BCA Semester 4 – Numerical Methods – Unit 2: Interpolation and Approximation (8Hrs.)

Question 1

Estimate the value of ln⁡(2.5) using Newton-Gregory forward difference formula given the following data:

Marks: 5

Year: 2021 Final TU FOHSS

Question 2

Given the data points: |i|0|1|2| |-|-|-|-| |\(Xi\)|9|16|25| |\(fi\)|3|4|5| Estimate the function value of fat \(x = 14\) using cubic splines.

Marks: 10

Year: 2021 Final TU FOHSS

Given data: \[ \begin{array}{c|ccc} i & 0 & 1 & 2 \\ \hline xi & 9 & 16 & 25 \\ fi & 3 & 4 & 5 \end{array} \] We are required to estimate \( f(x) \) at \[ x = 14 \] --- Step 1: Interval and step si

Question 3

What are the advantages of Lagrange's formula over Newton's formula? When Newton's forward interpolation formula is used?

Marks: 5

Year: 2022 Final TU FOHSS

1. Advantages of Lagrange’s Formula over Newton’s Formula 1. Simplicity of Form - Lagrange's formula gives the interpolating polynomial directly without computing divided differences. 2. No Need

Question 4

Find the polynomial that interpolates \(f(0) = 0, f(1) = 1, f(2) = 0, f(3) = 1, f(4) = 0 \)using Newton divided differences. Use the Newton table to generate the necessary divided differences.

Marks: 5

Year: 2022 Final TU FOHSS

Given Data: \[ \begin{array}{c|c} xi & fi \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ 3 & 1 \\ 4 & 0 \\ \end{array} \] We need to find the interpolating polynomial using Newton divided differences. --- Step

Question 5

Estimate the value of \(sin θ \) at \(θ = 45°\) using Newton's backward difference formula from the following set of data. |\(\theta\)|10|20|30|40|50|60| |-|-|-|-|-|-|-| \(sin \theta\)|0.1736|0.3420|0

Marks: 5

Year: 2023 Final TU FOHSS

Given Data: | \(\theta\) (°) | 10 | 20 | 30 | 40 | 50 | 60 | |----------------|-------|-------|-------|-------|-------|-------| | \(\sin \theta\)| 0.1736| 0.3420| 0.5000| 0.6428| 0.7660| 0

Question 6

a) Fit a straight line to the following set of data using Least Square Regression.\ b) Apply the factorization method (any) to solve the equations: \[ \quad \quad 2x + 3y + 7z =4 \] \[ \quad \quad2

Marks: 10

Year: 2023 Final TU FOHSS

Part (a) Fit a Straight Line Using Least Squares Goal: Fit a line \(y = a + bx\) to a set of data \((xi, yi)\). Steps: 1. Given Data: Suppose data points are: \(| xi | yi |\) (fill with act

i	0	1	2
$X_i$	9	16	25
$f_i$	3	4	5