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tutorial:polynomials_tutorial [2010/02/08 10:41] – herr | user_guide:tutorials:polynomials_tutorial [2019/01/25 09:38] – ↷ Page moved from user_guide:polynomials_tutorial to user_guide:tutorials:polynomials_tutorial oroehrig | ||
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- | ===== Basic Usage of Polynomials | + | A short note on variable naming up front: You can alter the settings for the names that are used for polynomial variables |
+ | < | ||
+ | > reset_custom %polynomial_var_names; | ||
+ | </ | ||
- | The following will be available starting from version 2.9.8. | + | ===== Usage of Polynomials in Perl ===== |
+ | ===Constructors=== | ||
+ | The easiest way to create a simple [[https:// | ||
+ | < | ||
+ | > $p = new Polynomial(" | ||
+ | </ | ||
+ | Sometimes it's convenient to use the constructor that takes a vector of coefficients and a matrix of exponents: | ||
+ | < | ||
+ | > $coeff = new Vector([9, | ||
+ | > $exp = new Matrix< | ||
+ | > $p2 = new Polynomial($coeff, | ||
+ | > print $p2; | ||
+ | -5*x_0^8*x_1^3 + 9*x_1^4 | ||
+ | </ | ||
+ | There is a seperate type for univariate polynomials, | ||
+ | < | ||
+ | > $up = new UniPolynomial(" | ||
+ | </ | ||
- | A polynomial always carries a reference | + | Polynomials (and UniPolynomials) are templated by their coefficient and exponent types, defaulting |
+ | < | ||
+ | > $pp = new UniPolynomial< | ||
+ | > print $pp; | ||
+ | (4*x^2 + 5)*y^3/2 + (-5/ | ||
+ | </ | ||
+ | ===Computations=== | ||
+ | The standard arithmetic functions " | ||
+ | < | ||
+ | > print $p + ($p^2); | ||
+ | 9*x_1^2 + 6*x_1*x_2^5 + 27*x_1 + x_2^10 + 9*x_2^5 + 20 | ||
+ | </ | ||
+ | However, note that due to the fact that their precedence is given in perl, it may be necessary to write more parentheses than expected at first sight. For example, as above, you always have to write " | ||
+ | < | ||
+ | > print $p + $p^2; | ||
+ | 36*x_1^2 + 24*x_1*x_2^5 + 96*x_1 + 4*x_2^10 + 32*x_2^5 + 64 | ||
+ | </ | ||
+ | For UniPolynomials, | ||
+ | < | ||
+ | > print (($up^2)/ | ||
+ | (2*x^2 + 3*x + 4)/(1) | ||
+ | </ | ||
+ | ===Example: Newton Polynomials=== | ||
+ | Here is one way to produce polytopes from polynomials (as the convex hull of the exponent vectors of all terms). | ||
< | < | ||
- | $r=new Ring(qw(x y)); | + | polytope > $np = newton($p*($p+$p)); |
+ | polytope > print $np-> | ||
+ | 1 0 0 0 | ||
+ | 1 0 2 0 | ||
+ | 1 0 0 10 | ||
+ | polytope > print equal_polyhedra($np, | ||
+ | 1 | ||
</ | </ | ||
- | It may be convenient | + | The final " |
+ | |||
+ | === Example: Toric Degeneration === | ||
+ | |||
+ | The following describes how to construct | ||
< | < | ||
- | polytope> | + | polytope > $points = new Matrix< |
+ | polytope > $height | ||
+ | polytope > $coefficients = new Vector< | ||
</ | </ | ||
- | Notice | + | The following is generic (assuming |
< | < | ||
- | polytope> | + | polytope > $p = new Polynomial($coefficients, |
- | polytope> | + | |
- | polytope> | + | |
- | 1 + 2*x + x^2 + 2*y + 3*x*y + y^2 + x^2*y + x*y^2 | + | |
</ | </ | ||
- | Standard arithmetic function " | + | Notice that the points are given in Euclidean coordinates; |
- | + | ||
- | Here is one way to produce polytopes from polynomials (as the convex hull of the exponent vectors of all terms). | + | |
< | < | ||
- | polytope> | + | polytope > print $p; |
- | polytope> | + | 1/ |
- | 1 0 0 | + | |
- | 1 2 0 | + | |
- | 1 0 2 | + | |
- | 1 2 1 | + | |
- | 1 1 2 | + | |
- | polytope> | + | |
- | 1 | + | |
</ | </ | ||
- | The final " | ||