Newtons method is an example of a fixed-point iteration scheme. Explain. What co

Question

Newtons method is an example of a fixed-point iteration scheme. Explain. What condition ensures that the bisection method will find a zero of a continuous nonlinear function f in the interval [a, b]? What is meant by quadratic convergence rate for an iterative method? What is meant by fixed point of a function f? In using the secant method for solving a one dimensional nonlinear equation, how many starting guesses for the solution are required? Given the nonlinear equation f(x) = 0, can you determine an equivalent fixed-point problem. i.e. a function g(x) such that a fixed point x of g is a solution to f(x) = 0? List one advantage and one disadvantage of the secant method compared with the bisection method for finding a simple zero of a nonlinear equation. List one advantage and one disadvantage of the secant method compared with Newton's method for finding a simple zero of a nonlinear equation. True or false: there are arbitrarily many different mathematical functions that interpolate a given set of data points. What is the basic difference between interpolation and approximation of a function? It is ever possible for two distinct polynomials to interpolate the same data points? If so under what conditions, and if not, why? Is interpolation an appropriate procedure for fitting a function to noisy data? Explain. What is a good alternative? Why is interpolation by a polynomial of high degree often unsatisfactory? In fitting a large number of data points, what is the main advantage of piecewise polynomial interpolation over interpolation by a single polynomial? What is precisely a cubic spline interpolant? How many times is a cubic spline interpolant differentiable? The continuity and smoothness requirements on a cubic spline interpolant still leave two

Explanation / Answer

Solution :- (1)Fixed point iteration method and a particular case of this method called Newton's method.
If f is continuous and (xn) converges to some 0 then
it is clear that 0 is a fixed point of g and hence it is a solution of the equation.

(2)

Given a closed interval [a,b] on which f changes sign,
we divide the interval in half and note that f must change sign on either the right
or the left half (or be zero at the midpoint of [a,b].) We then replace [a,b]
by the half-interval on which f changes sign.
This process is repeated until the interval has total length less than .
In the end we have a closed interval of length less than on which f changes sign.
The IVT guarantees that there is a zero of f in this interval.
The endpoints of this interval, which are known, must be within of this zero.

Initialization: The bisection method is initialized by specifying the function f(x),
the interval [a,b], and the tolerance > 0.

We also check whether f(a) = 0 or f(b) = 0, and if so return the value of a or b and exit.

Loop: Let m = (a + b)/2 be the midpoint of the interval [a,b].
Compute the signs of f(a), f(m), and f(b).
If any are zero, return the corresponding point and exit.
Assuming none are zero, if f(a) and f(m) have opposite sides, replace b by m,
else replace a by m.
If the length of the [a,b] is less than , return the value of a and exit.

(3)  The speed at which a convergent sequence approaches its limit is called the rate of convergence.

4) In mathematics, a fixed point of a function is an element of the function's
domain that is mapped to itself by the function. That is to say,
c is a fixed point of the function f(x) if and only if f(c) = c.

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