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last edited 11 years ago by rrogers |
Edit detail for ExampleGroebner revision 4 of 8
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Editor: hemmecke
Time: 2014/12/03 11:59:06 GMT+0 |
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changed: - Find a m and n (depending on the coefficients of p and q Find a m and n (depending on the coefficients of p and q)
Application of Groebner Bases
Problem
Let $$p(x) = a x^2 + b x + c, \qquad q(y) = u y^2 + v y + w.$$ Find a m and n (depending on the coefficients of p and q) such that for the transformaton $$y = m x + n$$ it holds $$p(x) = q(m x + n).$$
Setup of the problem
(1) -> ---- Ordered variable lists. Poly_to_Gauss:=[d,\begin{equation*} \label{eq1}\left[ d , \: n , \: m \right]?\end{equation*}n, m]
Gauss_to_Poly:=[x,\begin{equation*} \label{eq2}\left[ x , \: y , \: a , \: b , \: c \right]?\end{equation*}y, a, b, c]
----coefficient arrays. corg := d* matrix [[c,\begin{equation*} \label{eq3}\left[ \begin{array}{ccc} {c \ d}&{b \ d}&{a \ d} \end{array} \right]\end{equation*}b, a]]
---- Explicit target cgauss := matrix [[0,\begin{equation*} \label{eq4}\left[ \begin{array}{ccc} 0 & 1 & - 1 \end{array} \right]\end{equation*}1, -1]]
---- Generalized target ctar := matrix [[w,\begin{equation*} \label{eq5}\left[ \begin{array}{ccc} w & v & u \end{array} \right]\end{equation*}v, u]]
---- polynomial basis arrays. xorg := matrix ([[1,\begin{equation*} \label{eq6}\left[ \begin{array}{ccc} 1 & x &{{x}^{2}} \end{array} \right]\end{equation*}x, x^2]])
xgauss := matrix([[1,\begin{equation*} \label{eq7}\left[ \begin{array}{ccc} 1 & y &{{y}^{2}} \end{array} \right]\end{equation*}y, y^2]])
---- Example row(corg * transpose(xorg),\begin{equation*} \label{eq8}\left[{{a \ d \ {{x}^{2}}}+{b \ d \ x}+{c \ d}}\right]?\end{equation*}1)
---- Translation matrix Pascal Pa(n) for 3x3 case ---- see Aceto below for references. Pa(n) == matrix [[1,0, 0], [n, 1, 0], [n^2, 2*n, 1]]
---- Scalar matrix Sc(m) == diagonalMatrix [1,m, m^2]
---- Now define transform in matrix form D := corg -(cgauss * Pa(n) * Sc(m))
Compiling function Pa with type Variable(n) -> Matrix(Polynomial(
Integer))Compiling function Sc with type Variable(m) -> Matrix(Polynomial(
Integer))---- Now we do a more realistic solve in two steps ---- Step one disallow silly answers E:=groebnerFactorize(row(D,\begin{equation*} \label{eq10}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{{n}^{2}}- n +{c \ d}}, \:{{2 \ m \ n}- m +{b \ d}}, \: \right. \ \ \displaystyle \left.{{2 \ b \ d \ n}+{{\left(-{4 \ c \ d}+ 1 \right)}\ m}-{b \ d}}, \:{{2 \ a \ n}-{b \ m}- a}, \:{{{m}^{2}}+{a \ d}}, \: \right. \ \ \displaystyle \left.{{{\left({4 \ a \ c}-{{b}^{2}}\right)}\ d}- a}\right] \end{array} }, \right. \ \ \displaystyle \left.\:{\left[ 1 \right]?}\right] \end{array} \end{equation*}1), [b*d, m, a, b^2-3*a*c], true)
we found a groebner basis and check whether it contains reducible polynomials [1] factorGroebnerBasis: no reducible polynomials in this basis we found a groebner basis and check whether it contains reducible polynomials 2 [n - n + c d,2 m n - m + b d, 2 b d n + (- 4 c d + 1)m - b d, 2 2 2 a n - b m - a, m + a d, (4 a c - b )d - a] factorGroebnerBasis: no reducible polynomials in this basis
---- and clean it up (a lot). I wish these two steps could be one! solve(E.1,\begin{equation*} \label{eq11}\begin{array}{@{}l} \displaystyle \left[ \left[{d ={\frac{a}{{4 \ a \ c}-{{b}^{2}}}}}, \:{n ={\frac{{b \ m}+ a}{2 \ a}}}, \: \right. \ \ \displaystyle \left.{{{{\left({4 \ a \ c}-{{b}^{2}}\right)}\ {{m}^{2}}}+{{a}^{2}}}= 0}\right] \right] \end{array} \end{equation*}Poly_to_Gauss)
---- Now lets test the reasonableness the width to start with is ---- 2*sqrt(b^2-4*a*c)/(2*a) which the left hand term yields. There is a sign ambiguity ---- corresponding to whether the source quadratic is to the left or right. ---- I could swap n,\begin{equation*} \label{eq12}\left[ \begin{array}{ccc} {- w -{n \ v}-{{{n}^{2}}\ u}+{c \ d}}&{-{m \ v}-{2 \ m \ n \ u}+{b \ d}}&{-{{{m}^{2}}\ u}+{a \ d}} \end{array} \right]\end{equation*}m in solve() but then the n term (left hand one) is more obscure ---- Knowing the width m we can compute moving the center to 1/2 (for x*(1-x)) ---- It should amount to -b/(2*a)+1/2 ---- and in fact that is the answer n= m(scale factor)*(b/2a)+1/2 ---- d is required and in English is a "normalizing factor"
----General formulation Dorg := corg -(ctar * Pa(n) * Sc(m))
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