Provided by: libmath-algebra-symbols-perl_1.21-1_all 
Terms
Symbolic Algebra in Pure Perl: terms.
See user manual "NAME".
A term represents a product of: variables, coefficents, divisors, square roots,
exponentials, and logs.
PhilipRBrenan@yahoo.com, 2004, Perl License.
Constructors
new
Constructor
newFromString
New from String
n
Short name for "newFromString"
newFromStrings
New from Strings
gcd
Greatest Common Divisor.
lcm
Least common multiple.
isTerm
Confirm type
intCheck
Integer check
c
Coefficient
d
Divisor
timesInt
Multiply term by integer
divideInt
Divide term by integer
negate
Negate term
isZero
Zero?
notZero
Not Zero?
isOne
One?
notOne
Not One?
isMinusOne
Minus One?
notMinusOne
Not Minus One?
i
Get/Set i - sqrt(-1)
iby
i by power: multiply a term by a power of i
Divide
Get/Set divide by.
removeDivide
Remove divide
Sqrt
Get/Set square root.
removeSqrt
Remove square root.
Exp
Get/Set exp
Log
# Get/Set log
vp
Get/Set variable power.
On get: returns the power of a variable, or zero if the variable is not present in the
term.
On set: Sets the power of a variable. If the power is zero, removes the variable from the
term. =cut
v
Get all variables mentioned in the term. Variables to power zero should have been removed
by "vp".
clone
Clone a term. The existing term must be finalized, see "z": the new term will not be
finalized, allowing modifications to be made to it.
split
Split a term into its components
signature
Sign the term. Used to optimize addition. Fix the problem of adding different logs
getSignature
Get the signature of a term
add
Add two finalized terms, return result in new term or undef.
subtract
Subtract two finalized terms, return result in new term or undef.
multiply
Multiply two finalized terms, return the result in a new term or undef
divide2
Divide two finalized terms, return the result in a new term or undef
invert
Invert a term
power
Take power of term
sqrt2
Square root of a term
exp2
Exponential of a term
sin2
Sine of a term
cos2
Cosine of a term
log2
Log of a term
id
Get Id of a term
zz
# Check term finalized
z
Finalize creation of the term. Once a term has been finalized, it becomes readonly, which
allows optimization to be performed. =cut
print
Print
constants
Useful constants
import
Export "newFromStrings" to calling package with a name specifed by the caller, or as
term() by default. =cut
Operators
Operator Overloads
Operator Overloads
add3
Add operator.
negate3
Negate operator.
multiply3
Multiply operator.
divide3
Divide operator.
power3
Power operator.
equals3
Equals operator.
print3
Print operator.
sqrt3
Square root operator.
exp3
Exponential operator.
sin3
Sine operator.
cos3
Cosine operator.
log3
Log operator.
test
Tests
NAME
Math::Algebra::Symbols - Symbolic Algebra in Pure Perl.
User guide.
SYNOPSIS
Example symbols.pl
#!perl -w -I..
#______________________________________________________________________
# Symbolic algebra.
# Perl License.
# PhilipRBrenan@yahoo.com, 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,$f" eq '2,3');
DESCRIPTION
This package supplies a set of functions and operators to manipulate operator expressions
algebraically using the familiar Perl syntax.
These expressions are constructed from "Symbols", "Operators", and "Functions", and
processed via "Methods". For examples, see: "Examples".
Symbols
Symbols are created with the exported symbols() constructor routine:
Example t/constants.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: constants.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my ($x, $y, $i, $o, $pi) = symbols(qw(x y i 1 pi));
ok( "$x $y $i $o $pi" eq '$x $y i 1 $pi' );
The symbols() routine constructs references to symbolic variables and symbolic constants
from a list of names and integer constants.
The special symbol i is recognized as the square root of -1.
The special symbol pi is recognized as the smallest positive real that satisfies:
Example t/ipi.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: constants.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>2;
my ($i, $pi) = symbols(qw(i pi));
ok( exp($i*$pi) == -1 );
ok( exp($i*$pi) <=> '-1' );
Constructor Routine Name
If you wish to use a different name for the constructor routine, say S:
Example t/ipi2.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: constants.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols symbols=>'S';
use Test::Simple tests=>2;
my ($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.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my $z = symbols('1234567890987654321/1234567890987654321');
ok( eval $z eq '1');
Operators
"Symbols" can be combined with "Operators" to create symbolic expressions:
Arithmetic operators
Arithmetic Operators: + - * / **
Example t/x2y2.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplification.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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).
# PhilipRBrenan@yahoo.com, 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.
# PhilipRBrenan@yahoo.com, 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($x)' );
The exponential operator.
Logarithm Operator: log
Example t/logExp.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: log: need better example.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>1;
my ($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.
# PhilipRBrenan@yahoo.com, 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($x)**2 <=> '1' );
This famous trigonometric identity is not preprogrammed into Symbols as it is in
commercial products.
Instead: an expression for sin() is constructed using the complex exponential: "exp", said
expression is algebraically multiplied out to prove the identity. The proof steps involve
large intermediate expressions in each step, as yet I have not provided a means to neatly
lay out these intermediate steps and thus provide a more compelling demonstration of the
ability of Symbols to verify such statements from first principles.
Relational operators
Relational operators: ==, !=
Example t/x2y2.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplification.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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.
# PhilipRBrenan@yahoo.com, 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*$t' );
The relational operator eq is a synonym for the minus - operator, with the expectation
that later on the solve() function will be used to simplify and rearrange the equation.
You may prefer to use eq instead of - to enhance readability, there is no functional
difference.
Complex operators
Complex operators: the dot operator: ^
Example t/dot.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: dot operator. Note the low priority
# of the ^ operator.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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.
# PhilipRBrenan@yahoo.com, 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.
# PhilipRBrenan@yahoo.com, 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-$y**2' );
The ~ operator returns the complex conjugate of its right hand side.
Complex operators: the modulus operator: abs
Example t/abs.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: dot operator. Note the low priority
# of the ^ operator.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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*$x**2)' );
The abs operator returns the modulus (length) of its right hand side.
Complex operators: the unit operator: !
Example t/unit.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: unit operator.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>4;
my ($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.
# PhilipRBrenan@yahoo.com, 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.
# PhilipRBrenan@yahoo.com, 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 m@perlmeister.com.
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.
# PhilipRBrenan@yahoo.com, 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*$x))/2) );
The trigonometric functions cos, sin, tan, sec, csc, cot are available, either as exports
to the caller's name space, or as methods.
Hyperbolic functions
Example t/tanh.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols hyper=>1;
use Test::Simple tests=>1;
my ($x, $y) = symbols(qw(x y));
ok( tanh($x+$y)==(tanh($x)+tanh($y))/(1+tanh($x)*tanh($y)));
The hyperbolic functions cosh, sinh, tanh, sech, csch, coth are available, either as
exports to the caller's name space, or as methods.
Complex functions
Complex functions: re and im
use Math::Algebra::Symbols complex=>1;
Example t/reIm.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: methods.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>2;
my ($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.
# PhilipRBrenan@yahoo.com, 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.
# PhilipRBrenan@yahoo.com, 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.
# PhilipRBrenan@yahoo.com, 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+$x**7+1' );
Simplify() attempts to simplify an expression. There is no general simplification
algorithm: consequently simplifications are carried out on ad hoc basis. You may not even
agree that the proposed simplification for a given expressions is indeed any simpler than
the original. It is for these reasons that simplification has to be explicitly requested
rather than being performed automagically.
At the moment, simplifications consist of polynomial division: when the expression
consists, in essence, of one polynomial divided by another, an attempt is made to perform
polynomial division, the result is returned if there is no remainder.
The += operator may be used to simplify and assign an expression to a Perl variable. Perl
operator overloading precludes the use of = in this manner.
Substituting into equations: sub()
Example t/sub.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: expression substitution for a variable.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>2;
my ($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,
m@perlmeister.com, has proposed that substitutions for expressions should also be allowed,
as in:
$x/$y => $z
Solving equations: solve()
Example t/solve1.t
#!perl -w
#______________________________________________________________________
# Symbolic algebra: examples: simplify.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests=>3;
my ($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*$t' );
solve() assumes that the equation on the left hand side is equal to zero, applies various
simplifications, then attempts to rearrange the equation to obtain an equation for the
first variable in the parameter list assuming that the other terms mentioned in the
parameter list are known constants. There may of course be other unknown free variables in
the equation to be solved: the proposed solution is automatically tested against the
original equation to check that the proposed solution removes these variables, an error is
reported via die() if it does not.
Example t/solve.t
#!perl -w -I..
#______________________________________________________________________
# Symbolic algebra: quadratic equation.
# PhilipRBrenan@yahoo.com, 2004, Perl License.
#______________________________________________________________________
use Math::Algebra::Symbols;
use Test::Simple tests => 2;
my ($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.
# PhilipRBrenan@yahoo.com, 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 philiprbrenan@yahoo.com
Credits
Author
philiprbrenan@yahoo.com
Copyright
philiprbrenan@yahoo.com, 2004
License
Perl License.