Name
Math::Algebra::Symbols
Synopsis
Symbolic Algebra in Pure Perl
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,$f" eq '2,3');
Description
This package supplies a set of functions and operators to manipulate
operator expressions algebraically using the familiar Perl syntax.
These expressions are constructed
from L</Symbols>, L</Operators>, and L</Functions>, and processed via
L</Methods>. For examples, see: L</Examples>.
Symbols
Symbols are created with the exported B<symbols()> constructor routine:
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my ($x, $y, $i, $o, $pi) = symbols(qw(x y i 1 pi));
ok( "$x $y $i $o $pi" eq '$x $y i 1 $pi' );
The B<symbols()> routine constructs references to symbolic variables and
symbolic constants from a list of names and integer constants.
The special symbol B<i> is recognized as the square root of B<-1>.
The special symbol B<pi> is recognized as the smallest positive real
that satisfies:
use Math::Algebra::Symbols;
use Test::Simple tests=>2;
my ($i, $pi) = symbols(qw(i pi));
ok( exp($i*$pi) == -1 );
ok( exp($i*$pi) <=> '-1' );
Constructor Routine Name
If you wish to use a different name for the constructor routine, say
B<S>:
use Math::Algebra::Symbols symbols=>'S';
use Test::Simple tests=>2;
my ($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.
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my $z = symbols('1234567890987654321/1234567890987654321');
ok( eval $z eq '1');
Operators
L</Symbols> can be combined with L</Operators> to create symbolic
expressions:
Arithmetic operators
Arithmetic Operators: + - * / **
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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: B<+=> B<-=> B<*=> B</=> are overloaded to work symbolically
rather than numerically. If you need numeric results, you can always
B<eval()> the resulting symbolic expression.
Square root Operator: sqrt
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 B<i>, which allows complex
expressions to be processed by Math::Complex.
Exponential Operator: exp
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($x)' );
The exponential operator.
Logarithm Operator: log
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my ($x) = symbols(qw(x));
ok( log($x) <=> 'log($x)' );
Logarithm to base B<e>.
Note: the above result is only true for x > 0. B<Symbols> does not include
domain and range specifications of the functions it uses.
Sine and Cosine Operators: sin and cos
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($x)**2 <=> '1' );
This famous trigonometric identity is not preprogrammed into B<Symbols> as it
is in commercial products.
Instead: an expression for B<sin()> is constructed using the complex
exponential: L</exp>, said expression is algebraically multiplied out to
prove the identity. The proof steps involve large intermediate expressions in
each step, as yet I have not provided a means to neatly lay out these
intermediate steps and thus provide a more compelling demonstration of the
ability of B<Symbols> to verify such statements from first principles.
Relational operators
Relational operators: ==, !=
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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 B<==> compares two symbolic expressions
and returns TRUE(1) or FALSE(0) accordingly. B<!=> produces the opposite
result.
Relational operator: eq
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 / $t)->solve(qw(x in terms of v t)) <=> '$t*$v' );
The relational operator B<eq> is a synonym for the minus B<-> operator, with
the expectation that later on the L<solve()|/Solving equations> function will
be used to simplify and rearrange the equation. You may prefer to use B<eq>
instead of B<-> to enhance readability, there is no functional difference.
Complex operators
Complex operators: the dot operator: ^
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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' );
Please note the use of brackets: The B<^> operator has low priority.
The B<^> 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
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 B<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 B<x> operator returns the area of this
parallelogram.
Note the space before the B<x>, otherwise Perl is unable to disambiguate the
expression correctly.
Complex operators: the conjugate operator: ~
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-$y**2' );
The B<~> operator returns the complex conjugate of its right hand side.
Complex operators: the modulus operator: abs
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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*$x**2)' );
The B<abs> operator returns the modulus (length) of its right hand side.
Complex operators: the unit operator: !
use Math::Algebra::Symbols;
use Test::Simple tests=>4;
my ($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 B<!> 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: +=
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 B<+=> is a synonym for the
L<simplify()|/"simplifying_equations:_simplify()"> method, if and only if,
the target on the left hand side initially has a value of undef.
Admittedly this is very strange behaviour: it arises due to the shortage of
over-ride-able operators in Perl: in particular it arises due to the shortage
of over-ride-able unary operators in Perl. Never-the-less: this operator is
useful as can be seen in the L<Synopsis|/"synopsis">, and the desired
pre-condition can always achieved by using B<my>.
Equation Manipulation Operators: Solve operator: >
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 B<E<gt>> is a synonym for the
L<solve()|/"Solving_equations:_solve()"> method.
The priority of B<E<gt>> is higher than that of B<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 B<E<gt>> 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), B<E<gt>>
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 L</Symbols> construction routine, or they may be exported into the user's
name space as described in L</EXPORT>.
Trigonometric and Hyperbolic functions
Trigonometric functions
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my ($x, $y) = symbols(qw(x y));
ok( (sin($x)**2 == (1-cos(2*$x))/2) );
The trigonometric functions B<cos>, B<sin>, B<tan>, B<sec>, B<csc>, B<cot>
are available, either as exports to the caller's name space, or as methods.
Hyperbolic functions
use Math::Algebra::Symbols hyper=>1;
use Test::Simple tests=>1;
my ($x, $y) = symbols(qw(x y));
ok( tanh($x+$y)==(tanh($x)+tanh($y))/(1+tanh($x)*tanh($y)));
The hyperbolic functions B<cosh>, B<sinh>, B<tanh>, B<sech>, B<csch>,
B<coth> are available, either as exports to the caller's name space, or
as methods.
Complex functions
Complex functions: re and im
use Math::Algebra::Symbols;
use Test::Simple tests=>2;
my ($x, $i) = symbols(qw(x i));
ok( ($i*$x)->re <=> 0 );
ok( ($i*$x)->im <=> '$x' );
The B<re> and B<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
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 B<dot> and B<cross> operators are available as functions, either as
exports to the caller's name space, or as methods.
Complex functions: conjugate, modulus and unit
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 B<conjugate>, B<abs> and B<unit> operators are available as functions:
B<conjugate>, B<modulus> and B<unit>, either as exports to the caller's name
space, or as methods. The confusion over the naming of: the B<abs> operator
being the same as the B<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
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+$x**7+1' );
B<Simplify()> attempts to simplify an expression. There is no general
simplification algorithm: consequently simplifications are carried out on
ad-hoc basis. You may not even agree that the proposed simplification for a
given expressions is indeed any simpler than the original. It is for these
reasons that simplification has to be explicitly requested rather than being
performed auto-magically.
At the moment, simplifications consist of polynomial division: when the
expression consists, in essence, of one polynomial divided by another, an
attempt is made to perform polynomial division, the result is returned if
there is no remainder.
The B<+=> operator may be used to simplify and assign an expression to a Perl
variable. Perl operator overloading precludes the use of B<=> in this manner.
Substituting into equations: sub()
use Math::Algebra::Symbols;
use Test::Simple tests=>2;
my ($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 B<sub()> function example on line B<#1> demonstrates replacing variables
with expressions. The replacement specified for B<z> has no effect as B<z> is
not present in this equation.
Line B<#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 => $z
Solving equations: solve()
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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 / $t)->solve(qw(x in terms of v t)) <=> '$t*$v' );
B<solve()> assumes that the equation on the left hand side is equal to zero,
applies various simplifications, then attempts to rearrange the equation to
obtain an equation for the first variable in the parameter list assuming that
the other terms mentioned in the parameter list are known constants. There
may of course be other unknown free variables in the equation to be solved:
the proposed solution is automatically tested against the original equation
to check that the proposed solution removes these variables, an error is
reported via B<die()> if it does not.
use Math::Algebra::Symbols;
use Test::Simple tests => 2;
my ($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), B<solve()>
returns an array of symbolic expressions containing the solutions.
Methods for performing Calculus
Differentiation: d()
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)' );
B<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 B<t>, B<x>, B<y>, B<z> is present, then that variable is used in honour 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 B<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;
symbols=>'s'
Create a function with name B<s()> in the callers name space to create new
symbols. The default is B<symbols()>.
trig=>0
The default, do not export trigonometric functions.
trig=>1
Export trigonometric functions: B<tan>, B<sec>, B<csc>, B<cot> to the
caller's name space. B<sin>, B<cos> are created by default by overloading the
existing Perl B<sin> and B<cos> operators.
trigonometric
Alias of B<trig>
hyperbolic=>0
The default, do not export hyperbolic functions.
hyper=>1
Export hyperbolic functions: B<sinh>, B<cosh>, B<tanh>, B<sech>,
B<csch>, B<coth> to the caller's name space.
hyperbolic
Alias of B<hyper>
complex=>0
The default, do not export complex functions
complex=>1
Export complex functions: B<conjugate>, B<cross>, B<dot>, B<im>, B<modulus>,
B<re>, B<unit> to the caller's name space.
Packages
The B<Symbols> packages manipulate a sum of products representation of an
algebraic equation. The B<Symbols> package is the user interface to the
functionality supplied by the B<Symbols::Sum> and B<Symbols::Term> packages.
Math::Algebra::Symbols::Term
B<Symbols::Term> represents a product term. A product term consists of the
number B<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 B<Symbols>
package, including minus one: represented as B<i>.
Reciprocal
The multiplicative inverse of any symbolic expression representable by the
B<Symbols> package: i.e. a B<SymbolsTerm> may be divided by any symbolic
expression representable by the B<Symbols> package.
Exp
The number B<e> raised to the power of any symbolic expression representable
by the B<Symbols> package.
Log
The logarithm to base B<e> of any symbolic expression representable by the
B<Symbols> package.
Thus B<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 B<Symbols::Sum> is required.
Math::Algebra::Symbols::Sum
B<Symbols::Sum> represents a sum of product terms supplied by
B<Symbols::Term> and thus behaves as a polynomial. Operations such as
equation solving and differentiation are applied at this level.
Author
Philip R Brenan at B<
[email protected]>
Copyright
Philip R Brenan at B<
[email protected]> 2004-2016
License
Perl License.
