MEI Numerical Methods - Methods you need to know

A brief explanation of all the methods you need to know for OCR-MEI NM exam. No graphical interpretations but shart step by step guidelines to help with numerical methods.

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  • Created by: DanArthur
  • Created on: 14-05-13 18:31
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OCRMEI Numerical Methods
Methods You Need to Know
The Solution of Equations
1) The Bisection Method:
For a root in the interval [a, b] , find the midpoint c = a+b
2 . If c has the same sign as a , then let
a = c . If c has the same sign as b , then let b = c . Keep repeating this process until you have
reached the desired level of accuracy. c is the best estimate for the root.
a b Sign of f (a) Sign of f (b) c Sign of f (c)
2) The Method of False Position:
For a root in the interval [a, b] , find a variable c such that c = af f(b )-cf (a)
(b)-f (a) . Likewise with the
Bisection method, if c has the same sign as a , then let a = c . If c has the same sign as b ,
then let b = c . Keep repeating this process until you have reached the desired level of
accuracy. c is the best estimate for the root.
a b Sign of f (a) Sign of f (b) c Sign of f (c)
3) Fixed Point Iteration
Rearrange f (x) so that x = g(x) where g(x) is another function that contains x . From this, you
can create an iterative formula to put into your calculator.
For example:
f (x) = x2 - sin x - 1 has a root in the interval [1, 2] . Use a fixed point iteration to find a root to 4
decimal places:
f (x) = x2 - sin x - 1
x = 1 + sin x
The iterative formula is:
where x0 = 1+2
xr+1 = 1 + sin xr 3
2 = 2 (from interval)
Therefore, you can type 3
2 on your calculator and press = . Then type:

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OCRMEI Numerical Methods
Methods You Need to Know
1 + sin ANS and then press = . This will give a value for x1.
The Solution of Equations
x1 1
x2 1.357008101
x3 1.406141602
x4 1.409423644
x5 1.409612592
x6 1.409623354
4) The NewtonRaphson Method
The NewtonRaphson Method is an iterative formula to estimate the root of an equation.
The equation is as follows:
xr+1 = xr - ff((x
xr)
r)
where x0 = a point near the root (i.e.…read more

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OCRMEI Numerical Methods
Methods You Need to Know
Numerical Integration
1) The Midpoint Rule
The Midpoint Rule is a way of estimating the area under a curve (integration) and is found by
splitting the area underneath the curve into equals strips of width h . Here is the general
equation for the midpoint rule:
x0+x1 x1+x2 x2+x3 xn-1+xn
M n = h[f ( 2 ) + f ( 2 ) + f ( 2 ) + ...…read more

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OCRMEI Numerical Methods
Methods You Need to Know
Numerical Integration
There are also some other formulae that you should learn for the exam:
T 2n = T n+
2
Mn
4T 2n-T n
Sn = 3
4M 2n-M n
"Best Midpoint" = 3
4T 2n-T n
"Best Trapezium" = 3
16S2n-Sn
"Best Simpson's" = 15
These are the only methods you need to know for numerical integration.…read more

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OCRMEI Numerical Methods
Methods You Need to Know
Numerical Differentiation
1) The Forward Difference Method
This method of numerical differentiation finds the gradient of the tangent to the curve between
the points x and x + h . This is not the most reliable method but is very simple and is found by:
f (x)f (x+h)-
h
f (x)
2) The Central Difference Method
This method is slightly more reliable as the centre of the chord you are finding the gradient of
is x .…read more

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OCRMEI Numerical Methods
Methods You Need to Know
Approximating Functions
Finite Difference Tables:
This is where you find the difference between all of your f (x) values as this determines what
order you function is of.…read more

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OCRMEI Numerical Methods
Methods You Need to Know
NOTE: This formula is given in the exam under the name: "Newton's Forward Difference
Formula".…read more

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