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

[zaasmi]

If x is an eigen value corresponding to eigen value of V of a matrix A. If a is any constant, then x – a is an eigen value corresponding to eigen vector V is an of the matrix A - a I.
True
False

[zaasmi]

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

If x is an eigen value corresponding to eigen value of V of a matrix A. If a is any constant, then x – a is an eigen value corresponding to eigen vector V is an of the matrix A - a I.
True
False

True

Explanation:

• If ( x ) is an eigenvalue of a matrix ( A ) corresponding to an eigenvector ( V ), this means:
  [ A V = x V ]

• If ( a ) is any constant, then ( x - a ) will be an eigenvalue of the matrix ( A - aI ), where ( I ) is the identity matrix.

• To see why, consider:
  [ (A - aI) V = A V - a I V ]
  [ = x V - a V ]
  [ = (x - a) V ]

• This shows that ( V ) is still an eigenvector of ( A - aI ), but now with the eigenvalue ( x - a ).

So, the statement is:

True

[zaasmi]

Central difference method seems to be giving a better approximation, however it requires more computations.

True
False

[zaasmi]

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

Central difference method seems to be giving a better approximation, however it requires more computations.

True
False

True

Explanation:

• Central Difference Method: This method approximates derivatives by averaging the forward and backward differences:
  [ f’(x) \approx \frac{f(x + h) - f(x - h)}{2h} ]

• Accuracy: The central difference method is often more accurate than the forward or backward difference methods because it uses information from both sides of the point where the derivative is being approximated. It has a smaller truncation error and provides a better approximation to the derivative.

• Computations: While it is more accurate, the central difference method requires evaluating the function at two points (both ( x + h ) and ( x - h )), as opposed to just one point for forward or backward differences. This requires more function evaluations and, therefore, more computational effort.

So, the statement that the central difference method gives a better approximation but requires more computations is:

True

[zaasmi]

Iterative algorithms can be more rapid than direct methods.
True
False

[zaasmi]

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

Iterative algorithms can be more rapid than direct methods.
True
False

True

Explanation:

• Iterative Algorithms: These methods, such as Jacobi, Gauss-Seidel, and Conjugate Gradient, are often used for solving large systems of linear equations, especially when the matrix is sparse. They can be more efficient in terms of memory and computation time for very large problems.

• Direct Methods: These methods, such as Gaussian elimination or LU decomposition, provide exact solutions (within rounding errors) but can be computationally intensive and require significant memory, especially for large systems.

• Efficiency: For large-scale problems, iterative methods can converge more quickly to an approximate solution and are often preferred due to their lower computational complexity and memory requirements compared to direct methods.

Therefore, iterative algorithms can indeed be more rapid than direct methods in certain cases.

So, the statement is:

True

[zaasmi]

Central Difference method is the finite difference method.
True
False

[zaasmi]

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

Central Difference method is the finite difference method.
True
False

True

Explanation:

• Central Difference Method: This method is a type of finite difference method used for approximating derivatives. It estimates the derivative by considering the average of the forward and backward differences:
  [ f’(x) \approx \frac{f(x + h) - f(x - h)}{2h} ]

• Finite Difference Methods: These methods are used to approximate derivatives and solve differential equations by using discrete points. The central difference method is one of these approaches, specifically designed to improve the accuracy of derivative approximations.

Thus, the central difference method is indeed a finite difference method.

So, the statement is:

True

[zaasmi]

Back substitution procedure is used in …
Select correct option:
Gaussian Elimination Method
Jacobi’s method
Gauss-Seidel method
None of the given choices

[zaasmi]

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

Back substitution procedure is used in …
Select correct option:
Gaussian Elimination Method
Jacobi’s method
Gauss-Seidel method
None of the given choices

Gaussian Elimination Method

Explanation:

• Gaussian Elimination Method: This is a direct method for solving systems of linear equations. It involves transforming the system into an upper triangular form using row operations and then performing back substitution to find the solution.

• Jacobi’s Method: This is an iterative method for solving linear systems, and it does not use back substitution.

• Gauss-Seidel Method: This is another iterative method for solving linear systems, and it also does not use back substitution.

• None of the given choices: This option is incorrect because the back substitution procedure is indeed used in the Gaussian Elimination Method.

So, the correct option is:

Gaussian Elimination Method

[zaasmi]

The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal.
True
False

[zaasmi]

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

The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal.
True
False

False

Explanation:

• Jacobi’s Method: This iterative method for solving a system of linear equations can be applied to matrices that may have zeros on the main diagonal. However, for the method to converge, it is usually preferable for the matrix to be diagonally dominant (where the magnitude of each diagonal element is greater than the sum of the magnitudes of the other elements in the same row) or for the matrix to have certain other properties that ensure convergence.

• Matrix with Zeros on Diagonal: If a matrix has zeros on the main diagonal, the Jacobi method can still be used, but additional steps or modifications might be needed to handle the zero entries. For example, a diagonal element that is zero would require special treatment to ensure the method can proceed.

Therefore, the Jacobi method does not require the matrix to have no zeros along its main diagonal.

So, the statement is:

False

[zaasmi]

Power method is applicable if the eigen vectors corresponding to eigen values are linearly independent.
True
False

[zaasmi]

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

Power method is applicable if the eigen vectors corresponding to eigen values are linearly independent.
True
False

False

Explanation:

• Power Method: This iterative method is used to find the dominant eigenvalue (the one with the largest magnitude) and its corresponding eigenvector of a matrix. It is particularly effective when there is a single eigenvalue that is significantly larger in magnitude than the others.

• Eigenvectors Independence: The power method does not require the eigenvectors to be linearly independent. It works as long as there is a dominant eigenvalue with a unique largest magnitude. If the matrix has a dominant eigenvalue with a corresponding eigenvector, the power method will converge to that eigenvalue and its eigenvector.

• Linear Independence: The requirement for eigenvectors to be linearly independent is more relevant for methods that need a complete set of eigenvectors, such as when performing a full eigendecomposition or finding all eigenvalues and eigenvectors.

Therefore, the power method is applicable even if the eigenvectors corresponding to eigenvalues are not all linearly independent.

So, the statement is:

False

[zaasmi]

Power method is applicable if the eigen values are ______________.
realanddistinct
real and equal
positive and distinct
negative and distinct

[zaasmi]

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

Power method is applicable if the eigen values are ______________.
realanddistinct
real and equal
positive and distinct
negative and distinct

Positive and distinct

Explanation:

• Power Method: The power method is designed to find the eigenvalue with the largest magnitude (dominant eigenvalue) and its corresponding eigenvector.

• Requirements: The power method is most effective when the matrix has a single dominant eigenvalue that is larger in magnitude than the other eigenvalues. It is not specifically limited to eigenvalues being positive and distinct, but having a dominant eigenvalue (which can be positive, negative, or zero) significantly larger in magnitude than the others improves convergence.

• Real and Distinct, Real and Equal, Positive and Distinct, Negative and Distinct: While the method can work with various types of eigenvalues, the convergence is most straightforward when there is a clear dominant eigenvalue, which could be positive or negative.

Thus, while the power method is generally applicable, it is especially straightforward when the dominant eigenvalue is distinct and of a significantly larger magnitude compared to others. “Positive and distinct” is often mentioned as a favorable condition but not a strict requirement.

So the most fitting answer from the given options would be:

Positive and distinct