Numerical Methods Civil Engineering (CE) Notes | EduRev

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The document Numerical Methods Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Topic wise GATE Past Year Papers for Civil Engineering.
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Question 1: The value of the function f(x) is given at n distinct values of x and its value is to be interpolated at the point x*, using all the n points. The estimate is obtained first by the Lagrange polynomial, denoted by IL and then by the Newton polynomial, denoted by IN. Which one of the following statements is correct?
(a) IL is always greater than IN 

(b) No definite relation exists between IL and IN
(c) IL and IN are always equal
(d) IL is always less than IN        [2019 : 1 Mark, Set-ll]

Answer: (c)

Question 2: The quadratic equation 2x2 - 3x + 3 = 0 is to be solved numerically starting with an initial guess as x0 = 2. The new estimate of x after the first iteration using Newton-Raphson method is ______.    [2018 : 1 Mark, Set-II]
Solution:
Given
f(x) = 2x2 - 3x + 3, x0 = 2
f'(x) = 4x - 3
By Newton-Rapshon
Numerical Methods Civil Engineering (CE) Notes | EduRev


Question 3: Consider the equation du/dt = 3t2 + 1 with u = 0 at t = 0. This is numerically solved by using the forward Euler method with a step size. Δt = 2. The absolute error in the solution in the end of the first time step is _____________.    [2017 : 2 Marks, Set-I]
Solution:  du/dt = 3t2 + 1
f(u , f) = 3t2 + 1
u0 = 0
t0 = 0
Δt = 2
By Euler’s method
Numerical Methods Civil Engineering (CE) Notes | EduRev
After first iteration u = 2 when t = 2
Numerical Methods Civil Engineering (CE) Notes | EduRev
Absolute error = Exact value - approx value
= 10 -2
= 8

Question 4: Newton-Raphson method is to be used to find foot of equation 3x - ex + sinx = 0. If the initial trail value of the roots is taken as 0.333, the next approximation for the root would be __________.    [2016 : 1 Mark, Set-I]
Solution: According to Newton-Raphson Method:
Numerical Methods Civil Engineering (CE) Notes | EduRev
or, f' (x) = 3 - ex + cosx
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev
∴ X1 = 0.36

Question 5: For step-size, Δx = 0.4, the value of following integral using Simpson’s 1/3 rule is _________. Numerical Methods Civil Engineering (CE) Notes | EduRev
[2015 : 2 Marks, Set-II]
Solution: a = 0,b = 0.8, Δx = 0.4
f{x) = 0.2 + 25x - 200x2 + 675x3 - 900x4 + 400x5
Numerical Methods Civil Engineering (CE) Notes | EduRev
By Simpson’s 1/3 Rule
Numerical Methods Civil Engineering (CE) Notes | EduRev
0 = y(0) = 0.2
y1 = y (0.4) = 2.456
y2 = y(0.8) = 0.232
Numerical Methods Civil Engineering (CE) Notes | EduRev
= 1.367

Question 6: In Newton-Raphson iterative method, the initial guess value (xini) is considered as zero while finding the roots of the equation:
f(x) = -2 + 6x - 4X2 + 0.5.x3.
The correction, Δx, to be added to xini in the first iteration is ___________.
    [2015 : 1 Mark, Set-II]

Solution:
f (x) = - 2 + 6x - 4x2 + 0.5x3 
f '(x) = 6 - 8x + 1 .5x2 
xini = 0
By Newton Raphson Method,
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev

Question 7: The quadratic equation x2 - 4x + 4 = 0 is to be solved numerically, starting with the initial guess x0 = 3. The Newton-Raphson method is applied once to get a new estimate and then the Secant method is applied once using the initial guess and this new estimate. The estimated value of the root after the application of the Secant method is  _____.    [2015 : 2 Marks, Set-I]
Solution.
f(x) = x2 - 4x + 4
f'(x) = 2x - 4
x0 = 3
f (3) = 1, f'(3) = 2
By Newton Rapshon method,
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev

f(5/2) = 25/4 -10 + 4 = 1/4
By Secant method,
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev

Question 8: The integral Numerical Methods Civil Engineering (CE) Notes | EduRev  is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If I is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship? 
(a) J > I 
(b) J < I 
(c) J = I 
(d) insufficient data to determine the relationship    
[2015 : 1 Mark, Set-I]
Answer: (a)
Solution:
Numerical Methods Civil Engineering (CE) Notes | EduRev

Exact value is computed by integration which follows thee exact shape of graph while computing the area.
Whereas, in Trapezoidal rule, the lines joining each points are considered straight lines which is not the exact variation of graph all the time like as shown in figure.
∴ J > I
OR
Numerical Methods Civil Engineering (CE) Notes | EduRev
Here, f(x) = x2 
or,  f'(x) = 2x
or, f''(x) = 2 > 0
Since f''(x) is positive, the error is negative.
Since error = exact - approximate.
= I - J
and since error is negative in this case J > I is true.

Question 9: The magnitude as the error (correct to two decimal places) in the estimation of following integral using simpson 1/3 rule. Take the step length as 1.
Numerical Methods Civil Engineering (CE) Notes | EduRev    [2013 : 2 Marks]
Solution: Using Simpson’s Rule,
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev
= 245.33
The value of integral,
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev
∴Magnitude of error = 245.33 - 244.8 = 0.53

Question 10: The error in Numerical Methods Civil Engineering (CE) Notes | EduRev for a continuous function estimated with h = 0.03 using the central difference formula Numerical Methods Civil Engineering (CE) Notes | EduRev Numerical Methods Civil Engineering (CE) Notes | EduRev The values of  x0 and f(x0) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately. 
(a) 1.3 x 10-4 
(b) 3.0 x 10-4 
(c) 4.5 x 10-4 
(d) 9.0 x 10-4     
[2011 : 2 Marks]
Answer: (d)
Solution: Error in central difference formula is 0(h2)
This means,
error ∝ h2
If error for h = 0.03 is 2 x 10-3 then,
Error for h = 0.02 is approximately

Numerical Methods Civil Engineering (CE) Notes | EduRev

Question 11: The estimate of Numerical Methods Civil Engineering (CE) Notes | EduRev Obtained using Simpson's rule with three-point function evaluation exceeds the exact value by 
(a) 0.235 
(b) 0.068 
(c) 0.024 
(d) 0.012
   [2011 : 2 Marks]
Answer: (d)
Solution: 
Numerical Methods Civil Engineering (CE) Notes | EduRev
Approximate value by Simpson’s rule with 3 point is,
Numerical Methods Civil Engineering (CE) Notes | EduRev
(npt is the number of pts and ni is the number of intervals)
Hero Numerical Methods Civil Engineering (CE) Notes | EduRev
The tabic is
Numerical Methods Civil Engineering (CE) Notes | EduRev
So the estimate exceeds the exact value by,
Approximate value - Exact value
= 0.012499
≈ 0.012

Question 12: The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x2 - N = 0. If i denotes the iteration index, the correct iterative scheme will be
Numerical Methods Civil Engineering (CE) Notes | EduRev
[2011 : 1 Mark]
Answer: (a)
Solution: 
Numerical Methods Civil Engineering (CE) Notes | EduRev
Numerical Methods Civil Engineering (CE) Notes | EduRev

Question 13: The table below gives values of a function F(x) obtained for values of x at intervals of 0.25.
Numerical Methods Civil Engineering (CE) Notes | EduRev
The value of the integral of the function between the limits 0 to 1 using Simpson’s rule is 
(a) 0.7854 
(b) 2.3562 
(c) 3.1416 
(d) 7.5000
  [2010 : 2 Marks]
Answer: (a)
Solution: 
Numerical Methods Civil Engineering (CE) Notes | EduRev

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