Question

write a python code that applies the Newton divided difference algorithm for interpolation for the below function and in the specified interval, please do all steps including comments for task 6 tanh^{-1}(x)$ [-1, 1]$ 1. (2.5 points) Initially choose 6 points ($N = 6$) including both lower and upper limits of the interval ${x_i}_{i=1}^N$. Then evaluate the original function $f(x_i)$ on these points. For example, if your interval is $[0, 5]$ then your points will be ${0, 1, 2, 3, 4, 5}$. 2. (2.5 points) Use newton_interpolation to obtain the interpolated polynomial $P$ using $\{x_i, f(x_i)\}$ you obtained above. 3. (2.5 points) Use evaluate_newton_interpolation to evaluate both the orig- inal function $f$ and the polynomial $P$ on a much finer grid, say $x =$ np.linspace(x_l, x_u, 1000, endpoint=True)$, where $x_l$ and $x_u$ are the lower and upper limits of your interval. 4. (2.5 points) Plot both $f$ and $P$ on this finer grid; i.e., against $x$. Also plot $f - P$ against $x$. Don't forget to put labels on $x$ and $y$ axis of your plot, and add legends as well. 5. (10 points) Repeat the tasks (1-4) with $N = 12$. 6. (5 points) Give useful comments about these simulations.

          write a python code that applies the Newton divided difference algorithm for
interpolation for the below function and in the specified interval, please do
all steps including comments for task 6
tanh^{-1}(x)$	[-1, 1]$
1. (2.5 points) Initially choose 6 points ($N = 6$) including both lower and upper limits of the
interval ${x_i}_{i=1}^N$. Then evaluate the original function $f(x_i)$ on these points. For example, if
your interval is $[0, 5]$ then your points will be ${0, 1, 2, 3, 4, 5}$.
2. (2.5 points) Use newton_interpolation to obtain the interpolated polynomial $P$ using
$\{x_i, f(x_i)\}$ you obtained above.
3. (2.5 points) Use evaluate_newton_interpolation to evaluate both the orig-
inal function $f$ and the polynomial $P$ on a much finer grid, say $x =$
np.linspace(x_l, x_u, 1000, endpoint=True)$, where $x_l$ and $x_u$ are the lower and
upper limits of your interval.
4. (2.5 points) Plot both $f$ and $P$ on this finer grid; i.e., against $x$. Also plot $f - P$ against $x$.
Don't forget to put labels on $x$ and $y$ axis of your plot, and add legends as well.
5. (10 points) Repeat the tasks (1-4) with $N = 12$.
6. (5 points) Give useful comments about these simulations.

$write a python code that applies the Newton divided difference algorithm for interpolation for the below function and in the specified interval, please do all steps including comments for task 6 tanh^-1(x)[-1, 1] 1. (2.5 points) Initially choose 6 points (N = 6) including both lower and upper limits of the interval xii=1^N. Then evaluate the original function f(xi) on these points. For example, if your interval is [0, 5] then your points will be 0, 1, 2, 3, 4, 5. 2. (2.5 points) Use newtoninterpolation to obtain the interpolated polynomial P using {xi, f(xi)} you obtained above. 3. (2.5 points) Use evaluatenewtoninterpolation to evaluate both the orig- inal function f and the polynomial P on a much finer grid, say x = np.linspace(xl, xu, 1000, endpoint=True), wherexlandxuare the lower and upper limits of your interval. 4. (2.5 points) Plot bothfandPon this finer grid; i.e., againstx. Also plotf - Pagainstx. Don't forget to put labels onxandyaxis of your plot, and add legends as well. 5. (10 points) Repeat the tasks (1-4) withN = 12. 6. (5 points) Give useful comments about these simulations.$

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