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→Detailed discussion: named the application of the equioscillation theorem |
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<math>p_2(x)</math> to the ordinates <math>(-1)^i</math>
:<math>p_1(x_i) = f(x_i), p_2(x_i) = (-1)^i, i = 0, ..., n.</math>
To this end, use each time [[Newton polynomial|Newton's interpolation formula]] with the [[divided differences]] of order <math>0, ...,n</math> and <math>O(n^2)</math> arithmetic operations.
The polynomial <math>p_2(x)</math> has its ''i''-th zero between <math>x_{i-1}</math> and <math>x_i,\ i=1, ...,n</math>, and thus no further zeroes between <math>x_n</math> and <math>x_{n+1}</math>: <math>p_2(x_n)</math> and <math>p_2(x_{n+1})</math> have the same sign <math>(-1)^n</math>.
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