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October 23rd, 2016, 12:27 PM   #1
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How derive new class of polynomials?

We have $\displaystyle g_n(x_0)$ which is composition of functions $\displaystyle f_i(x_0,x_1,...x_{i-1}), i=1..n$ where $\displaystyle f_i(...)$ are basic arithmetic operations (add, sub, mul) on $\displaystyle x_k$ and constants. For example:
$\displaystyle x_1 = 2\cdot x_0*x_0 + x_0+0.567$
$\displaystyle x_2 = x_0x_1 + x_1 + 0.341$
$\displaystyle x_3 = 3.2x_2x_2+x_1+x_0$
this way is simply to reach polynomial degree of $\displaystyle 2^n$ using only n multiplications.
Where this can be used ? In approximation of non-rational functions like tangent, arcus tangent or cosinus.
Algorithm of Remez enable to find optimal polynomial degree n, but if is required big accuracy (>double) these polynomials have high degrees.
How is possible restrict all polynomials of degree 60 to polynomials derived by above formula with max 6 or 7 multiplications? and how to find optimal approximation polynomial only inside this narrow class of polynomials?
Generalization of this wil is adding operation "divide" and obtaining rational expressions.
Fast formula of sqrt just generates high degree rational expressions.
If would be possible find optimals rational expressions narorwed to n-div, k-mul, computing of non-rational functions like sinus or tangent would be fast as computing sqrt.
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