Nonlinear Equation True and False Nonlinear Equations Consid

Nonlinear Equation, True and False:

Nonlinear Equations Consider the following statements about nonlinear equation solving. Mark each statement as true or false. T/F A small residual ||f(x)|| guarantees an accurate solution of a system of nonlinear equations f(x) = 0. T/F Newton\'s method is an example of a fixed point iteration. T/F If an iterative method for solving a nonlinear equation gains more than one bit of accuracy per iteration, then it is said to have a super linear convergence rate. T/F Newton\'s method always converges quadratic ally. T/F A fixed point of a function f(x) is a point x* such that f(x*) = 0.

Solution

Answers.

1. False. A small residual applies for condition numbers as f(x) = y but do not applies for f(x) = 0, due to solutions are an approximation of a (uknown) real value.

2. True. Related for finding roots of a given differentiable function and f(x)/f ´(x) = 0

3. True. In example, a quadratically convergent method doubles the number of correct digits with each iteration

4. True. The method will usually converge quadratically, provided this initial guess is close enough to the unknown zero, and that ƒ\'(x₀) 0. However, there are some difficulties with the method. If the method fails to converge it is because the assumptions made in this proof are not met.

5. False. For given equation f (x) = 0, there may be many equivalent fixed-point problems x = g(x) with different choices for g.