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Math::Algebra::Symbols::Sum.3pm
Langue: en
Version: 2004-06-14 (debian - 07/07/09)
Section: 3 (Bibliothèques de fonctions)
Sums
Symbolic Algebra using Pure Perl: sums.See user manual ``NAME''.
Operations on sums of terms.
PhilipRBrenan@yahoo.com, 2004, Perl License.
Constructors
new
Constructor
newFromString
New from String
n
New from Strings
sigma
Create a sum from a list of terms.
makeInt
Construct an integer
Methods
isSum
Confirm type
t
Get list of terms from existing sum
count
Count terms in sum
st
Get the single term from a sum containing just one term
negate
Multiply each term in a sum by -1
add
Add two sums together to make a new sum
subtract
Subtract one sum from another
Conditional Multiply
Multiply two sums if both sums are defined, otherwise return the defined sum. Assumes that at least one sum is defined.
multiply
Multiply two sums together
divide
Divide one sum by another
sub
Substitute a sum for a variable.
isEqual
Check whether one sum is equal to another after multiplying out all divides and divisors.
normalizeSqrts
Normalize sqrts in a sum.
This routine needs fixing.
It should simplify square roots.
isEqualSqrt
Check whether one sum is equal to another after multiplying out sqrts.
isZero
Transform a sum assuming that it is equal to zero
powerOfTwo
Check that a number is a power of two
solve
Solve an equation known to be equal to zero for a specified variable.
power
Raise a sum to an integer power or an integer/2 power.
d
Differentiate.
simplify
Simplify just before assignment.
There is no general simplification algorithm. So try various methods and see if any simplifications occur. This is cheating really, because the examples will represent these specific transformations as general features which they are not. On the other hand, Mathematics is full of specifics so I suppose its not entirely unacceptable.
Simplification cannot be done after every operation as it is inefficient, doing it as part of += ameliorates this inefficiency.
Note: += only works as a synonym for simplify() if the left hand side is currently undefined. This can be enforced by using my() as in: my $z += ($x**2+5x+6)/($x+2);
polynomialDivide
Polynomial divide - divide one polynomial (a) by another (b) in variable v
eigenValue
Eigenvalue check
polynomialDivision
Polynomial division.
Sqrt
Square root of a sum
Exp
Exponential (e raised to the power) of a sum
Log
Log to base e of a sum
Sin
Sine of a sum
Cos
Cosine of a sum
tan, Ssc, csc, cot
Tan, sec, csc, cot of a sum
sinh
Hyperbolic sine of a sum
cosh
Hyperbolic cosine of a sum
Tanh, Sech, Csch, Coth
Tanh, Sech, Csch, Coth of a sum
dot
Dot - complex dot product of two complex sums
cross
The area of the parallelogram formed by two complex sums
unit
Intersection of a complex sum with the unit circle.
re
Real part of a complex sum
im
Imaginary part of a complex sum
modulus
Modulus of a complex sum
conjugate
Conjugate of a complexs sum
clone
Clone
signature
Signature of a sum: used to optimize add(). # Fix the problem of adding different logs
getSignature
Get the signature (see ``signature'') of a sum
id
Get Id of sum: each sum has a unique identifying number.
zz
Check sum finalized. See: ``z''.
z
Finalize creation of the sum: Once a sum has been finalized it becomes read only.
Print sum
constants
Useful constants
factorize
Factorize a number.
import
Export ``n'' with either the default name sums, or a name supplied by the caller of this package.
Operators
Operator Overloads
Overload Perl operators. Beware the low priority of ^.
add3
Add operator.
negate3
Negate operator. Used in combination with the ``add3'' operator to perform subtraction.
multiply3
Multiply operator.
divide3
Divide operator.
power3
Power operator.
equals3
Equals operator.
nequal3
Not equal operator.
tequals
Evaluate the expression on the left hand side, stringify it, then compare it for string equality with the string on the right hand side. This operator is useful for making examples written with Test::Simple more readable.
solve3
Solve operator.
print3
Print operator.
sqrt3
Sqrt operator.
exp3
Exp operator.
sin3
Sine operator.
cos3
Cosine operator.
tan3
Tan operator.
log3
Log operator.
dot3
Dot Product operator.
cross3
Cross operator.
unit3
Unit operator.
modulus3
Modulus operator.
conjugate3
Conjugate.
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.comCredits
Author
philiprbrenan@yahoo.com
Copyright
philiprbrenan@yahoo.com, 2004
License
Perl License.
Contenus ©2006-2024 Benjamin Poulain
Design ©2006-2024 Maxime Vantorre