dy/dx - = 1 - y,y(0) = 0 is an example of

[zaasmi]

While using the relaxation method for finding the solution of the following system
8x1+3x2-2x3=5
4x1+7×2+2x3=9 with the initial vector(0, 0, 0), the residuals would be
3X1+5x2+9x3=2

R1 = 3, R2 = 7, R3 = 6
R1 = 2, R2 = 6, R3 = 3
R1 = 5, R2 = 9, R3 = 2
R1=-4, R2 = 8, R3 = 9

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

While using the relaxation method for finding the solution of the following system
8x1+3x2-2x3=5
4x1+7×2+2x3=9 with the initial vector(0, 0, 0), the residuals would be
3X1+5x2+9x3=2

R1 = 3, R2 = 7, R3 = 6
R1 = 2, R2 = 6, R3 = 3
R1 = 5, R2 = 9, R3 = 2
R1=-4, R2 = 8, R3 = 9

R1 = 3, R2 = 7, R3 = 6

[zaasmi]

Gauss-Seidel method is also known as method of …

False position

Iterations

None of the given choices

Successive displacement

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Gauss-Seidel method is also known as method of …

False position

Iterations

None of the given choices

Successive displacement

Successive displacement

[raza rajpoot]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Gauss-Seidel method is also known as method of …

Gauss–Seidel method is an improved form of Jacobi method, also known as the successive displacement method. This method is named after Carl Friedrich Gauss (Apr.

[zaasmi]

Sparse matrix is a matrix with …

Select the correct option:
Many elements are one
Some elements are one
Some elements are zero
Many elements are zero

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Sparse matrix is a matrix with …

Select the correct option:
Many elements are one
Some elements are one
Some elements are zero
Many elements are zero

Many elements are zero

[sanwal mazhar]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Sparse matrix is a matrix with …

A sparse matrix is a special case of a matrix in which the number of zero elements is much higher than the number of non-zero elements. As a rule of thumb, if 2/3 of the total elements in a matrix are zeros, it can be called a sparse matrix.

[zaasmi]

While using Gaussian Elimination method, the following augmented matrix

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

While using Gaussian Elimination method, the following augmented matrix

Correct Answer: A

[zaasmi]

While using the relaxation method for finding the solution of the following system
5x1+12-3x3=5
x1 +5x2 +2x3 = 7 with the initial vector(0, 0, 0), the residuals would be
x1 +2×2+4x3=3

Select the correct option
R1 = 3, R2 = 2, R3 = 1
R1 = 2, R2 = 1, R3 = 1
R1 = 5, R2 = 7, R3 = 3
R1 = 1,R2 = 3, R3 = 2

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

While using the relaxation method for finding the solution of the following system
5x1+12-3x3=5
x1 +5x2 +2x3 = 7 with the initial vector(0, 0, 0), the residuals would be
x1 +2×2+4x3=3

Select the correct option
R1 = 3, R2 = 2, R3 = 1
R1 = 2, R2 = 1, R3 = 1
R1 = 5, R2 = 7, R3 = 3
R1 = 1,R2 = 3, R3 = 2

R1 = 5, R2 = 7, R3 = 3

[zaasmi]

While solving by Gauss-Seidel method, which of the following is the first Iterative solution for the
system; x-2y =1, x+4y=4 ?
Select correct option:
(1, 0.75)
(0,0)
(1,0)
(0,1)

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

While solving by Gauss-Seidel method, which of the following is the first Iterative solution for the
system; x-2y =1, x+4y=4 ?
Select correct option:
(1, 0.75)
(0,0)
(1,0)
(0,1)

To solve the system of equations using the Gauss-Seidel method, we first need to rearrange each equation to solve for one variable in terms of the other. Given the system:

1. ( x - 2y = 1 )
2. ( x + 4y = 4 )

Rearrange these equations:

1. ( x = 1 + 2y )
2. ( x = 4 - 4y )

At the first iteration, we start with initial guesses. A common starting point is ( (0, 0) ), though other choices might be used depending on the problem.

Using ( (x_0, y_0) = (0, 0) ):

1. For ( x ): ( x = 1 + 2y )
   [ x_1 = 1 + 2 \cdot 0 = 1 ]

2. For ( y ): Use the updated ( x ) value from step 1.
   [ y = \frac{4 - x}{4} ]
   [ y_1 = \frac{4 - 1}{4} = \frac{3}{4} = 0.75 ]

Thus, the first iterative solution with the initial guess ( (0,0) ) is ( (1, 0.75) ).

So the correct option is:

(1, 0.75).

[zaasmi]

While using Relaxation method, which of the following is the Residuals for 1st iteration on the system;
2x+3y = 1, 3x +2y =4 ?
Select correct option:
(2,3)
(3,-2)
(-2,3)
(1,4)

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

While using Relaxation method, which of the following is the Residuals for 1st iteration on the system;
2x+3y = 1, 3x +2y =4 ?
Select correct option:
(2,3)
(3,-2)
(-2,3)
(1,4)

Let’s calculate the residuals for the given system using the Relaxation method (or Gauss-Seidel method, with ( \omega = 1 )).

System of Equations:

1. ( 2x + 3y = 1 )
2. ( 3x + 2y = 4 )

Initial guesses:
( x_0 = 0 ) and ( y_0 = 0 )

First Iteration:

1. Update ( x ):

   [ x = \frac{1 - 3y}{2} ]

   With ( y_0 = 0 ):
   [ x_1 = \frac{1 - 3 \cdot 0}{2} = \frac{1}{2} = 0.5 ]

2. Update ( y ):

   Substitute ( x_1 = 0.5 ) into the second equation:
   [ 3x + 2y = 4 ]
   [ 3 \cdot 0.5 + 2y = 4 ]
   [ 1.5 + 2y = 4 ]
   [ 2y = 4 - 1.5 ]
   [ 2y = 2.5 ]
   [ y_1 = \frac{2.5}{2} = 1.25 ]

Residual Calculation:

1. For (2x + 3y = 1):
   [ 2 \cdot 0.5 + 3 \cdot 1.25 = 1 + 3.75 = 4.75 ]
   Residual: ( 4.75 - 1 = 3.75 )

2. For (3x + 2y = 4):
   [ 3 \cdot 0.5 + 2 \cdot 1.25 = 1.5 + 2.5 = 4 ]
   Residual: ( 4 - 4 = 0 )

Residuals:

• For the first equation: ( 3.75 )
• For the second equation: ( 0 )

Among the given options, the residuals that match our calculated values are:

(3, 0)

However, since this option is not listed, it is likely there was a misinterpretation or a mistake in the options. Given the closest match from the provided options, (3, -2) seems to be the option that aligns best with the calculations, despite a discrepancy in the exact residual values.