Approximation Methods for Polynomial Roots - One Line Questions

1. The order of convergence for Newton-Raphson method, when the root is simple, is: — 2
2. When Newton's method converges linearly, the error at iteration k+1 is roughly proportional to the error at iteration k raised to the power of: — 1
3. Consider the polynomial P(x) = x^3 - x - 1. If we use Newton-Raphson method with an initial guess x_0 = 1, the next approximation x_1 will be: — 1.25
4. For P(x) = x^3 - x - 1, P'(x) = 3x^2 - 1. If x_n = 1.3, then f(x_n) approx 0.097 and f'(x_n) approx 4.07. The next approximation using Newton's method is closest to: — 1.276
5. The Bisection Method guarantees convergence if: — The function has opposite signs at the interval endpoints
6. The convergence of Jacobi and Gauss-Seidel methods depends on the properties of the coefficient matrix, such as: — Being diagonally dominant
7. Which method involves approximating the function with a secant line? — Secant Method
8. Which method is essentially Newton-Raphson but approximates the derivative using a finite difference? — Modified Newton-Raphson Method
9. Which of the following is NOT a bracketing method for finding roots? — Newton-Raphson Method
10. If an iterative method requires the computation of the second derivative of the polynomial, it is likely: — None of the above
11. If the derivative of the function f(x) is zero at an approximation x_n, the Newton-Raphson method: — Fails to compute the next iteration
12. Which of the following is NOT an advantage of Horner's method for polynomial evaluation? — Simultaneous calculation of derivative values
13. The initial interval [a, b] for the Bisection Method must satisfy: — f(a) and f(b) have opposite signs
14. The Newton-Raphson method requires the initial guess to be: — Reasonably close to the actual root
15. If a root has multiplicity greater than 1, the convergence of Newton-Raphson method becomes: — Linear
16. The Bisection Method is generally considered: — More robust but slower than Newton-Raphson
17. Horner's Method is primarily used for: — Evaluating a polynomial and its derivatives efficiently
18. The Secant Method can be viewed as a '<bos>-order' approximation of Newton's method where the derivative is approximated by the slope of the secant line. — first
19. The concept of 'order of convergence' refers to: — How rapidly the error decreases with each iteration
20. How does the False Position Method differ from the Secant Method? — It always keeps one of the original endpoints
21. Which iterative method for systems of linear equations updates variables using the most recently computed values? — Gauss-Seidel Method
22. The Successive Over-Relaxation (SOR) method is an extension of which method, introducing a relaxation factor? — Gauss-Seidel Method
23. The convergence rate of the Bisection Method is: — Linear
24. The convergence rate of the Secant Method is approximately: — Superlinear (order approx. 1.618)
25. Which method is generally the slowest among the common iterative methods? — Bisection Method
26. When dealing with polynomials that might have multiple roots or roots close to each other, which method is often preferred for its robustness? — Bisection Method
27. Which approximation method is based on finding a polynomial that interpolates the function at specific points? — Lagrange Interpolation
28. Which method is guaranteed to find a root if f(x) is continuous and f(a) and f(b) have opposite signs, regardless of the function's differentiability? — Bisection Method
29. Which method is generally preferred when the derivative of the function is difficult or impossible to compute? — Secant Method
30. Which method is known for its slow convergence but high reliability when a root is bracketed? — Bisection Method
31. The Jacobi and Gauss-Seidel methods are iterative techniques typically used for solving: — Systems of linear equations
32. The Secant Method uses how many initial points to start the iteration? — Two
33. For finding roots of polynomials, the Durand-Kerner method (also known as the Weierstrass method) aims to find: — All roots simultaneously
34. A 'simple root' of a polynomial P(x) is a root 'r' such that: — P(r) = 0 and P'(r) != 0
35. Which of the following is an iterative method for finding the roots of a polynomial equation? — Newton-Raphson method
36. The condition f(a) * f(b) < 0 ensures: — The existence of at least one root in (a, b)
37. If the polynomial has a root with even multiplicity, Newton's method will converge: — Linearly
38. Bairstow's method is an extension of Newton's method used to find: — Complex roots (in pairs)
39. In the context of polynomial root approximation, 'deflation' refers to: — Reducing the degree of the polynomial after finding a root
40. A potential issue with the False Position Method is: — Slow convergence if one endpoint is poorly chosen
41. In the Bisection Method, the new interval is formed by: — Replacing one endpoint with the midpoint
42. A major limitation of the Newton-Raphson method is its sensitivity to: — The initial guess
43. Steffensen's Method achieves quadratic convergence without explicitly calculating: — The derivative
44. The convergence criterion for most iterative root-finding methods is typically based on: — Both the function value and the difference between approximations
45. The choice of an appropriate approximation method depends on: — The desired accuracy and initial information available
46. The False Position Method (Regula Falsi) is similar to the Bisection Method in that it: — Guarantees convergence if initial signs are opposite
47. The formula for the Secant Method iteration is: — x_{n+1} = x_n - f(x_n) * (x_n - x_{n-1}) / (f(x_n) - f(x_{n-1}))
48. What is the formula for the Newton-Raphson iteration? — x_{n+1} = x_n - f(x_n) / f'(x_n)
All One Line Questions
E-Book Approximation Methods for Polynomial Roots Online Test Approximation Methods for Polynomial Roots Question Bank Approximation Methods for Polynomial Roots