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In the rational case, we are interested to find polynomials $P(x)$ and $Q(x)$ s.t. $f(x_k)-P(x_k)/Q(x_k)=(-1)^kE$ for $k=1,2,\ldots, N$ where $N=deg(P)+deg(Q)+2$

This can be rewritten as $$ (1)~~~~~~(f(x_k)-(-1)^kE)Q(x_k)-P(x_k)=0 $$

The equation (1) is non-linear. The notes in this document suggest using the following variation of the system of equations for linearizing (1). $$ (2)~~~~~~((-1)^k E_0 - f(x_k))\sum_{i=1}^{q}b_i x_k^i+\sum_{i=1}^{p}a_i x_k^i + (-1)^kE=f(x_k) $$

It seems quite different from (1). Is (2) above is the correct interpretation of (1)? How this was derived from the definition? Where the additional $f(x_k)$ and $(-1)^kE$ come from?

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I have found that this is Fraser and Hart's variant of the Remez algorithm.

Let us fix the constant term of $Q(x_k)$ to $1$:

$$ f(x_k) - \frac{\sum_{i=0}^{p}a_i x_k^i}{\sum_{i=1}^{q}b_i x_k^i + 1}=(-1)^kE $$

from which we obtain $$ f(x_k) + f(x_k)\sum_{i=1}^{q}b_i x_k^i - \sum_{i=0}^{p}a_i x_k^i=(-1)^kE\sum_{i=1}^{q}b_i x_k^i + (-1)^kE $$

regrouping, we get

$$ (*)~~~~~~~\left((-1)^kE- f(x_k)\right)\sum_{i=1}^{q}b_i x_k^i + \sum_{i=0}^{p}a_i x_k^i + (-1)^kE = f(x_k) $$

Since we don't know $E$, $(*)$ is not linear. The approach of Fraser and Hart is to make an initial guess $E_0$ so that $(*)$ becomes $(2)$ which is a linear system.

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