Numerical Method
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Numerical Method
2075 Boards
Section A
Answer any two questions.
1
What is a non-linear equation? Derive the required expression to calculate the root of non-linear equation using the secant method. Using this expression, find a root of the following equation.
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2
What is matrix factorization? Factorize the given matrix A into LU using Doolittle algorithm and also solve Ax = b for given b using L and U matrices.
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3
What is initial value problem and boundary value problem? Write an algorithm and program to solve the boundary value problem using shooting method.
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Section B
Answer any eight questions.
4
Calculate a real negative root of following equation using Newton’s method for polynomial.
(x^4 + 2x^3 + 3x^2 + 4x = 5)
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5
What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.
5
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6
Find the lowest degree polynomial, which passes through the following points:
| X | F(x) |
|---|---|
| -2 | -19 |
| -1 | 0 |
| 1 | 2 |
| 2 | -3 |
| 3 | -4 |
| 4 | 5 |
Using this polynomial estimate f(x) at x = 0
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7
The fit function of type y = a + bx for the following points using the least squares method.
| X | F(x) |
| -1 | 1 |
| 1.2 | 20 |
| 2 | 27 |
| 2.7 | 33 |
| 3.6 | 41 |
| 4 | 45 |
5
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8
Calculate the integral value of the function given below from x = 1.8 to x = 3.4 using Simpson’s 1/3 rule.
| X | F(X) |
| 1.8 | 0.003 |
| 2.0 | 0.778 |
| 2.2 | 1.632 |
| 2.4 | 2.566 |
| 2.6 | 3.579 |
| 2.8 | 4.672 |
| 3.0 | 7.097 |
| 3.4 | 8.429 |
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9
Evaluate the following integration using Romberg integration.
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10
Solve the following set of equations using Gauss Seidel method.
x + 2y + 3z = 4
6x – 4y + 5z = 10
5x + 2y + 2z = 25
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11
From the following differential equation estimate y(1) using RK 4th order method.
[Take h = 0.5]
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12
Solve the Poison’s equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.
5
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