Statistics Solutions

[See Steps] Interpolation of rational functions. A rational function of degree two has the form f(t)=(c_1+c_2 t+c_3 t^2)/(1+d_1) t+d_2 t^2 where c_1, c_2,

Question: Interpolation of rational functions. A rational function of degree two has the form

\[f(t)=\frac{c_{1}+c_{2} t+c_{3} t^{2}}{1+d_{1} t+d_{2} t^{2}}\]

where \(c_{1}, c_{2}, c_{3}, d_{1}, d_{2}\) are coefficients. ('Rational' refers to the fact that \(f\) is a ratio of polynomials. Another name for \(f\) is bi-quadratic.) Consider the interpolation conditions

\[f\left(t_{i}\right)=y_{i}, \quad i=1, \ldots, K\]

where \(t_{i}\) and \(y_{i}\) are given numbers. Express the interpolation conditions as a set of linear equations in the vector of coefficients \(\theta=\left(c_{1}, c_{2}, c_{3}, d_{1}, d_{2}\right)\), as \(A \theta=b\). Give \(A\) and \(b\), and their dimensions.

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