Math::Algebra::Symbols - Symbolic Algebra using Perl
use symbols hyper=>1;
($n, $x, $y) = symbols(qw(n x y));
$a = sin($x)**2 + cos($x)**2; $b = ($x**8-1) / ($x-1); $c = (sin($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($y));
print "$a\n$b\n$c\n$d\n";
# 1 # 1+$x+$x**2+$x**3+$x**4+$x**5+$x**6+$x**7 # $n**4 # 1
This package supplies a set of functions and operators to manipulate operator expressions algebraically using the familiar Perl syntax.
These expressions are constructed from Symbols, Operators, and Functions, and processed via Methods. For examples, see: Examples.
Symbols are created with the exported symbols() constructor routine:
use symbols;
my ($x, $y, $i, $o, $pi) = symbols(qw(x y i 1 pi));
print "$x $y $i $o\n";
# $x $y $i 1
The symbols() routine constructs references to symbolic variables and symbolic constants from a list of names and integer constants.
The special symbol i is recognized as the square root of -1.
The special symbol pi is recognized as the smallest positive real that satisfies:
use symbols;
($i, $pi) = symbols(qw(i pi));
print exp($i*$pi), "\n";
# -1
If you wish to use a different name for the constructor routine, say S:
use symbols symbols=>'S';
my ($x, $y, $i, $o) = S(qw(x y i 1));
print "$x $y $i $o\n";
# $x $y $i 1
If you wish to use symbols constructed with big integers from the Math::BigInt manpage:
use symbols BigInt=>1;
my $z = symbols('1234567890987654321/1234567890987654321');
print "$z\n";
# 1
Symbols can be combined with Operators to create symbolic expressions:
use symbols;
($x, $y) = symbols(qw(x y));
$z = ($x**2-$y**2)/($x-$y);
print "$z\n";
# $x+$y
The auto assign versions of these operators: += -= *= /= all work courtesy of Perl Auto-Magical Operator Generation.
use symbols;
$x = symbols(qw(x));
$z = sqrt(-$x**2);
print "$z\n";
# $i*$x
use symbols;
$x = symbols(qw(x));
$z = exp($x)->d($x);
print "$z\n";
# exp($x)
use symbols;
$x = symbols(qw(x));
$z = log(exp($x)*exp($x));
print "$z\n";
# 2*$x
use symbols;
$x = symbols(qw(x));
$z = sin($x)**2 + cos($x)**2;
print "$z\n";
# 1
use symbols;
($x, $y) = symbols(qw(x y));
$z = ($x**2-$y**2)/($x+$y) == $x - $y;
print "$z\n";
# 1
The relational equality operator == compares two symbolic expressions
and returns TRUE(1)
or FALSE(0)
accordingly.
use symbols;
($x, $v, $t) = symbols(qw(x v t));
$z = ($v eq $x / $t)->solve(qw(x in terms of v t));
print "x=$z\n";
# x=$v*$t
The relational operator eq is in fact a synonym for the minus - operator, with the expectation that later on the solve() function will be used to simplify and rearrange the equation.
use symbols;
($a, $b, $i) = symbols(qw(a b i));
$z = ($a+$i*$b)^($a-$i*$b);
print "$z\n";
# $a**2-$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.
use symbols;
($x, $i) = symbols(qw(x i));
$z = $i*$x x $x;
print "$z\n";
# $x**2
The x operator treats its left hand and right hand arguments as complex numbers, which in turn are regarded as two dimensional vectors defining the sides of a parallelogram. The x operator returns the area of this parallelogram.
Note the space before the x, otherwise Perl is unable to disambiguate the expression correctly.
use symbols;
($x, $y, $i) = symbols(qw(x y i));
$z = $x+$i*$y;
print ~$z, "\n";
# $x-$i*$y
The ~ operator returns the complex conjugate of its right hand side.
use symbols;
($x, $i) = symbols(qw(x i));
$z = abs($x+$i*$x);
print "$z\n";
# sqrt(2)*$x
The abs operator returns the modulus (length) of its right hand side.
use symbols;
$i = symbols(qw(i));
$z = !($i+1);
print "$z\n";
# $i*sqrt(1/2)+sqrt(1/2)
The ! operator returns a complex number of unit length pointing in the same direction as its right hand side.
Perl operator overloading is very useful for producing compact representations of algebraic expressions. Unfortunately there are only a small number of operators that Perl allows to be overloaded. The following functions are used to provide capabilities not easily expressed via Perl operator overloading.
These functions may either be called as methods from symbols constructed by the Symbols construction routine, or they may be exported into the user's namespace as described in EXPORT.
use symbols trig=>1;
($x, $y) = symbols(qw(x y));
$z = sin($x)**2 == (1-cos(2*$x))/2;
print "$z\n";
# 1
The trigonometric functions cos, sin, tan, sec, csc, cot are available, either as exports to the caller's name space, or as methods.
use symbols hyper=>1;
($x, $y) = symbols(qw(x y));
$z = tanh($x+$y)==(tanh($x)+tanh($y))/(1+tanh($x)*tanh($y));
print "$z\n";
# 1
The hyperbolic functions cosh, sinh, tanh, sech, csch, coth are available, either as exports to the caller's name space, or as methods.
use symbols complex=>1;
($x, $i) = symbols(qw(x i));
$R = re($i*$x); $I = im($i*$x);
print "$R $I\n";
# 0 $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.
use symbols complex=>1;
$i = symbols(qw(i));
$c = cross($i+1, $i-1); $d = dot ($i+1, $i-1);
print "$c $d\n";
# 2 0
The dot and cross operators are available as functions, either as exports to the caller's name space, or as methods.
use symbols complex=>1;
$i = symbols(qw(i));
$x = unit($i+1); $y = modulus($i+1); $z = conjugate($i+1);
print "$x\n$y\n$z\n";
# $i*sqrt(1/2)+sqrt(1/2) # sqrt(2) # 1-$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.
use symbols; ($x, $y) = symbols(qw(x y)); $e = 1+$x+$x**2/2+$x**3/6+$x**4/24+$x**5/120;
$e2 = $e->sub(x=>$y**2, z=>2); #1 $e3 = $e->sub(x=>1); #2
print "$e2\n\n$e3\n\n";
# 1+$y**2+1/2*$y**4+1/6*$y**6+1/24*$y**8+1/120*$y**10
# 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 acuracy, however:
$e3 =~ /^(\d+)\/(\d+)$/; $result = $1/$2;
or similar will produce approximate results.
use symbols;
($x, $v, $t) = symbols(qw(x v t));
$z = ($v eq $x / $t)->solve(qw(x in terms of v t)); #1
print "x=$z\n";
# x=$v*$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.
use symbols;
($x, $i) = S(qw(x i));
$z = exp($x)->d->d('x')->d($x)->d();
print "$z\n";
# exp($x)
d() differentiates the equation on the left hand side by the named variable.
The variable to be differentiated by may be explicitly specifed, either as a string or as single symbol; or it may be heuristically guessed as follows:
If the equation to be differentiated refers to only one symbol, then that symbol is used. If several symbols are present in the equation, but only one of t, x, y, z is present, then that variable is used in honor of Newton, Leibnitz, Cauchy.
use symbols;
($a, $b, $x, $y, $i) = symbols(qw(a b x y i));
print $i x 1, "\n"; # Cross product # 1
print $i^1, "\n"; # Dot product - different vectors # 0
print $i^$i, "\n"; # Dot product - same vector # 1
print abs $i, "\n"; # Length of unit vector # 1
print ~($a+$b) == ~$a+~$b, "\n"; # Conjugation is distributive # 1 over addition
print ~($a*$b) == ~$a*~$b, "\n"; # Conjugation is distributive # 1 over multiplication
print ~($a**2) == (~$a)**2,"\n"; # Conjugation is distributive # 1 over power
print abs(!($x+$y*$i))==1,"\n"; # Length of unit vector # 1
print # Length of product = product of lengths abs($a+$i*$b)*abs($x+$i*$y) == abs(($a+$i*$b)* ($x+$i*$y)), "\n"; # 1
use symbols; ($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 eq $B)->solve(qw(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.
use symbols;
$x = symbols(qw(x));
$x->test();
The test() method performs many tests which are useful in validating this package and as examples of the capabilities of this package. These tests may also be run as:
perl symbols.pm
use Maths::Algebra::Symbols symbols=>'S', BigInt=>0, trig=>1 hyper=>1, complex=>1;
Philip R Brenan at philiprbrenan@yahoo.com