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user_guide:polynomials_tutorial [2019/01/25 09:27] – ↷ Page moved from tutorial:polynomials_tutorial to user_guide:polynomials_tutorial oroehrig | user_guide:tutorials:polynomials_tutorial [2019/02/04 22:55] (current) – external edit 127.0.0.1 | ||
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- | A short note on variable naming up front: You can alter the settings for the names that are used for polynomial variables in parsing them from strings or for pretty-printing using the '' | + | {{page>.:latest:@FILEID@}} |
- | <code> | + | |
- | > reset_custom %polynomial_var_names; | + | |
- | </ | + | |
- | + | ||
- | ===== 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(" | + | |
- | </ | + | |
- | + | ||
- | Polynomials (and UniPolynomials) are templated by their coefficient and exponent types, defaulting to Rational for coefficients and Int for exponents. You can even have polynomials of polynomials (of polynomials...). | + | |
- | < | + | |
- | > $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). | + | |
- | < | + | |
- | 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 | + | |
- | </ | + | |
- | + | ||
- | The final " | + | |
- | + | ||
- | === Example: Toric Degeneration === | + | |
- | + | ||
- | The following describes how to construct the polynomial which describes the toric deformation with respect to a point configuration and a height function. | + | |
- | + | ||
- | < | + | |
- | polytope > $points = new Matrix< | + | |
- | polytope > $height = new Vector< | + | |
- | polytope > $coefficients = new Vector< | + | |
- | </ | + | |
- | + | ||
- | The following is generic (assuming that the dimensions of the objects above match). | + | |
- | + | ||
- | < | + | |
- | polytope > $p = new Polynomial($coefficients, | + | |
- | </ | + | |
- | + | ||
- | Notice that the points are given in Euclidean coordinates; | + | |
- | + | ||
- | < | + | |
- | polytope > print $p; | + | |
- | 1/3*s^3*x2 -1/ | + | |
- | </ | + | |