Which of the following is NOT an advantage of Gauss-Seidel method over Newton-Raphson method? 

Explanation:

Gauss-Seidel Method vs. Newton-Raphson Method

The Gauss-Seidel method and the Newton-Raphson method are both iterative techniques used for solving systems of nonlinear equations. Each method has its own advantages and disadvantages, depending on the specific problem and context in which they are applied. Below, we will delve into the detailed analysis of the Gauss-Seidel method and the Newton-Raphson method, and why option 4 is the correct answer to the question regarding the advantages of the Gauss-Seidel method over the Newton-Raphson method.

Gauss-Seidel Method:

The Gauss-Seidel method is an iterative technique for solving a system of linear equations. It is a relaxation method that improves the solution iteratively by considering each equation one at a time and using the latest available values for the variables.

• Single iteration of computation is faster: The Gauss-Seidel method often requires less computational effort per iteration compared to the Newton-Raphson method because it deals with linear equations and updates the solution incrementally.
• Ease of programming: The Gauss-Seidel method is simpler to implement programmatically as it involves straightforward iterative updates without the need for complex derivative calculations.
• Most efficient use of core memory: The Gauss-Seidel method can be more memory efficient as it updates the solution in place and does not require storage of large matrices or additional data structures.

Newton-Raphson Method:

The Newton-Raphson method is a powerful and widely-used iterative technique for solving systems of nonlinear equations. It uses the first-order Taylor series expansion to approximate the solution and requires the computation of Jacobian matrices and their inverses.

• Quadratic convergence: The Newton-Raphson method typically converges faster than the Gauss-Seidel method because it has a quadratic rate of convergence, meaning that the error decreases quadratically as the iterations proceed.
• Robustness: The Newton-Raphson method is more robust for solving nonlinear equations as it can handle more complex and nonlinear systems effectively.

Correct Option Analysis:

The correct option is:

Option 4: Number of iterations required is lesser.

This option is NOT an advantage of the Gauss-Seidel method over the Newton-Raphson method. In fact, the Newton-Raphson method generally requires fewer iterations to converge to a solution compared to the Gauss-Seidel method. This is due to the quadratic convergence property of the Newton-Raphson method, which allows it to reach the solution more quickly as the iterations progress.

To further understand the analysis, let’s evaluate the other options:

Option 1: Single iteration of computation is faster.

This is an advantage of the Gauss-Seidel method. The computational effort per iteration is generally lower in the Gauss-Seidel method because it deals with linear equations and updates the solution incrementally without the need for complex derivative calculations.

Option 2: Ease of programming.

This is also an advantage of the Gauss-Seidel method. The method is simpler to implement programmatically, as it involves straightforward iterative updates and does not require the calculation of Jacobian matrices or their inverses, which can be complex and computationally intensive.

Option 3: Most efficient use of core memory.

This is another advantage of the Gauss-Seidel method. It can be more memory efficient as it updates the solution in place and does not require storage of large matrices or additional data structures, making it suitable for problems with limited memory resources.

Conclusion:

In summary, the Gauss-Seidel method is advantageous in terms of computational speed per iteration, ease of programming, and memory efficiency. However, it generally requires more iterations to converge to a solution compared to the Newton-Raphson method, which is known for its faster convergence due to its quadratic convergence property. Therefore, the statement that the number of iterations required is lesser is NOT an advantage of the Gauss-Seidel method over the Newton-Raphson method, making option 4 the correct answer.