# Polynomial Taylor Shift

The calculator evaluates Taylor shift of the given polynomial using Shaw and Traub algorithm.

To isolate polynomial roots by VAS method we need to evaluate Taylor shifts,

i.e. to transform the polyomial: p(x)->p(x+x_{0}). There are few methods for Taylor shifts. One of the most optimal^{1} Taylor shift algorithms is the method, described by Shaw and Traub^{2}, which we use in this calculator. The algorithm description is just below the calculator:

### Shaw and Traub method for the Taylor shift

q(x) = p(x+x_{0}) transformation is accomplished by the three simpler transformation:

- g(x) = p(x
_{0}x) - f(x) = g(x+1)
- q(x) = f(x/x
_{0})

#### The algorithm, step by step

Given the n-degree polynomial: p(x) = a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}

We must obtain new polynomial coefficients q_{i}, by Taylor shift q(x) = p(x+ x_{0}).

We'll use the matrix *t* of dimensions m x m, m=n+1 to store data.

- Compute t
_{i,0}= a_{n-i-1}x_{0}^{n-i-1}for i=0..n-1 - Store t
_{i,i+1}= a_{n}x_{0}^{n}for i=0..n-1 - Compute t
_{i,j+1}= t_{i-1,j}+t_{i-1,j+1}for j=0..n-1, i=j+1..n - Compute the coefficients: q
_{i}= t_{n,i+1}/x_{0}^{i}for i=0..n-1 - The highest degree coefficient is the same: q
_{n}= a_{n}

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