Interpolation
We will cover following topics
PYQs
Newton’s (Forward And Backward) Interpolation
1) Using Newton’s forward difference formula find the lowest degree polynomial \(u_x\) when it is given that \(u_1=1\), \(u_2=9\), \(u_3=25\), \(u_4=55\),\(u_5=105\).
[2018, 10M]
2) In an examination, the number of students who obtained marks between certain limits were given in the following table:
\(\begin{array}{|c|c|c|c|c|}\hline \text { Marks } & {30-40} & {40-50} & {50-60} & {60-70} & {70-80} \\ \hline \text { No. of students } & {31} & {42} & {51} & {35} & {31} \\ \hline\end{array}\).
Using Newton forward interpolation formula, find the number of students whose marks lie between 45 and 50.
[2013, 10M]
Lagrange’s Interpolation
1) For given equidistant values \(u_{-1}\), \(u_{0}\), \(u_{1}\) and \(u_{2}\), a value is interpolated by Lagrange’s formula. Show that it may be written in the form \(u_{x}=y u_{0}+\dfrac{y\left(y^{2}-1\right)}{3 !} \Delta^{2} u_{-1}+\dfrac{x\left(x^{2}-1\right)}{3 !} \Delta^{2} u_{0}\), where \(x+y=1\).
[2017, 15M]
2) Let \(f(x)=e^{2 x} \cos 3 x\) for \(x \in[0,1]\). Estimate the value of \(f(0.5)\) using Lagrange interpolating polynomial of degree 3 over the nodes \(x=0\), \(x=0.3\), \(x=0.6\) and \(x=1\). Also compute the error bound over the interval \([0,1]\) and the actual error \(E(0.5)\).
[2016, 20M]
3) Find the Lagrange interpolating polynomial that fits the following data:
\[\begin{array}{|c|c|c|c|c|}\hline x & {-1} & {2} & {3} & {4}\\ \hline f(x) & {-1} & {11} & {31} & {69}\\ \hline \end{array}\]Find \(f(1.5)\).
[2015, 20M]
4) For the given set of data points \((x_1, f(x_1))\), \((x_2, f(x_2))\), \(\cdots\) \((x_n, f(x_n))\), write an algorithm to find the value of \(f(x)\) by using Lagrange’s interpolation formula.
[2010, 10M]
5) Using Lagrange interpolation formula, calculate the value of \(f(3)\) from the following table of values of \(x\) and \(f(x)\):
\[\begin{array}{|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {4} & {5} & {6}\\ \hline f(x) & {1} & {14} & {15} & {5} & {6} & {19}\\ \hline \end{array}\][2009, 15M]
6) The following values of the function \(f(x)=\sin x+\cos x\) are given:
| $$\begin{array}{ | c | c | c | c | c | c | c | }\hline x & {10^{0}} & {20^{0}} & {30^{0}}\ \hline f(x) & {1.1585} & {1.2817} & {1.3360}\ \hline \end{array}$$. |
Construct the quadratic interpolating polynomial that fits the data. Hence calculate \(f\left(\dfrac{\pi}{12}\right)\). Compare with exact value.
[2008, 15M]
7) If \(Q\) is a polynomial with simple roots \(\alpha_{1}, \alpha_{2}, \ldots \alpha_{n}\) and if \(P\) is a polynomial of \(degree < n\), show that \(\dfrac{P(x)}{Q(x)}=\sum_{k=1}^{n} \dfrac{P\left(\alpha_{k}\right)}{Q^{\prime}\left(\alpha_{k}\right)\left(x-\alpha_{k}\right)}\). Hence, prove that there exists a unique polynomial of degree with given values \(c_{k}\) at the point \(\alpha_{k}, k=1,2, \ldots n\).
[2006, 30M]
8) Find the unique polynomial \(P(x)\) of degree 2 or less such that \(P(1)=1\), \(P(3)=27\), \(P(4)=64\). Using the Lagrange’s interpolation formula and the Newton’s divided difference formula, evaluate \(P(1.5)\).
[2005, 30M]
9) Find the cubic polynomial which takes the following values: \(y(0)=1\), \(y(1)=0\), \(y(2)=1\) and \(y(3)=10\). Hence, or otherwise, obtain \(y(4)\).
[2002, 10M]
10) Show that the truncation error associated with linear interpolation of \(f(x)\), using ordinates at \(x_{0}\) and \(x_{1}\) with \(x_{0} \leq x \leq x_{1}\) is not larger in magnitude than \(\dfrac{1}{8} M_{2}\left(x_{1}-x_{0}\right)^{2}\) where \(M_{2}=\max \left \vert f^{\prime \prime}(x)\right \vert\) in \(x_{0} \leq x \leq x_{1}\). Hence, show that if \(f(x)=\dfrac{2}{\sqrt{\pi}} \int_{0}^{\pi} e^{-t^{2}} d t\), the truncation error corresponding to linear interpolation of \(f(x)\) in \(x_{0} \leq x \leq x_{1}\) cannot exceed \(\dfrac{\left(x_{1}-x_{0}\right)^{2}}{2 \sqrt{2 \pi e}}\).
[2001, 12M]