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Math::Algebra::SymbolsUser Contributed Perl DocumentaMath::Algebra::Symbols(3)NameMath::Algebra::SymbolsSynopsisSymbolic Algebra in Pure Perl use Math::Algebra::Symbols hyper=>1; use Test::Simple tests=>5; ($n, $x, $y) = symbols(qw(n x y)); $a += ($x**8 - 1)/($x-1); $b += sin($x)**2 + cos($x)**2; $c += (sin($n*$x) + cos($n*$x))->d->d->d->d / (sin($n*$x)+cos($n*$x)); $d = tanh($x+$y) == (tanh($x)+tanh($y))/(1+tanh($x)*tanh($y)); ($e,$f) = @{($x**2 eq 5*$x-6) > $x}; print "$a\n$b\n$c\n$d\n$e,$f\n"; ok("$a" eq '$x+$x**2+$x**3+$x**4+$x**5+$x**6+$x**7+1'); ok("$b" eq '1'); ok("$c" eq '$n**4'); ok("$d" eq '1'); ok("$e,$f" eq '2,3'); The focii to locus round trip of an ellipse has a length of twice the major radius: use Math::Algebra::Symbols; use Test::More tests => 1; my ($R, $f, $x, $i) = symbols(qw(R f x i)); # Major radius, focii my $y = sqrt($R*$R-$f*$f - $x*$x +$f*$f*$x*$x / ($R*$R)); # Ellipse: rr=RR-ff my $a = $x+$i*$y - $f; # Vector from focus1 to a point on the locus my $b = $x+$i*$y + $f; # Vector from focus2 to same point on the locus ok(abs($a) + abs($b) == 2*$R, 'Focus trip is constant 2R'); Floating point calculations on a triangle with angles of 22.5, 45, 112.5 degrees to determine whether two of the diameters of the nine point circle are at right angles yield (on my computer) the following inconclusive result when the dot product between the diameters is formed numerically: my $o = 1; my $a = sqrt($o/2); # X position of apex my $b = $o - $a; # Y position of apex my $s = ($a*$a+$b*$b-$a)/2/$b; my ($nx, $ny) = ($o/4 + $a/2, $b/2 - $s/2); # Nine point centre my ($px, $py) = ($o/2, 0); # Diameter from mid point my ($qx, $qy) = ($o/2 + $a/2, $b/2); # Diameter from mid point my $d = ($px-$nx)*($qx-$nx)+($py-$ny)*($qy-$ny); # Dot product should be zero print +($d == 0)||0, "\n$d\n"; # Definitively zero if 1 # 0 # Not exactly zero # -6.93889390390723e-18 # Is this significant or not? By contrast with Math::Algebra::Symbols I get the much more convincing: my ($o, $i) = symbols(qw(1 i)); # Units in x,y my $a = sqrt($o/2); # X position of apex my $b = $o - $a; # Y position of apex my $s = ($a*$a+$b*$b-$a)/2/$b; my $n = $o/4 + $a/2 +$i*($b/2 - $s/2); # Nine point centre my $p = $o/2; # Diameter from mid point my $q = $o/2 + $a/2 +$i* $b/2; # Diameter from mid point my $d = (($p-$n) ^ ($q-$n)); # Dot product should be zero print +($d == 0)||0, "\n$d\n"; # Definitively zero if 1 # 1 # 17/32/(-2*sqrt(1/2)+3/2)-3/4/(-2*sqrt(1/2)+3/2)*sqrt(1/2)-7/16/(-sqrt(1/2)+1) # +5/8/(-sqrt(1/2)+1)*sqrt(1/2)-1/8*sqrt(1/2)+1/16DescriptionThis package supplies a set of functions and operators to manipulate operator expressions algebraically using the familiar Perl syntax. These expressions are constructed from L</Symbols>, L</Operators>, and L</Functions>, and processed via L</Methods>. For examples, see: L</Examples>.SymbolsSymbols are created with the exported B<symbols()> constructor routine: use Math::Algebra::Symbols; use Test::Simple tests=>1; my ($x, $y, $i, $o, $pi) = symbols(qw(x y i 1 pi)); ok( "$x $y $i $o $pi" eq '$x $y i 1 $pi' ); The B<symbols()> routine constructs references to symbolic variables and symbolic constants from a list of names and integer constants. It is often useful to declare a variable B<$o> to contain the unit B<1> with which to start symbolic expressions, thus: $o/2 is a symbolic expression for one half where-as: 1/2 == 0.5 is merely the numeric representation of one half. The special symbol B<i> is recognized as the square root of B<-1>. The special symbol B<pi> is recognized as the smallest positive real that satisfies: use Math::Algebra::Symbols; use Test::Simple tests=>2; my ($i, $pi) = symbols(qw(i pi)); ok( exp($i*$pi) == -1 ); ok( exp($i*$pi) <=> '-1' );ConstructorRoutineNameIf you wish to use a different name for the constructor routine, say B<S>: use Math::Algebra::Symbols symbols=>'S'; use Test::Simple tests=>2; my ($i, $pi) = S(qw(i pi)); ok( exp($i*$pi) == -1 ); ok( exp($i*$pi) <=//> '-1' );BigIntegersSymbols automatically uses big integers if needed. use Math::Algebra::Symbols; use Test::Simple tests=>1; my $z = symbols('1234567890987654321/1234567890987654321'); ok( eval $z eq '1');OperatorsL</Symbols> can be combined with L</Operators> to create symbolic expressions:ArithmeticoperatorsArithmetic Operators:+-*/**use Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x, $y) = symbols(qw(x y)); ok( ($x**2-$y**2)/($x-$y) == $x+$y ); ok( ($x**2-$y**2)/($x-$y) != $x-$y ); ok( ($x**2-$y**2)/($x-$y) <=> '$x+$y' ); The operators: B<+=> B<-=> B<*=> B</=> are overloaded to work symbolically rather than numerically. If you need numeric results, you can always B<eval()> the resulting symbolic expression. Square root Operator:sqrtuse Math::Algebra::Symbols; use Test::Simple tests=>2; my ($x, $i) = symbols(qw(x i)); ok( sqrt(-$x**2) == $i*$x ); ok( sqrt(-$x**2) <=> 'i*$x' ); The square root is represented by the symbol B<i>, which allows complex expressions to be processed by Math::Complex. Exponential Operator:expuse 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($x)' ); The exponential operator. Logarithm Operator:loguse Math::Algebra::Symbols; use Test::Simple tests=>1; my ($x) = symbols(qw(x)); ok( log($x) <=> 'log($x)' ); Logarithm to base B<e>. Note: the above result is only true for x > 0. B<Symbols> does not include domain and range specifications of the functions it uses. Sine and Cosine Operators:sinandcosuse Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x) = symbols(qw(x)); ok( sin($x)**2 + cos($x)**2 == 1 ); ok( sin($x)**2 + cos($x)**2 != 0 ); ok( sin($x)**2 + cos($x)**2 <=> '1' ); This famous trigonometric identity is not preprogrammed into B<Symbols> as it is in commercial products. Instead: an expression for B<sin()> is constructed using the complex exponential: L</exp>, said expression is algebraically multiplied out to prove the identity. The proof steps involve large intermediate expressions in each step, as yet I have not provided a means to neatly lay out these intermediate steps and thus provide a more compelling demonstration of the ability of B<Symbols> to verify such statements from first principles.RelationaloperatorsRelational operators:==,!=use Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x, $y) = symbols(qw(x y)); ok( ($x**2-$y**2)/($x-$y) == $x+$y ); ok( ($x**2-$y**2)/($x-$y) != $x-$y ); ok( ($x**2-$y**2)/($x-$y) <=> '$x+$y' ); The relational equality operator B<==> compares two symbolic expressions and returns TRUE(1) or FALSE(0) accordingly. B<!=> produces the opposite result. Relational operator:eqmy ($x, $v, $t) = symbols(qw(x v t)); ok( ($v eq $x / $t)->solve(qw(x in terms of v t)) == $v*$t ); ok( ($v eq $x / $t)->solve(qw(x in terms of v t)) != $v+$t ); ok( ($v eq $x / $t)->solve(qw(x in terms of v t)) <=> '$t*$v' ); The relational operator B<eq> is a synonym for the minus B<-> operator, with the expectation that later on the L<solve()|/Solving equations> function will be used to simplify and rearrange the equation. You may prefer to use B<eq> instead of B<-> to enhance readability, there is no functional difference.ComplexoperatorsComplex operators: thedotoperator:^use Math::Algebra::Symbols; use Test::Simple tests=>3; my ($a, $b, $i) = symbols(qw(a b i)); ok( (($a+$i*$b)^($a-$i*$b)) == $a**2-$b**2 ); ok( (($a+$i*$b)^($a-$i*$b)) != $a**2+$b**2 ); ok( (($a+$i*$b)^($a-$i*$b)) <=> '$a**2-$b**2' ); Please note the use of brackets: The B<^> operator has low priority. The B<^> operator treats its left hand and right hand arguments as complex numbers, which in turn are regarded as two dimensional vectors to which the vector dot product is applied. Complex operators: thecrossoperator:xuse Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x, $i) = symbols(qw(x i)); ok( $i*$x x $x == $x**2 ); ok( $i*$x x $x != $x**3 ); ok( $i*$x x $x <=> '$x**2' ); The B<x> operator treats its left hand and right hand arguments as complex numbers, which in turn are regarded as two dimensional vectors defining the sides of a parallelogram. The B<x> operator returns the area of this parallelogram. Note the space before the B<x>, otherwise Perl is unable to disambiguate the expression correctly. Complex operators: theconjugateoperator:~use Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x, $y, $i) = symbols(qw(x y i)); ok( ~($x+$i*$y) == $x-$i*$y ); ok( ~($x-$i*$y) == $x+$i*$y ); ok( (($x+$i*$y)^($x-$i*$y)) <=> '$x**2-$y**2' ); The B<~> operator returns the complex conjugate of its right hand side. Complex operators: themodulusoperator:absuse Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x, $i) = symbols(qw(x i)); ok( abs($x+$i*$x) == sqrt(2*$x**2) ); ok( abs($x+$i*$x) != sqrt(2*$x**3) ); ok( abs($x+$i*$x) <=> 'sqrt(2*$x**2)' ); The B<abs> operator returns the modulus (length) of its right hand side. Complex operators: theunitoperator:!use Math::Algebra::Symbols; use Test::Simple tests=>4; my ($i) = symbols(qw(i)); ok( !$i == $i ); ok( !$i <=> 'i' ); ok( !($i+1) <=> '1/(sqrt(2))+i/(sqrt(2))' ); ok( !($i-1) <=> '-1/(sqrt(2))+i/(sqrt(2))' ); The B<!> operator returns a complex number of unit length pointing in the same direction as its right hand side.EquationManipulationOperatorsEquation Manipulation Operators:Simplifyoperator:+=use Math::Algebra::Symbols; use Test::Simple tests=>2; my ($x) = symbols(qw(x)); ok( ($x**8 - 1)/($x-1) == $x+$x**2+$x**3+$x**4+$x**5+$x**6+$x**7+1 ); ok( ($x**8 - 1)/($x-1) <=> '$x+$x**2+$x**3+$x**4+$x**5+$x**6+$x**7+1' ); The simplify operator B<+=> is a synonym for the L<simplify()|/"simplifying_equations:_simplify()"> method, if and only if, the target on the left hand side initially has a value of undef. Admittedly this is very strange behaviour: it arises due to the shortage of over-ride-able operators in Perl: in particular it arises due to the shortage of over-ride-able unary operators in Perl. Never-the-less: this operator is useful as can be seen in the L<Synopsis|/"synopsis">, and the desired pre-condition can always achieved by using B<my>. Equation Manipulation Operators:Solveoperator: > use Math::Algebra::Symbols; use Test::Simple tests=>2; my ($t) = symbols(qw(t)); my $rabbit = 10 + 5 * $t; my $fox = 7 * $t * $t; my ($a, $b) = @{($rabbit eq $fox) > $t}; ok( "$a" eq '1/14*sqrt(305)+5/14' ); ok( "$b" eq '-1/14*sqrt(305)+5/14' ); The solve operator B<E<gt>> is a synonym for the L<solve()|/"Solving_equations:_solve()"> method. The priority of B<E<gt>> is higher than that of B<eq>, so the brackets around the equation to be solved are necessary until Perl provides a mechanism for adjusting operator priority (cf. Algol 68). If the equation is in a single variable, the single variable may be named after the B<E<gt>> operator without the use of [...]: use Math::Algebra::Symbols; my $rabbit = 10 + 5 * $t; my $fox = 7 * $t * $t; my ($a, $b) = @{($rabbit eq $fox) > $t}; print "$a\n"; # 1/14*sqrt(305)+5/14 If there are multiple solutions, (as in the case of polynomials), B<E<gt>> returns an array of symbolic expressions containing the solutions. This example was provided by Mike Schilli m@perlmeister.com.FunctionsPerl operator overloading is very useful for producing compact representations of algebraic expressions. Unfortunately there are only a small number of operators that Perl allows to be overloaded. The following functions are used to provide capabilities not easily expressed via Perl operator overloading. These functions may either be called as methods from symbols constructed by the L</Symbols> construction routine, or they may be exported into the user's name space as described in L</EXPORT>.TrigonometricandHyperbolicfunctionsTrigonometric functions use Math::Algebra::Symbols; use Test::Simple tests=>1; my ($x, $y) = symbols(qw(x y)); ok( (sin($x)**2 == (1-cos(2*$x))/2) ); The trigonometric functions B<cos>, B<sin>, B<tan>, B<sec>, B<csc>, B<cot> are available, either as exports to the caller's name space, or as methods. Hyperbolic functions use Math::Algebra::Symbols hyper=>1; use Test::Simple tests=>1; my ($x, $y) = symbols(qw(x y)); ok( tanh($x+$y)==(tanh($x)+tanh($y))/(1+tanh($x)*tanh($y))); The hyperbolic functions B<cosh>, B<sinh>, B<tanh>, B<sech>, B<csch>, B<coth> are available, either as exports to the caller's name space, or as methods.ComplexfunctionsComplex functions:reandimuse Math::Algebra::Symbols; use Test::Simple tests=>2; my ($x, $i) = symbols(qw(x i)); ok( ($i*$x)->re <=> 0 ); ok( ($i*$x)->im <=> '$x' ); The B<re> and B<im> functions return an expression which represents the real and imaginary parts of the expression, assuming that symbolic variables represent real numbers. Complex functions:dotandcrossuse Math::Algebra::Symbols; use Test::Simple tests=>2; my $i = symbols(qw(i)); ok( ($i+1)->cross($i-1) <=> 2 ); ok( ($i+1)->dot ($i-1) <=> 0 ); The B<dot> and B<cross> operators are available as functions, either as exports to the caller's name space, or as methods. Complex functions:conjugate,modulusandunituse Math::Algebra::Symbols; use Test::Simple tests=>3; my $i = symbols(qw(i)); ok( ($i+1)->unit <=> '1/(sqrt(2))+i/(sqrt(2))' ); ok( ($i+1)->modulus <=> 'sqrt(2)' ); ok( ($i+1)->conjugate <=> '1-i' ); The B<conjugate>, B<abs> and B<unit> operators are available as functions: B<conjugate>, B<modulus> and B<unit>, either as exports to the caller's name space, or as methods. The confusion over the naming of: the B<abs> operator being the same as the B<modulus> complex function; arises over the limited set of Perl operator names available for overloading.MethodsMethodsformanipulatingEquationsSimplifying equations:simplify()Example t/simplify2.t use Math::Algebra::Symbols; use Test::Simple tests=>2; my ($x) = symbols(qw(x)); my $y = (($x**8 - 1)/($x-1))->simplify(); # Simplify method my $z += ($x**8 - 1)/($x-1); # Simplify via += ok( "$y" eq '$x+$x**2+$x**3+$x**4+$x**5+$x**6+$x**7+1' ); ok( "$z" eq '$x+$x**2+$x**3+$x**4+$x**5+$x**6+$x**7+1' ); B<Simplify()> attempts to simplify an expression. There is no general simplification algorithm: consequently simplifications are carried out on ad-hoc basis. You may not even agree that the proposed simplification for a given expressions is indeed any simpler than the original. It is for these reasons that simplification has to be explicitly requested rather than being performed auto-magically. At the moment, simplifications consist of polynomial division: when the expression consists, in essence, of one polynomial divided by another, an attempt is made to perform polynomial division, the result is returned if there is no remainder. The B<+=> operator may be used to simplify and assign an expression to a Perl variable. Perl operator overloading precludes the use of B<=> in this manner. Substituting into equations:sub()use Math::Algebra::Symbols; use Test::Simple tests=>2; my ($x, $y) = symbols(qw(x y)); my $e = 1+$x+$x**2/2+$x**3/6+$x**4/24+$x**5/120; ok( $e->sub(x=>$y**2, z=>2) <=> '$y**2+1/2*$y**4+1/6*$y**6+1/24*$y**8+1/120*$y**10+1' ); ok( $e->sub(x=>1) <=> '163/60'); The B<sub()> function example on line B<#1> demonstrates replacing variables with expressions. The replacement specified for B<z> has no effect as B<z> is not present in this equation. Line B<#2> demonstrates the resulting rational fraction that arises when all the variables have been replaced by constants. This package does not convert fractions to decimal expressions in case there is a loss of accuracy, however: my $e2 = $e->sub(x=>1); $result = eval "$e2"; or similar will produce approximate results. At the moment only variables can be replaced by expressions. Mike Schilli, m@perlmeister.com, has proposed that substitutions for expressions should also be allowed, as in: $x/$y => $z Solving equations:solve()use Math::Algebra::Symbols; use Test::Simple tests=>3; my ($x, $v, $t) = symbols(qw(x v t)); ok( ($v eq $x / $t)->solve(qw(x in terms of v t)) == $v*$t ); ok( ($v eq $x / $t)->solve(qw(x in terms of v t)) != $v/$t ); ok( ($v eq $x / $t)->solve(qw(x in terms of v t)) <=> '$t*$v' ); B<solve()> assumes that the equation on the left hand side is equal to zero, applies various simplifications, then attempts to rearrange the equation to obtain an equation for the first variable in the parameter list assuming that the other terms mentioned in the parameter list are known constants. There may of course be other unknown free variables in the equation to be solved: the proposed solution is automatically tested against the original equation to check that the proposed solution removes these variables, an error is reported via B<die()> if it does not. use Math::Algebra::Symbols; use Test::Simple tests => 2; my ($x) = symbols(qw(x)); my $p = $x**2-5*$x+6; # Quadratic polynomial my ($a, $b) = @{($p > $x )}; # Solve for x print "x=$a,$b\n"; # Roots ok($a == 2); ok($b == 3); If there are multiple solutions, (as in the case of polynomials), B<solve()> returns an array of symbolic expressions containing the solutions.MethodsforperformingCalculusDifferentiation:d()use Math::Algebra::Symbols; use Test::More tests => 5; $x = symbols(qw(x)); ok( sin($x) == sin($x)->d->d->d->d); ok( cos($x) == cos($x)->d->d->d->d); ok( exp($x) == exp($x)->d($x)->d('x')->d->d); ok( (1/$x)->d == -1/$x**2); ok( exp($x)->d->d->d->d <=> 'exp($x)' ); B<d()> differentiates the equation on the left hand side by the named variable. The variable to be differentiated by may be explicitly specified, either as a string or as single symbol; or it may be heuristically guessed as follows: If the equation to be differentiated refers to only one symbol, then that symbol is used. If several symbols are present in the equation, but only one of B<t>, B<x>, B<y>, B<z> is present, then that variable is used in honour of Newton, Leibnitz, Cauchy.ExampleofEquationSolving:thefociiofahyperbola:use Math::Algebra::Symbols; my ($a, $b, $x, $y, $i, $o) = symbols(qw(a b x y i 1)); print "Hyperbola: Constant difference between distances from focii to locus of y=1/x", "\n Assume by symmetry the focii are on ", "\n the line y=x: ", $f1 = $x + $i * $x, "\n and equidistant from the origin: ", $f2 = -$f1, "\n Choose a convenient point on y=1/x: ", $a = $o+$i, "\n and a general point on y=1/x: ", $b = $y+$i/$y, "\n Difference in distances from focii", "\n From convenient point: ", $A = abs($a - $f2) - abs($a - $f1), "\n From general point: ", $B = abs($b - $f2) + abs($b - $f1), "\n\n Solving for x we get: x=", ($A - $B) > $x, "\n (should be: sqrt(2))", "\n Which is indeed constant, as was to be demonstrated\n"; This example demonstrates the power of symbolic processing by finding the focii of the curve B<y=1/x>, and incidentally, demonstrating that this curve is a hyperbola.Exportsuse Math::Algebra::Symbols symbols=>'s', trig => 1, hyper => 1, complex=> 1; symbols=>'s' Create a function with name B<s()> in the callers name space to create new symbols. The default is B<symbols()>. trig=>0 The default, do not export trigonometric functions. trig=>1 Export trigonometric functions: B<tan>, B<sec>, B<csc>, B<cot> to the caller's name space. B<sin>, B<cos> are created by default by overloading the existing Perl B<sin> and B<cos> operators.trigonometricAlias of B<trig> hyperbolic=>0 The default, do not export hyperbolic functions. hyper=>1 Export hyperbolic functions: B<sinh>, B<cosh>, B<tanh>, B<sech>, B<csch>, B<coth> to the caller's name space.hyperbolicAlias of B<hyper> complex=>0 The default, do not export complex functions complex=>1 Export complex functions: B<conjugate>, B<cross>, B<dot>, B<im>, B<modulus>, B<re>, B<unit> to the caller's name space.PackagesThe B<Symbols> packages manipulate a sum of products representation of an algebraic equation. The B<Symbols> package is the user interface to the functionality supplied by the B<Symbols::Sum> and B<Symbols::Term> packages.Math::Algebra::Symbols::TermB<Symbols::Term> represents a product term. A product term consists of the number B<1>, optionally multiplied by: Variables Any number of variables raised to integer powers. Coefficient An integer coefficient optionally divided by a positive integer divisor, both represented as BigInts if necessary. Sqrt The sqrt of of any symbolic expression representable by the B<Symbols> package, including minus one: represented as B<i>. Reciprocal The multiplicative inverse of any symbolic expression representable by the B<Symbols> package: i.e. a B<SymbolsTerm> may be divided by any symbolic expression representable by the B<Symbols> package. Exp The number B<e> raised to the power of any symbolic expression representable by the B<Symbols> package. Log The logarithm to base B<e> of any symbolic expression representable by the B<Symbols> package. Thus B<SymbolsTerm> can represent expressions like: 2/3*$x**2*$y**-3*exp($i*$pi)*sqrt($z**3) / $x but not: $x + $y for which package B<Symbols::Sum> is required.Math::Algebra::Symbols::SumB<Symbols::Sum> represents a sum of product terms supplied by B<Symbols::Term> and thus behaves as a polynomial. Operations such as equation solving and differentiation are applied at this level.InstallationStandard Module::Build process for building and installing modules: perl Build.PL ./Build ./Build test ./Build installCopyrightPhilip R Brenan at B<PhilipRBrenan@gmail.com> 2004-2016LicensePerl License. perl v5.32.0 2020-08-10 Math::Algebra::Symbols(3)

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