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Numerical Methods Solutions

The document is an internal assessment test for the Department of Mathematics at Government College of Engineering, covering topics in Statistics and Numerical Methods. It includes questions on various numerical methods such as Newton-Raphson, Gauss elimination, and interpolation, along with specific problems to solve using these methods. The test is structured into two parts, with Part A consisting of short answer questions and Part B containing detailed problem-solving tasks.

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79 views4 pages

Numerical Methods Solutions

The document is an internal assessment test for the Department of Mathematics at Government College of Engineering, covering topics in Statistics and Numerical Methods. It includes questions on various numerical methods such as Newton-Raphson, Gauss elimination, and interpolation, along with specific problems to solve using these methods. The test is structured into two parts, with Part A consisting of short answer questions and Part B containing detailed problem-solving tasks.

Uploaded by

Leotta Princy
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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**Government College of Engineering - Srirangam, Trichy - 12**

**Department of Mathematics - Second Semester**

**Internal Assessment Test - I**

**MA3251 - Statistics and Numerical Methods**

**CIVIL / CSE / EEE / MECH**

PART A - Answer All the Questions (6 x 2 = 12 Marks)

1. State the convergence condition for Newton-Raphson method.

- The method converges if the initial guess is close to the actual root and if f(x), f'(x), and f''(x) are

continuous near the root. Also, f'(x) 0 in that interval.

2. Compare Gauss elimination method and Gauss-Jordan method.

- Gauss Elimination: Converts matrix to upper triangular form and then uses back-substitution.

- Gauss-Jordan: Reduces matrix to diagonal form for direct solution.

3. Why Gauss-Seidel is better than Gauss-Jacobis method?

- Gauss-Seidel uses the most recently updated values in each iteration, hence converges faster.

4. Define interpolation.

- Interpolation is the method of estimating unknown values within the range of a discrete set of

known data points.

5. State Newtons forward difference interpolation formula.

- f(x) = f(x0) + f(x0) (x - x0)/h + f(x0)/2! (x - x0)(x - x1)/h +


6. What is the order of the error in Trapezoidal rule?

- The order of error is O(h).

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PART B (3 x 16 = 48 Marks)

7(a)(i) Solve by Gauss-Seidel method:

Given system:

4x + 2y + z = 14

x + 5y - z = 10

x + y + 8z = 20

Iteration form:

x = (1/4)(14 - 2y - z)

y = (1/5)(10 - x + z)

z = (1/8)(20 - x - y)

Initial guess: x0 = 0, y0 = 0, z0 = 0

Iteration 1:

x1 = 3.5, y1 = 1.3, z1 = 1.525

Iteration 2:

x2 = 2.96875, y2 = 1.71125, z2 = 1.665

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7(a)(ii) Newton-Raphson Method:

Find root of x logx - 1.2 = 0

f(x) = x logx - 1.2

f'(x) = logx + 1/ln10

Initial guess x0 = 2

f(2) -0.5979, f'(2) 0.7353, x1 = 2.813

f(2.813) -0.026, x2 = 2.8424

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7(b)(i) Power Method:

Find the numerically largest Eigen value of:

A = [ [1, 4], [4, -1] ]

Initial vector: x(0) = [1, 0]

x(1) = Ax(0) = [1, 4] normalized = [0.25, 1]

Next [4.25, 0]

Converges to eigenvalue 4.123

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7(b)(ii) Solve 3x = (1 + sin x)

Iterative form: x = (1/3) (1 + sin x)


Initial guess x0 = 0.5

x1 = 0.4304

x2 = 0.4253

Converges to 0.4235

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8(a)(i) Jacobi Method:

A = [ [5, 0, 1], [-2, 1, 0], [0, 1, 5] ]

Jacobi iteration involves zeroing off-diagonal elements using rotations.

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8(a)(ii) Gauss-Jordan Method:

Equations:

2x + y + 4z = 12

8x - 3y + 2z = 20

4x + 11y - z = 33

Use row operations to get solution:

x = 1, y = 2, z = 2

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End of Solutions

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