Provided by: libmath-algebra-symbols-perl_1.21-1_all bug

Symbols

       Symbolic Algebra in Pure Perl.

       See user manual "NAME".

       PhilipRBrenan@yahoo.com, 2004, Perl License.

       Synopsis

       This package delivers the public components of package sum.

       import

       Export components as requested by caller.

        use Math::Algebra::Symbols symbols=>'s' trig=>1 hyper=>1 complex=>0;

       Valid options are:

       symbols=>'s'
           Create a function with name s() in the callers namespace to create new symbols. The
           default is symbols().

           item trig=>0

           The default, no trigonometric functions  are exported.

           item trig=>1

           Export trigonometric functions: tan, sec, csc, cot. sin, cos are created by default by
           overloading the existing Perl sin and cos operators.

       hyper=>0
           The default, no hyperbolic functions

       hyper=>1
           Export hyperbolic functions: sinh, cosh, tanh, sech, csch, coth.

       complex=>0
           The default, no complex functions

       complex=>1
           Export complex functions:  conjugate, cross, dot, im, modulus, re,  unit.

       Trigonometric can be used instead of trig.

       Hyperbolic can be used instead of hyper.

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.