Polynomial Taylor Shift

To isolate polynomial roots by VAS method we need to evaluate Taylor shifts,
i.e. to transform the polyomial: p(x)->p(x+x₀). There are few methods for Taylor shifts. One of the most optimal¹ Taylor shift algorithms is the method, described by Shaw and Traub², which we use in this calculator. The algorithm description is just below the calculator:

Polynomial shift

Polynomial

Polynomial coefficients, space separated.

Shift

Calculation precision

Exact

Rounded

Digits after the decimal point: 2

Calculate

Input polynomial

 

Polynomial with Taylor shift

 

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Shaw and Traub method for the Taylor shift

q(x) = p(x+x₀) transformation is accomplished by the three simpler transformation:

• g(x) = p(x₀x)
• f(x) = g(x+1)
• q(x) = f(x/x₀)

The algorithm, step by step

Given the n-degree polynomial: p(x) = a_nxⁿ+a_n-1x^n-1+...+a₁x+a₀
We must obtain new polynomial coefficients q_i, by Taylor shift q(x) = p(x+ x₀).
We'll use the matrix t of dimensions m x m, m=n+1 to store data.

1. Compute t_i,0 = a_n-i-1x₀^n-i-1 for i=0..n-1
2. Store t_i,i+1 = a_nx₀ⁿ for i=0..n-1
3. Compute t_i,j+1 = t_i-1,j+t_i-1,j+1 for j=0..n-1, i=j+1..n
4. Compute the coefficients: q_i = t_n,i+1/x₀ⁱ for i=0..n-1
5. The highest degree coefficient is the same: q_n = a_n