Numerical Methods

Q1: Which of the following is TRUE about the convergence of the Newton-Raphson method for finding roots of f(x) = 0?

It has quadratic convergence for simple roots It has linear convergence for all roots It always converges regardless of initial guess It has cubic convergence for multiple roots

Q2: In Gaussian elimination with partial pivoting, what is the main purpose of pivoting?

To reduce round-off errors To make the matrix symmetric To ensure the matrix is invertible To reduce the number of operations

Q3: What is the order of convergence of the Secant method?

1 1.618 2 3

Q4: In numerical integration, which of the following is TRUE about the Trapezoidal rule?

It is exact for polynomials of degree 1 It is exact for polynomials of degree 2 It is exact for all polynomials It has error O(h⁴)

Q5: What is the condition number of a matrix A in the context of solving linear systems?

κ(A) = ||A|| ||A⁻¹|| κ(A) = ||A|| + ||A⁻¹|| κ(A) = ||A||/||A⁻¹|| κ(A) = det(A)

Q6: Which of the following is TRUE about the Runge-Kutta methods for solving ODEs?

The classical RK4 method has local truncation error O(h⁵) The classical RK4 method has local truncation error O(h⁴) The classical RK4 method is unconditionally stable The classical RK4 method has order 3

Q7: In finite difference methods, what is the central difference approximation for f'(x)?

[f(x+h) - 2f(x) + f(x-h)]/h² [f(x+h) - f(x-h)]/(2h) [f(x+h) - f(x)]/h [f(x) - f(x-h)]/h

Q8: What is the main advantage of the Gauss-Seidel method over the Jacobi method?

It converges faster for most problems It requires less memory It always converges It can solve non-linear systems

Q9: In polynomial interpolation, what is the degree of the interpolating polynomial for n+1 distinct points?

At most n At most n+1 At most n-1 Exactly n+1

Q10: What is the error bound for Simpson's rule when approximating ∫_a^b f(x) dx?

|E| ≤ (b-a)M₄/180 h⁴ |E| ≤ (b-a)M₂/12 h² |E| ≤ (b-a)M₄/180 h² |E| ≤ (b-a)M₃/180 h⁴

Q11: Which of the following methods is used to solve systems of non-linear equations?

Newton's method for systems Gaussian elimination LU decomposition Crout's method

Q12: What is the order of convergence of the Bisection method?

Linear with rate 1/2 Quadratic Cubic Superlinear

Q13: In solving ODEs, what is the characteristic of an explicit method?

The solution at the next step depends only on known values The solution depends on future unknown values It is always stable It requires solving a non-linear system

Q14: What is the condition for the convergence of the fixed-point iteration x_{n+1} = g(x_n)?

|g'(x)| < 1 near the fixed point |g(x)| < 1 |g'(x)| > 1 near the fixed point g(x) is continuous

Q15: Which of the following is TRUE about the power method for finding eigenvalues?

It finds the dominant eigenvalue and eigenvector It finds all eigenvalues simultaneously It finds the smallest eigenvalue It works only for symmetric matrices