Chapter 3 (Numerical Solution Of Equations Of A Single Variable)

Question 1

Describe bisection method

Answer 1

The sinplest bracketing method to find a root. If f(a) and f(b) have opposite signs on the interval [a,b] we locate the midpoint c1 such that it is set as the new endpoint of opposite sign; i.e.[a,c1] or [c1,b]. The interval with the root is kept and the process is repeated.

Question 2

Name both bracketing methods

Answer 2

Bisection and Regula Falsi

Question 3

Name all three open methods

Answer 3

Fixed point, Newton’s, and secant

Question 4

What MATLAB function solves the numerical soln. of f(x)=0

Answer 4

fzero

Question 5

What are the three general “methods” are used to solve the numerical soln. of f(x)=0

Answer 5

Bracketing, open, MATLAB fzero function

Question 6

Describe the Regula Falsi method

Answer 6

False point. Bracketing method. On the interval [a,b] = [a1,b1] a straight line is drawn between A and B which are defined as A:(a1,f(a1)) and B:(b1,f(b1)). C1 is found where that line intersects the x-axis. C1 is then set as the new interval end; i.e. [a1, c1 = b2] or [c1 = a2, b1]. The next point on the line is set as c1: (c1, f(c1)) and the process is repeated.

Question 7

Describe fixed-point method

Answer 7

Open method. Iteration function. The function f(x)=0 is rewritten as g(x)=x. The “fixed-point” is found where y=g(x) and y=x intersect. This is also the root of f(x)=0. The process begins by guessing an x1 which is then plugged into y=g(x) that y value is used to set x2 via y=x. The process is then repeated until the tolerance is met. Convergence is quadratic

Question 8

Describe Newton’s Method (Newton-Raphson)

Answer 8

Most common method of solving f(x)=0. Find an initial point x1 and find the point (x1,f(x1)) on the curve. Take the tangent of that point and find its x intercept. That point becomes x2. Repeat the process. Sequence may not converge if x1 isnt close to the root. If f’(x) is too difficult to compute, the secant method may be used.

Question 9

Describe Secant Method

Answer 9

Starts with two initial points x1 and x2, find ghe points (x1,f(x1)) and (x2,f(x2)) then draw a secant line connecting them. The x intercept becomes x3. Repeat this process with x2 and x3, x3 and x4, and so on. Not guaranteed to convege. If it does converge its faster than linear but slower than quadratic i.e. Newtons.