Which of the following is the primary advantage of using the Runge-Kutta method over the Euler method?
(A) The Runge-Kutta method is simpler to implement.
(B) The Runge-Kutta method provides higher-order accuracy.
(C) The Runge-Kutta method converges slower than the Euler method.
(D) The Runge-Kutta method requires fewer function evaluations.
Answer: (B) The Runge-Kutta method provides higher-order accuracy.
What is the main disadvantage of the Jacobi method for solving linear systems?
(A) It converges more slowly than Gauss-Seidel.
(B) It requires fewer iterations than Gauss-Seidel.
(C) It is less accurate than the Gauss-Jordan method.
(D) It cannot be applied to non-linear systems.
Answer: (A) It converges more slowly than Gauss-Seidel.
The convergence rate of the Newton-Raphson method for solving non-linear equations depends on which of the following?
(A) The second derivative of the function at the root.
(B) The first derivative of the function at the root.
(C) The initial guess for the root.
(D) The exact value of the root.
Answer: (C) The initial guess for the root.
In the finite difference method for solving partial differential equations, what does the truncation error represent?
(A) The difference between the true solution and the numerical solution.
(B) The error introduced by discretizing the differential equation.
(C) The error caused by numerical round-off during calculations.
(D) The difference between two successive iterations.
Answer: (B) The error introduced by discretizing the differential equation.
Which method is generally used to solve large sparse systems of linear equations?
(A) Direct methods (like Gauss-Jordan elimination).
(B) Iterative methods (like Conjugate Gradient or GMRES).
(C) Runge-Kutta methods.
(D) Finite difference method.
Answer: (B) Iterative methods (like Conjugate Gradient or GMRES).
In the context of numerical integration, what is the primary advantage of Simpson’s rule over the Trapezoidal rule?
(A) Simpson’s rule has a higher order of accuracy.
(B) Simpson’s rule is easier to compute.
(C) Simpson’s rule requires fewer subintervals.
(D) Simpson’s rule is more suitable for irregular functions.
Answer: (A) Simpson’s rule has a higher order of accuracy.
In the context of solving boundary value problems using finite difference methods, what is the primary challenge in applying these methods?
(A) Finite difference methods do not converge.
(B) Discretization introduces errors in both the solution and the boundary conditions.
(C) Finite difference methods require a very fine grid.
(D) Boundary conditions are difficult to enforce.
Answer: (B) Discretization introduces errors in both the solution and the boundary conditions.
Which of the following statements is true for the stability of an explicit method in numerical analysis?
(A) Explicit methods are always stable regardless of the time step.
(B) Explicit methods are stable only for sufficiently small time steps.
(C) Explicit methods are always unstable.
(D) Explicit methods do not require a time step for stability.
Answer: (B) Explicit methods are stable only for sufficiently small time steps.
What is the main idea behind the method of least squares in numerical analysis?
(A) To minimize the sum of squared residuals between the approximate and exact solution.
(B) To maximize the residual error between the approximate and exact solution.
(C) To minimize the error between the initial guess and the true solution.
(D) To minimize the absolute difference between the approximate and exact solution.
Answer: (A) To minimize the sum of squared residuals between the approximate and exact solution.