|
|
Mathomatic 14.2.6 (Default branch)
12-15-2008
Mathomatic is a portable general-purpose CAS (Computer Algebra System) and calculator software that can symbolically solve, simplify, combine, and compare equations; perform general complex number and polynomial arithmetic; do simple calculus transformations, etc. License: GNU Lesser General Public License (LGPL) Changes:|
|
Math::Algebra::Symbols::Sum(3pm) User Contributed Perl Documentation Math::Algebra::Symbols::Sum(3pm) 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 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 $b $c $d $e,$f "; 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 "; # 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,$b "; # Roots ok($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", " Assume by symmetry the focii are on ", " the line y=x: ", $f1 = $x + $i * $x, " and equidistant from the origin: ", $f2 = -$f1, " Choose a convenient point on y=1/x: ", $a = $o+$i, " and a general point on y=1/x: ", $b = $y+$i/$y, " Difference in distances from focii", " From convenient point: ", $A = abs($a - $f2) - abs($a - $f1), " From general point: ", $B = abs($b - $f2) + abs($b - $f1), " Solving for x we get: x=", ($A - $B) > $x, " (should be: sqrt(2))", " Which is indeed constant, as was to be demonstrated "; 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. perl v5.10.0 2004-06-14 Math::Algebra::Symbols::Sum(3pm)