# Bessel functions often arise in advanced engineering and scientific analyses such as the study of electric fields.

Bessel functions often arise in advanced engineering and scientific analyses such as the study of electric fields. These functions are usually not amenable to straightforward evaluation and, therefore, are often compiled in standard mathematical tables.
$$\begin{array}\\ x&1.8&2.0&2.2&2.4&2.6\\ J_1(x)&0.5815&0.5767&0.5560&0.5202&0.4708 \end{array}$$
Estimate $$J_1(2.1)$$, (a) using an interpolating polynomial and (b) using cubic splines. Note that the true value is 0.5683

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a} By using a fourth-order Newton's interpolating polynomial we found that $$\displaystyle{7}{\left({2.1}\right)}_{{{1}}}$$ = 0.5683.
b) Using « cubic spline with not-a-knot end conditions yielded $$\displaystyle{J}{\left({2.1}\right)}_{{{1}}}={0.5683}$$, however, using a. piecewise cubic Hermite interpolation resulted in $$\displaystyle{J}{\left({2.1}\right)}_{{{1}}}={0.5687}$$.