Computational Methods MCQs

1. Which of the following is NOT a numerical method? A) Newton-Raphson method B) Euler’s method C) Binary search D) Gaussian elimination Answer: C) Binary search 2. Which method is used to find the roots of nonlinear equations? A) Gauss-Seidel method B) Bisection method C) LU decomposition D) Cramer’s rule Answer: B) Bisection method 3. The process of finding a numerical solution by successively refining an initial estimate is known as: A) Integration B) Iteration C) Extrapolation D) Interpolation Answer: B) Iteration 4. Which method is used to solve systems of linear equations by transforming the augmented matrix into row-echelon form? A) Gauss-Seidel method B) Jacobi iteration C) Gaussian elimination D) Successive over-relaxation (SOR) Answer: C) Gaussian elimination 5. Which numerical method is suitable for solving ordinary differential equations (ODEs) where the step size is relatively small? A) Euler’s method B) Runge-Kutta method C) Adams-Bashforth method D) Trapezoidal rule Answer: B) Runge-Kutta method 6. What is the order of the Runge-Kutta method typically used for accurate solutions of ODEs? A) First order B) Second order C) Third order D) Fourth order Answer: D) Fourth order 7. The trapezoidal rule is used for: A) Numerical integration B) Solving linear equations C) Finding eigenvalues D) Curve fitting Answer: A) Numerical integration 8. Which method is used for solving systems of nonlinear equations iteratively? A) Newton-Raphson method B) Gauss-Seidel method C) Simpson’s rule D) Secant method Answer: A) Newton-Raphson method 9. Which numerical method involves dividing the interval into smaller sub-intervals to approximate the integral of a function? A) Simpson’s rule B) Trapezoidal rule C) Monte Carlo method D) Romberg integration Answer: B) Trapezoidal rule 10. Romberg integration improves the accuracy of numerical integration by: A) Using random sampling B) Iteratively refining the estimate C) Dividing the interval into equal sub-intervals D) Using higher-order polynomials Answer: B) Iteratively refining the estimate 11. Which method is used to find the inverse of a matrix? A) Cramer’s rule B) Gaussian elimination C) LU decomposition D) Eigenvalue decomposition Answer: C) LU decomposition 12. Which iterative method is used for solving large sparse systems of linear equations? A) Jacobi iteration B) QR decomposition C) Singular Value Decomposition (SVD) D) Cholesky decomposition Answer: A) Jacobi iteration 13. Which method is used to minimize a function by iterative steps that adjust the parameters? A) Gradient descent B) LU decomposition C) Householder transformation D) QR algorithm Answer: A) Gradient descent 14. The Eigenvalue decomposition of a matrix is also known as: A) Cholesky decomposition B) Singular Value Decomposition (SVD) C) QR decomposition D) Spectral decomposition Answer: D) Spectral decomposition 15. Which method is used to solve partial differential equations (PDEs) by discretizing both time and space? A) Finite difference method B) Finite element method C) Boundary element method D) Monte Carlo method Answer: A) Finite difference method 16. In numerical methods, h is often used to denote: A) The step size B) The number of iterations C) The size of the matrix D) The tolerance level Answer: A) The step size 17. Which method is used to interpolate values between known data points? A) Gaussian elimination B) Lagrange interpolation C) Householder transformation D) Singular Value Decomposition (SVD) Answer: B) Lagrange interpolation 18. The condition number of a matrix measures its: A) Trace B) Determinant C) Sensitivity to rounding errors D) Eigenvalues Answer: C) Sensitivity to rounding errors 19. Which numerical method is used to approximate the solution of integral equations? A) Monte Carlo method B) Simpson’s rule C) Euler’s method D) Boundary element method Answer: D) Boundary element method 20. Which method is used for solving nonlinear optimization problems without requiring derivatives? A) Newton’s method B) Genetic algorithms C) LU decomposition D) Cholesky decomposition Answer: B) Genetic algorithms 21. Which method is commonly used to find the roots of a polynomial equation with complex coefficients? A) Newton-Raphson method B) Durand-Kerner method C) Gauss-Seidel method D) Spline interpolation Answer: B) Durand-Kerner method 22. Which numerical method is based on random sampling to compute the integral of a function? A) Monte Carlo method B) Gaussian elimination C) Simpson’s rule D) Bisection method Answer: A) Monte Carlo method 23. The LU decomposition of a matrix A can be written as A = LU, where L is lower triangular and U is upper triangular. This decomposition is useful for: A) Finding the determinant of A B) Solving systems of linear equations C) Minimizing functions D) Performing eigenvalue analysis Answer: B) Solving systems of linear equations 24. Which method is used for solving nonlinear optimization problems when the objective function is not smooth? A) Conjugate gradient method B) Newton’s method C) Secant method D) Householder transformation Answer: A) Conjugate gradient method 25. In numerical methods, what does convergence refer to? A) The number of significant digits in the result B) The process of refining a solution to approach the true solution C) The stability of the algorithm D) The accuracy of the initial guess Answer: B) The process of refining a solution to approach the true solution 26. Which method is used for finding the eigenvalues and eigenvectors of a matrix? A) Singular Value Decomposition (SVD) B) QR algorithm C) Cholesky decomposition D) Power iteration method Answer: D) Power iteration method 27. Which method is used to approximate the derivative of a function at a point using function evaluations? A) Trapezoidal rule B) Simpson’s rule C) Forward difference method D) Backward difference method Answer: C) Forward difference method 28. Which numerical method is used for smoothing data and making predictions based on polynomial interpolation? A) Spline interpolation B) Newton’s divided difference method C) Lagrange interpolation D) Neville’s algorithm Answer: A) Spline interpolation 29. Which method is used to solve linear programming problems by iteratively improving feasible solutions? A) Simplex method B) Newton’s method C) Steepest descent method D) Gram-Schmidt process Answer: A) Simplex method 30. Which method is used to solve simultaneous nonlinear equations by transforming them into a sequence of linear equations? A) Secant method B) Newton’s method C) Householder transformation D) Gram-Schmidt process Answer: B) Newton’s method 31. Which method is used to solve systems of linear equations by iteratively refining the solution based on successive approximations? A) Jacobi iteration B) QR decomposition C) Singular Value Decomposition (SVD) D) Cholesky decomposition Answer: A) Jacobi iteration 32. The QR decomposition of a matrix decomposes it into: A) Lower and upper triangular matrices B) Orthogonal and upper triangular matrices C) Diagonal and orthogonal matrices D) Symmetric and orthogonal matrices Answer: B) Orthogonal and upper triangular matrices 33. Which numerical method is used for solving stiff systems of ordinary differential equations (ODEs)? A) Forward Euler method B) Backward Euler method C) Implicit methods D) Explicit methods Answer: C) Implicit methods 34. Which method is used for approximating the solution of a differential equation by treating derivatives as differences? A) Finite difference method B) Finite element method C) Boundary element method D) Monte Carlo method Answer: A) Finite difference method 35. Which numerical method is used for finding the zeros of a function by using both function values and derivative information? A) Secant method B) Newton-Raphson method C) Bisection method D) Jacobi iteration Answer: B) Newton-Raphson method 36. Which method is used for finding the roots of a polynomial equation by constructing a sequence of polynomials that have the same roots? A) Durand-Kerner method B) Lagrange interpolation C) Neville’s algorithm D) Horner’s method Answer: A) Durand-Kerner method 37. Which method is used to approximate the area under a curve by dividing it into small trapezoids? A) Simpson’s rule B) Trapezoidal rule C) Gauss-Seidel method D) Gaussian elimination Answer: B) Trapezoidal rule 38. Which numerical method involves solving a series of linear equations to approximate the solution of a differential equation? A) Finite difference method B) Finite element method C) Monte Carlo method D) Simpson’s rule Answer: A) Finite difference method 39. Which method is used for minimizing a function by iteratively moving in the direction of the negative gradient? A) Gradient descent B) Newton’s method C) Conjugate gradient method D) Romberg integration Answer: A) Gradient descent 40. Which numerical method is used for solving nonlinear systems of equations by approximating the Jacobian matrix? A) Newton-Raphson method B) Gauss-Seidel method C) Jacobi iteration D) LU decomposition Answer: A) Newton-Raphson method 41. Which method is used for finding the approximate solution of a system of linear equations by iterating over each equation and updating the variables? A) Gauss-Seidel method B) LU decomposition C) QR decomposition D) Cholesky decomposition Answer: A) Gauss-Seidel method 42. The Householder transformation is used in numerical methods for: A) Finding the inverse of a matrix B) Solving nonlinear equations C) Reducing a matrix to upper triangular form D) Matrix factorization Answer: D) Matrix factorization 43. Which method is used for solving linear programming problems by iteratively improving feasible solutions at each step? A) Simplex method B) Gradient descent C) Newton’s method D) Bisection method Answer: A) Simplex method 44. Which numerical method is used for solving differential equations by discretizing the solution domain into small elements? A) Finite element method B) Monte Carlo method C) Boundary element method D) Romberg integration Answer: A) Finite element method 45. Which method is used to solve systems of linear equations by iteratively refining an initial guess until convergence? A) Jacobi iteration B) QR decomposition C) Cholesky decomposition D) LU decomposition Answer: A) Jacobi iteration 46. The SVD (Singular Value Decomposition) of a matrix decomposes it into: A) Lower and upper triangular matrices B) Orthogonal and upper triangular matrices C) Diagonal and orthogonal matrices D) Symmetric and orthogonal matrices Answer: C) Diagonal and orthogonal matrices 47. Which numerical method is used for finding the inverse of a matrix by decomposing it into simpler components? A) LU decomposition B) QR decomposition C) Cholesky decomposition D) Singular Value Decomposition (SVD) Answer: A) LU decomposition 48. Which method is used for finding the approximate solution of a differential equation by treating derivatives as finite differences? A) Finite difference method B) Finite element method C) Monte Carlo method D) Simpson’s rule Answer: A) Finite difference method 49. Which method is used for finding the roots of a polynomial equation by constructing a sequence of polynomials that have the same roots? A) Durand-Kerner method B) Lagrange interpolation C) Neville’s algorithm D) Horner’s method Answer: A) Durand-Kerner method 50. Which method is used for solving systems of linear equations by iteratively refining the solution based on successive approximations? A) Jacobi iteration B) QR decomposition C) Singular Value Decomposition (SVD) D) Cholesky decomposition Answer: A) Jacobi iteration

Computational Methods MCQs

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