PYQs

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 $u_x=y{u}_0+{\displaystyle \frac{y(y^2-1)}{3!}}{\mathrm{\Delta }}^2{u}_{-1}+{\displaystyle \frac{x(x^2-1)}{3!}}{\mathrm{\Delta }}^2{u}_0$, where $x+y=1$.

[2017, 15M]

2) Let $f(x)=e^{2x}\cos 3x$ 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:

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 $f(x)$:

[2009, 15M]

6) The following values of the function $f(x)=\sin x+\cos x$ are given:

Construct the quadratic interpolating polynomial that fits the data. Hence calculate $f\left({\displaystyle \frac{\pi }{12}}\right)$. Compare with exact value.

[2008, 15M]

7) If $Q$ is a polynomial with simple roots ${\alpha }_1,{\alpha }_2,\dots {\alpha }_n$ and if $P$ is a polynomial of $degree<n$, show that ${\displaystyle \frac{P(x)}{Q(x)}}=\sum _{k=1}^n{\displaystyle \frac{P\left({\alpha }_k\right)}{Q^{\mathrm{\prime }}\left({\alpha }_k\right)(x-{\alpha }_k)}}$. Hence, prove that there exists a unique polynomial of degree with given values $c_k$ at the point ${\alpha }_k,k=1,2,\dots n$.

[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 $x_0\le x\le x_1$ is not larger in magnitude than ${\displaystyle \frac{1}{8}}{M}_2{(x_1-x_0)}^2$ where $M_2=max|f^{\mathrm{\prime }\mathrm{\prime }}(x)|$ in $x_0\le x\le x_1$. Hence, show that if $f(x)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^{\pi }{e}^{-t^2}dt$, the truncation error corresponding to linear interpolation of $f(x)$ in $x_0\le x\le x_1$ cannot exceed ${\displaystyle \frac{{(x_1-x_0)}^2}{2\sqrt{2\pi e}}}$.

[2001, 12M]