numerical methods

Question 1

use of numerical methods

Answer 1

• often it is difficult or even impossible to find the exact value of a root. When this happens can use a numerical method to estimate its value.

Question 2

change of sign method

Answer 2

Simplest method for detecting a root is the change of sign method. If two real numbers c and d are such that f(c) and f(d) have opposite signs, you say f(x) changes sign between x=c and x=d
if f(x) is CONTINUOUS and changes sign between x=c and x=d then equation f(x)=0 has a root a, where c <a <d
Can also calculate root on calculator using table function and finding out to the required decimal places, when the answer changes sign
This is possible because y- coordinates change sign either side of the root

Question 3

what do you do when asked to prove that a number is a root to a certain amount of decimal places?

Answer 3

Find upper and lower bound, based on given decimal places.
then plug in both upper and lower bound to show a change in sign

Question 4

continuous function

Answer 4

A continuous function doesnt change sign in an interval which contains an even number of roots (counting repetitions) of the equation f(x)=0

Question 5

iterative root finding

Answer 5

a sequence x1, x2, x3 ….. converges to a if, as n increases, xn, gets ever closer to a

Question 6

divergent sequences

Answer 6

sequences are describes as divergent, when values get further away and reach infinity

Question 7

staircase or cobweb diagrams

Answer 7

can be used to display iterates given by xn+1= g(xn)
to do this, draw the curve C with equation y=g(x) and the straight line L with the equation y=x
Label point of intersection P and mark x- coordinate of P as a. This is solution to the equation x= g(x)
Start at point (x1,0) given that x1 is your initial guess for the root of the equation. Draw a vertical line until you reach the curve C, then draw a horizontal line until you reach the line L. Repeat this process for the required number of iterations
As the points c1, c2, c3 … on the curve C converge to the point P1. the sequence x1, x2, x3 ….. converges to the solution a. This is an example of a staircase diagram

Question 8

using gradient of the curve to determine if xn+1 = g(xn) converges to the solution a

Answer 8

if -1< g’(x) <1 for all x in an interval which contains a and the starting value x1 then xn+1 = g(xn) converges

Question 9

newton- raphson root finding

Answer 9

if a is a root of the equation f(x)=- then iterative sequence given by xn+1= xn- f(xn)
———
f’(xn)
converges to a, if it converges
When using this method, you must choose a suitable value for the first approximation. The method may not converge to the required root if the first approximation is close to the x- coordinate of a stationary point on the curve y= f(x). The newton- raphson method will also fail to converge if the derivative is zero at one of the values for xn

Question 10

numerical integration

Answer 10

often when working out area under the curve the integral cant be found so cant calculate the exact value of the definite integral
You can use trapezium rule to find an approximation
This involves splitting the area under the curve into three trapeziums of equal width.
Each trapezium has width h and their vertical sides are the y values
- increasing the number of intervals improves the accuracy of the estimate. This is because increasing the number of intervals decreases the width of each trapezium. The thinner each trapezium, the more accurately they approximate to the curve

Question 11

is answer an overestimate or an underestimate

Answer 11

always answer in terms of concave or convex
if concave answer will be underestimate and if convex answer will be overestimate