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

[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

[zaasmi]

Simpson’s rule is a numerical method that approximates the value of a definite
integral by using polynomials.
Quadratic
Linear
Cubic
Quartic

[zaasmi]

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

Simpson’s rule is a numerical method that approximates the value of a definite
integral by using polynomials.
Quadratic
Linear
Cubic
Quartic

Quadratic

Explanation:

• Simpson’s Rule: This numerical method approximates the value of a definite integral by using quadratic polynomials. Specifically, it uses parabolic segments to estimate the area under the curve, effectively fitting a second-degree polynomial (quadratic) to the points.

• Quadratic Polynomial: In Simpson’s rule, the integral is approximated using a quadratic polynomial that interpolates through three points. This polynomial fits the function being integrated and provides a more accurate approximation than linear methods.

Thus, Simpson’s rule is based on using quadratic polynomials.

So the correct option is:

Quadratic

[zaasmi]

In Simpson’s Rule, we use parabolas to approximating each part of the curve. This proves to be very efficient as compared to Trapezoidal rule.
True
False

[zaasmi]

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

In Simpson’s Rule, we use parabolas to approximating each part of the curve. This proves to be very efficient as compared to Trapezoidal rule.
True
False

True

Explanation:

• Simpson’s Rule: This method uses parabolic segments (quadratic polynomials) to approximate the integral of a function. By fitting a parabola to segments of the curve, Simpson’s rule often provides a more accurate approximation compared to linear methods.

• Trapezoidal Rule: This method approximates the integral by fitting straight lines (trapezoids) to the segments of the curve. While it is simpler, it generally requires more intervals to achieve the same level of accuracy as Simpson’s rule.

• Efficiency: Because parabolas can better capture the curvature of the function compared to straight lines, Simpson’s rule often achieves better accuracy with fewer intervals than the trapezoidal rule.

Therefore, Simpson’s rule is considered more efficient in terms of accuracy compared to the trapezoidal rule.

So, the statement is:

True

[zaasmi]

The predictor-corrector method an implicit method. (multi-step methods)
True
False

[zaasmi]

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

The predictor-corrector method an implicit method. (multi-step methods)
True
False

False

Explanation:

• Predictor-Corrector Methods: These are a type of multi-step methods used for solving ordinary differential equations. They involve predicting the solution at a future point using an explicit method and then correcting it using an implicit method or vice versa.

• Implicit vs. Explicit: Predictor-corrector methods are not exclusively implicit. The predictor step is usually explicit, while the corrector step can be either implicit or explicit, depending on the specific method used.

Therefore, the statement that the predictor-corrector method is an implicit method is:

False

[zaasmi]

Generally, Adams methods are superior if output at many points is needed.
True
False

[zaasmi]

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

Generally, Adams methods are superior if output at many points is needed.
True
False

True

Explanation:

• Adams Methods: These are multi-step methods for solving ordinary differential equations. Adams-Bashforth (explicit) and Adams-Moulton (implicit) methods are designed to use information from previous steps to provide higher accuracy.

• Efficiency: Adams methods can be more efficient when many output points are needed because they leverage information from multiple previous points to compute each new point. This can reduce the overall computational effort compared to methods that recalculate at each step without using past information.

• Performance: The performance of Adams methods improves with the number of points because they use past computed values effectively, which makes them advantageous for solving problems over many intervals.

Thus, Adams methods are generally superior when many output points are required.

So, the statement is:

True

[zaasmi]

The Trapezoidal rule is a numerical method that approximates the value of a.______________.
Indefinite integral
Definiteintegral
Improper integral
Function

[zaasmi]

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

The Trapezoidal rule is a numerical method that approximates the value of a.______________.
Indefinite integral
Definiteintegral
Improper integral
Function

Definite integral

Explanation:

• Trapezoidal Rule: This numerical method approximates the value of a definite integral. It works by dividing the area under a curve into trapezoids and summing their areas to estimate the total integral.

• Indefinite Integral: This represents a family of functions and is not computed using numerical methods like the trapezoidal rule.

• Improper Integral: This type of integral involves infinite limits or unbounded integrands, and while numerical methods can be adapted for improper integrals, the trapezoidal rule itself is typically used for definite integrals.

• Function: This is a broader concept and not directly related to the specific numerical approximation provided by the trapezoidal rule.

Thus, the trapezoidal rule specifically approximates the value of a definite integral.

So, the correct option is:

Definite integral

[zaasmi]

The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose anti derivative is not easy to obtain.

Antiderivative
Derivatives.