Numerical Method
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Numerical Method
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Section A
Answer any two questions.
1
What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve
[x^2 – 5x + 6 = 0]. Calculate upto 3 decimal places.
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2
What are the limitations of direct methods for solving a system of linear equations? How Gauss Seidel method differs from Jacobi iteration? Solve the following system of linear equation using Jacobi iteration method.
2x-7y-10z=-17
5x+y+3z=14
x+10y+9z=7
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3
Write an algorithm and program to implement Lagrange interpolation method.
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Section B
Answer any eight questions.
4
Consider the following data points estimate the f(0.6) using Newton’s interpolation formula.
| x | f(x) |
|---|---|
| 0.1 | 2.68 |
| 0.2 | 3.04 |
| 0.3 | 3.38 |
| 0.4 | 3.69 |
| 0.5 | 3.97 |
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Solution:
Given Data:
| x | f(x) |
|---|---|
| 0.1 | 2.68 |
| 0.2 | 3.04 |
| 0.3 | 3.38 |
| 0.4 | 3.69 |
| 0.5 | 3.97 |
Divided Difference Table:
| x | f(x) | 1st DD | 2nd DD | 3rd DD | 4th DD |
|---|---|---|---|---|---|
| 0.1 | 2.68 | 3.6 | -1.0 | -1.6667 | 4.1667 |
| 0.2 | 3.04 | 3.4 | -1.5 | 0 | |
| 0.3 | 3.38 | 3.1 | -1.5 | ||
| 0.4 | 3.69 | 2.8 | |||
| 0.5 | 3.97 |
Newton's Interpolation Formula:
f(x) = f(x0) + (x-x0)f[x0,x1] + (x-x0)(x-x1)f[x0,x1,x2] + ...
Calculation for x = 0.6:
f(0.6) ≈ 2.68 + (0.6-0.1)(3.6) + (0.6-0.1)(0.6-0.2)(-1.0) + (0.6-0.1)(0.6-0.2)(0.6-0.3)(-1.6667) + (0.6-0.1)(0.6-0.2)(0.6-0.3)(0.6-0.4)(4.1667)
f(0.6) ≈ 2.68 + 1.8 - 0.2 - 0.1000 + 0.1000
Final Answer:
f(0.6) ≈ 4.38
5
What is regression analysis? Fit a second order polynomial for the following data values.
| x | y |
|---|---|
| 2 | 1.4 |
| 4 | 2.0 |
| 6 | 2.4 |
| 8 | 2.6 |
| 10 | 2.8 |
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Regression analysis is a statistical method used to estimate the relationship between a dependent variable and one or more independent variables. It models the relationship by fitting a curve or line to the data points, allowing for prediction and inference about the relationship.
Second Order Polynomial Fit
The general form of a second-order polynomial is:
y = a + bx + cx2
We need to find the coefficients a, b, and c that best fit the given data. This involves solving a system of linear equations derived from minimizing the sum of squared errors.
Data:
| x | y |
|---|---|
| 2 | 1.4 |
| 4 | 2.0 |
| 6 | 2.4 |
| 8 | 2.6 |
| 10 | 2.8 |
Let n be the number of data points (n = 5). We need to calculate the following sums:
Σx, Σy, Σx2, Σy2, Σxy, Σx3, Σx2y
| x | y | x2 | y2 | xy | x3 | x2y |
|---|---|---|---|---|---|---|
| 2 | 1.4 | 4 | 1.96 | 2.8 | 8 | 5.6 |
| 4 | 2.0 | 16 | 4.00 | 8.0 | 64 | 32.0 |
| 6 | 2.4 | 36 | 5.76 | 14.4 | 216 | 86.4 |
| 8 | 2.6 | 64 | 6.76 | 20.8 | 512 | 166.4 |
| 10 | 2.8 | 100 | 7.84 | 28.0 | 1000 | 280.0 |
| Sum | 30 | 220 | 26.32 | 74 | 1480 | 570.4 |
Normal Equations for Second-Order Polynomial (y = a + bx + cx2)
- n·a + (Σx)b + (Σx2)c = Σy
- (Σx)a + (Σx2)b + (Σx3)c = Σxy
- (Σx2)a + (Σx3)b + (Σx4)c = Σx2y
Plugging in the sums:
- 5a + 30b + 220c = 11.2
- 30a + 220b + 1800c = 74
- 220a + 1800b + 14800c = 570.4
Solving the System:
From (1): a = (11.2 - 30b - 220c) / 5 = 2.24 - 6b - 44c
Substitute into (2):
30(2.24 - 6b - 44c) + 220b + 1800c = 74
67.2 - 180b - 1320c + 220b + 1800c = 74
40b + 480c = 6.8
b + 12c = 0.17 → b = 0.17 - 12c
Substitute into (3):
220(2.24 - 6b - 44c) + 1800b + 14800c = 570.4
492.8 - 1320b - 9680c + 1800b + 14800c = 570.4
480b + 5120c = 77.6
480(0.17 - 12c) + 5120c = 77.6
81.6 - 5760c + 5120c = 77.6
-640c = -4 → c = 0.00625
Now find b:
b = 0.17 - 12(0.00625) = 0.17 - 0.075 = 0.095
Now find a:
a = 2.24 - 6(0.095) - 44(0.00625) = 2.24 - 0.57 - 0.275 = 1.395
Second-Order Polynomial:
y = 1.395 + 0.095x + 0.00625x2
6
What is numerical differentiation? The table below gives the values of distance travelled by a vehicle at various time interval, estimate the velocity and acceleration at x = 4.
| Time | Distance |
| 1 | 0 |
| 2 | 1 |
| 4 | 5 |
| 8 | 21 |
| 10 | 27 |
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7
What is an application of numerical integration? Find the value of the integral
using Simpson’s ( 3/8 ) rule with ( n = 6 ).
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8
Solve the following system of linear equations using Gauss-Jordan elimination method.
x+2y-3z=4
2x+4y-6z=8
x-2y+5z=4
5
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9
Given the data points below
| X | F(X) |
| 1 | 1.5 |
| 3 | 4.5 |
| 4 | 9 |
Find cubic spline which belongs to 1<=x<=3 and estimate f(2) using cubic splines.
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10
What is differential equation? Differentiate between ODE and PDE with example.
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11
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12
Solve the Poisson equation
with boundary conditions:
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