$B = abs(\$b - \$f2) + abs(\$b - \$f1),
"\n\n  Solving for x we get:            x=", (\$A - \$B) > \$x,
"\n                         (should be: sqrt(2))",
"\n  Which is indeed constant, as was to be demonstrated\n";
```

This example demonstrates the power of symbolic processing by finding the focii of the curve y=1/x, and incidentally, demonstrating that this curve is a hyperbola.

## EXPORTS

``` use Math::Algebra::Symbols
symbols=>'S',
trig   => 1,
hyper  => 1,
complex=> 1;
```
trig=>0
The default, do not export trigonometric functions.
trig=>1
Export trigonometric functions: tan, sec, csc, cot to the caller's namespace. sin, cos are created by default by overloading the existing Perl sin and cos operators.
trigonometric
Alias of trig
hyperbolic=>0
The default, do not export hyperbolic functions.
hyper=>1
Export hyperbolic functions: sinh, cosh, tanh, sech, csch, coth to the caller's namespace.
hyperbolic
Alias of hyper
complex=>0
The default, do not export complex functions
complex=>1
Export complex functions: conjugate, cross, dot, im, modulus, re, unit to the caller's namespace.

## PACKAGES

The Symbols packages manipulate a sum of products representation of an algebraic equation. The Symbols package is the user interface to the functionality supplied by the Symbols::Sum and Symbols::Term packages.

Math::Algebra::Symbols::Term

Symbols::Term represents a product term. A product term consists of the number 1, optionally multiplied by:

Variables
any number of variables raised to integer powers,
Coefficient
An integer coefficient optionally divided by a positive integer divisor, both represented as BigInts if necessary.
Sqrt
The sqrt of of any symbolic expression representable by the Symbols package, including minus one: represented as i.
Reciprocal
The multiplicative inverse of any symbolic expression representable by the Symbols package: i.e. a SymbolsTerm may be divided by any symbolic expression representable by the Symbols package.
Exp
The number e raised to the power of any symbolic expression representable by the Symbols package.
Log
The logarithm to base e of any symbolic expression representable by the Symbols package.

Thus SymbolsTerm can represent expressions like:

```  2/3*\$x**2*\$y**-3*exp(\$i*\$pi)*sqrt(\$z**3) / \$x
```

but not:

```  \$x + \$y
```

for which package Symbols::Sum is required.

Math::Algebra::Symbols::Sum

Symbols::Sum represents a sum of product terms supplied by Symbols::Term and thus behaves as a polynomial. Operations such as equation solving and differentiation are applied at this level.

The main benefit of programming Symbols::Term and Symbols::Sum as two separate but related packages is Object Oriented Polymorphism. I.e. both packages need to multiply items together: each package has its own multiply method, with Perl method lookup selecting the appropriate one as required.

Math::Algebra::Symbols

Packaging the user functionality alone and separately in package Symbols allows the internal functions to be conveniently hidden from user scripts.

## AUTHOR

Philip R Brenan at [email protected]

Credits

Author