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 ux when it is given that u1=1, u2=9, u3=25, u4=55,u5=105.
[2018, 10M]
2) In an examination, the number of students who obtained marks between certain limits were given in the following table:
Marks 30−4040−5050−6060−7070−80 No. of students 3142513531.
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, u0, u1 and u2, a value is interpolated by Lagrange’s formula. Show that it may be written in the form ux=yu0+y(y2−1)3!Δ2u−1+x(x2−1)3!Δ2u0, where x+y=1.
[2017, 15M]
2) Let f(x)=e2xcos3x for x∈[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:
x−1234f(x)−1113169Find f(1.5).
[2015, 20M]
4) For the given set of data points (x1,f(x1)), (x2,f(x2)), ⋯ (xn,f(xn)), 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):
x012456f(x)114155619[2009, 15M]
6) The following values of the function f(x)=sinx+cosx 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(π12). Compare with exact value.
[2008, 15M]
7) If Q is a polynomial with simple roots α1,α2,…αn and if P is a polynomial of degree<n, show that P(x)Q(x)=∑nk=1P(αk)Q′(αk)(x−αk). Hence, prove that there exists a unique polynomial of degree with given values ck at the point αk,k=1,2,…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 x0 and x1 with x0≤x≤x1 is not larger in magnitude than 18M2(x1−x0)2 where M2=max|f′′(x)| in x0≤x≤x1. Hence, show that if f(x)=2√π∫π0e−t2dt, the truncation error corresponding to linear interpolation of f(x) in x0≤x≤x1 cannot exceed (x1−x0)22√2πe.
[2001, 12M]