Numerical Methods Coursework


OCR MEI GCE Unit 4753/02 Methods for Advanced Mathematics




OCR MEI GCE Unit 4753/02 Methods for Advanced Mathematics



The solution of equations are commonly found analytically, using a predetermined formula such as thequadratic formula or by algebraic manipulation, with the roots of a cubic, quartic or quintic polynomialequation being denoted by letters of the Greek alphabet. However with far more complex equations,analytical methods prove to be inadequate and this is where numerical methods become of value.Numerical methods are methods of finding numerical approximations of solutions. Numericalmethods should be used where analytical methods are unavailable.

I will use three distinct examples of numerical methodsi.

Systematic search for change of sign, using decimal searchii.

Fixed point iteration using the Newton-Raphson methodiii.

Fixed point iteration after rearranging the equation f (x) = 0 into the form x = g (x)I will use these three different methods to find the roots of complex equations, in order to show howeach deem successful. I will also show instances where each method fails comparing it numerically andgraphically. After using these methods, I will then compare the method by applying them on the sameequation to show which is faster and more reliable.

Numerical Methods| Change of sign (Decimal Search)

The decimal search method works by giving values for x for which values of f (x) change. When thevalues of 

f(x) =



7x+5 change from a positive to a negative or from a positive to a negative, thisindicates that the interval contains a root. Using the decimal search method I am going to find theroots of the equation x


7x+5 = 0 where f(x)=0

As you can see from the table, the highlighted cells indicate that the equation x


7x+5 = 0 has rootsbetween the intervals [-2,-1],[0,1] and [1,2]x


Хоть и не очень охотно, она все же улыбнулась: - Что будем делать. Стратмор просиял и, протянув руку, коснулся ее плеча. - Спасибо.

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