BCA DCA2101 COPUTER ORIENTED NUMERICAL METHOD

Description

SESSION	april 2024
PROGRAM	Bachelorof CoMPUTER APPLICATIONS (BCA)
SEMESTER	III
course CODE & NAME	DCA2101 – Computer Oriented Numerical Methods

 

 

 

SET-I

 

 

1. Show that

(a)   δμ= 1/2(∆+∇)

(b)   ∆-∇=∆∇

 

Ans 1.

1. a) To show that δμ=1/2(∆+∇), we can start with the definition of the Laplacian (δ) and the gradient (∇):

δf = ∇²f = ∇•∇f

Now, let’s consider the Laplacian of a function μ:

δμ = ∇²μ = ∇•∇μ

Next, we can use the identity that relates the Laplacian and the gradient:

∇•(∇μ) = ∇²μ Its Half solved only

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2. Find Lagrange’s interpolation polynomial fitting the pointsy(1) = -3,y(3)= 0,y(4)= 30, y(6) = 132 Hence find y(5).

 

Ans 2.

Step 1: Find the Lagrange basis polynomials

For each point, we need to create a Lagrange basis polynomial that has a value of 1 at that point and a value of 0 at all other points. We’ll use these polynomials to build the final interpolation polynomial.

 

For example, for the point (1, -3), the Lagrange basis polynomial is:

 

L1(x) = (x – 3)(x – 4)(x – 6) / ((1 – 3)(1 – 4)(1 – 6))

 

Simplifying this expression, we get:

 

L1(x) = (x – 3)(x – 4)(x

 

 

3. Evaluate , given the following table of values:

 

	10	20	30	40	50
	46	66	81	93	101

 

Ans 3.

To evaluate f(15) given the table of values, we need to interpolate the value of y corresponding to x = 15.

Looking at the table, we can observe that the values of x are evenly spaced by 10 units. To interpolate, we can use linear interpolation.

First, let’s find the interval in which 15 lies:

– x = 10 corresponds to y = 46

– x = 20

 

SET-II

 

 

4. Find the equation of the best fitting straight line for the data:

 

X	1	3	4	6	8	9	11	14
Y	1	2	4	4	5	7	8	9

 

Ans 4.

Linear Regression is a method to model the relationship between two continuous variables, X and Y. In this case, we have a set of data points (X, Y) and we want to find the best straight line that fits these data points.

 

To find the

 

 

5. Calculate the intercept (b): Finally, calculate the intercept (b) using the formula:

 

b = μY – m * μX

 

Substituting the values, we get:

 

b = 5 – m * 7

= 5 – approximately m * 7

= approximately b ≈ 5 – approximately m * 7

= approximately b ≈ 5 – approximately (32.00 * 7)

= approximately b ≈ -15

 

5. For what value of λ & μ the following system of equations:

x + y + z = 6

x +2y+3z =10

x+2y +λz =μ   may have

(i)  Unique solution

(ii) Infinite number of solutions

(iii)  No solution

 

Ans 5.

To determine the values of λ and μ that result in each type of solution, we can use the properties of systems of equations.

(i) For a unique solution:

• The rank of the coefficient matrix should be equal to the rank of the augmented matrix.
• The

 

 

6. Find the solution for x=0.2 taking interval length 0.1 using Euler’s method to solve: dy/dx=1-y given y(0)=0.

 

Ans 6.

 

Step 1: Understand the problem

 

We have the differential equation:

 

dy/dx = 1 – y

 

with the initial condition:

 

y(0) = 0