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How can Horner's rule be used to evaluate the f(x) and f'(x) of a polynomial at given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner's rule.
What is matrix factorization? How can it be used to solve system of linear equations? Factorize the given matrix A and solve the system of equations Ax = b for given b using L and U matrices.
⎡1 2 3⎤ 4
A = ⎢2 8 11⎥ and b = 12
⎣3 22 36⎦ 28
What is higher order differential equation? How can you solve the higher order differential equation? Explain. Solve the following differential equation for 1 ≤ x ≤ 2, taking h = 0.25.
d²y/dx² + 3 dy/dx + 5y = 0, with y(1) = 1 and y'(1) = 2 .
Attempt any Eight questions
[8x5=40]How the half-interval method can estimate a root of non-linear equation? Find a real root of following equation using half-interval method correct up to two decimal places.
x² − e⁻ˣ − x = 1
Calculate a real root of the given equation using fixed point iteration correct up to 3 significant figures.
2x³ − 2x = 5
What is Newton's interpolation? Obtain the divided difference table from the following data set and estimate the f(x) at x = 2 and x = 5.
| x | 3.2 | 2.7 | 1.0 | 4.8 | 5.6 |
| f(x) | 22.0 | 17.8 | 14.2 | 38.3 | 51.7 |
What is linear regression? Fit the linear function to the following data
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 2.4 |
| f(x) | 2.0 | 2.6 | 3.9 | 6.0 | 9.3 | 15 | 20.6 | 30.4 |
What are the problems with polynomial interpolation for large number of data set? How such problems are addressed? Explain with example.
Evaluate the following integration using Romberg integration.
∫₀¹ (sin² x)/x dx
Solve the following set of linear equations using Gauss Jordan method.
x₂ + 2x₃ + 3x₄ = 9
7x₁ + 6x₂ + 5x₃ + 4x₄ = 33
8x₁ + 9x₂ + x₄ = 27
2x₁ + 5x₂ + 4x₃ + 3x₄ = 23
Solve the following differential equation for 1 ≤ x ≤ 2, taking h = 0.25 using Heun's method.
y'(x) + x²y = 3x, with y(1) = 1
Consider a metallic plate of size 90 cm by 90 cm. The two adjacent sides of the plate are maintained at temperature of 100°C and remaining two adjacent sides are held at 200°C. Calculate the steady state temperature at interior points assuming a grid size of 30 cm by 30 cm.