Numerical Computing all previous/ past question papers

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Numerical Computing Fall 21 past paper

Numerical computing final paper

1. Solve the following system using Jacobi method (with 4 digu roues arithmetic, at least). Assume the error tolerance as 0.0001.

15x₁-2X2-6Xj+0x₁ -2x)+12×2-4X3-X4=300

0 -6×1-4×2+19×3-9×4 =0

0x1-X2-9X3+21x₁=0

(Marks 10)

2. Obtain the first and second derivative at x-7 using Stirling formula. Where

Xp= 12

f(x)

196

394

686

1090

1624

2306

(Marks-10)

3. Use the table in Question no. 2 of values by Newton forward differentiation formula. To compute f (0.25), (0.50), (0.75).

2125 (Marks= 15)

4. Use Question no. lequations, to find X1, X2, X3, and x, values using Gauss

Jordan Elimination Method.

(Marks 10)

5. Compute ff(x) dx based on Trapezoidal rule. 0.6

0.1

0.2

0.3

0.4

0.5

0.6

f(x)

0.0998

0.1987 0.2955 0.38994

0.4794

0.5646

(Marks-05)

Numerical Computing Mid term paper

Q1:

Find the Error if found then remove that error?

x	F(x)
2.5	24.145
2.0	22.043
2.5	20.225
2.5	18.644
2.2	17.262
3.0	16.047

Q2:

Find the approximate value of following integral for step, h = 0.25

Over the interval [3,9] where f(x) = x/x+3.

Composite Bool’s rule
Composite Weddle’s rule

Q3:

Construct the Newton’s Forward difference interpolating polynomial passing through the points (-1,-3), (1, 5), (3, 17), and (5, 21).

paper 2

Q1:

It is suspected that there is an error in one of the values of f(x) in the following table:

X	0	1	2	3	4	5	6	7	8	9	10
f(x)	0	2	20	90	272	605	1332	2450	4160	6642	10100

Construct the differences-table, detect and correct the error.

Q2:

Q3:

The following table represents the time x and the corresponding velocity f(x) of the particle moving with non-uniform velocity.

X	0.0	1.0	1.5	2.0
f(x)	2.5	1.0	4.6	5.3

From this table, using Lagrange’s formula inversely, determine the time when the velocity of the particle becomes 2.75.

Numerical Computing Sessional 1 question paper

NC S1

Numerical Computing Sessional 2 question paper

Filling the blanks (Score: 2 5=10)

(1) Let ξ=g(ξ)∈[a, b] be a fixed point of the real-valued and continuous function g(x). If g(x) has a continuous derivative in some neighborhood of ξ with ________.

Then the sequence (x_k) defined by x_k+1=g(x_k) converges to ξ as k→∞, provided that x₀ is sufficiently close to ξ.

(2) Assume that .Let be distinct real numbers and suppose that are real numbers. Then, there exists a unique Lagrange polynomial such that ____________.

(3) Suppose that a real-valued function g(x) has a fixed point ξ in [a, b]. Then, the corresponding expression can be written as________.

(4) Given that 4x²-a=0, where a is positive real number. Then, the iteration for solving this equation by the secant method can be expressed as ____________with x₀ and x₁ being the starting values.

(5) During the process of finding the single solution to the equation f(x) =0 in [a₀, b₀] with f(a₀)f(b₀)<0, the first step is to consider the midpoint . If (the given tolerance), then we need to choose the new interval:[a₁, b₁]=[a₀, c₀] in the case of _____;or [a₁, b₁]=[c₀, b₀] in the case of _____.

Find solution using Bessel’s formula Score= 10

Find Solution using Newton’s Backward Difference formula Score=10

x	f(x)
3.5	34.145
4.0	32.043
4.5	30.225
5.0	28.644
5.5	27.262
6.0	26.047

x = 4.75

Numerical Computing Final question paper

MCQ’s (Marks=10)
1. How much significant digits in this number 204.020050?
  1. 5
  2. 7

11
In which of the following method, we approximate the curve of solution by the tangent in each interval.
1. Hermite’s method
2. Euler’s method

Newton’s method

Lagrange’s method

When do we apply Lagrange’s interpolation?
1. Evenly spaced
2. Unevenly spaced

Both

None of above
When Newton’s backward formula is used?
1. To interpolate values
2. To calculate difference

To find approximate error

None of above

What are the errors in Trapezoidal rule of numerical integration?
1. E<Y
2. E>Y

None of above

Consider the initial value problem; give the condition of the absolute stability for the Euler method. (Marks= 10)

Suppose that the function values of f(x) are given in the following table.

X	-1	1	2
ln X	-3	0	4

(Marks= 10)
Find the approximate function value at the point by the Lagrange’s interpolation polynomial of degree 2.

Use the predictor-corrector method to get the approximate solution of the following initial value problem (Marks= 10)

with h = 0.1, 0 < x < 0.5.

Consider the function f(x) = cos x−x=0. Approximate a root of f(x) via Newton’s Method. (Marks= 10)

Numerical Computing Final Paper 2022

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Numerical Computing past paper