# Polynomial Taylor Shift

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

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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}

#### Similar calculators

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**PLANETCALC, Polynomial Taylor Shift

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