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extquad.md

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converges at the rate 6 and is same as Boole's rule (integral of cubic interpolant).
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(sec:simprichexterr)=
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##### Richardson error estimate for $I_n^{(1)}$ (Simpson rule)
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### Richardson error estimate for $I_n^{(1)}$ (Simpson rule)
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hermite.md

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H_N(x) = \sum_{i=0}^N y_i r_i(x) + \sum_{i=0}^N y_i' s_i(x)
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$$
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##### Uniqueness
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### Uniqueness
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Suppose there are two polynomials $H(x), G(x) \in \poly_{2N+1}$ that interpolate the same set of data. Then define
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f(x) - p_{2N+1}(x) = (x-x_0)^2 \ldots (x-x_N)^2 f[x_0,x_0,\ldots,x_N,x_N,x]
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$$
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##### Claim: $p_{2N+1} = H_N$
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### Claim: $p_{2N+1} = H_N$
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Assume that $f \in \cts^{2N+3}$. Due to the Newton form, we have
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myst.yml

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site:
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nav:
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- title: Notes on Numerical Analysis
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url: https://cpraveen.github.io/numa
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url: /numa
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template: book-theme
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title: Numerical Analysis
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options:

ode/stiff1.md

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plot(t,y,t,yexact(t))
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title("Forward Euler, h ="+str(h))
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legend(("Euler","Exact"))
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legend(("Euler","Exact"));
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```
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```{code-cell} ipython3

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