C3 Numerical Solutions to Equations

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Adam Arstall

Numerical Solutions to Equations Coursework

Change of sign method

The change of sign method involves finding the interval in which a root of an equation lies by taking two values of x and showing that the root lies between them as the value of f(x) for each case has a different sign. A change of sign will always indicate a root if the function is continuous.

This method will be used to find a root of the equation f(x)=x⁵+x⁴−2x³+5x²−7x−2=0. As the function f(x) is continuous, a change of sign will always indicate a root.

The method will be used to find the root which lies between -2 and -3

As the root lies in the interval [-2.7211575,-2.7211574], x=-2.72115745 ± 0.000000005

Here it is shown that f(x) changes from negative to positive between -2.7211575 and-2.7211574 and the root to f(x)=0 is between these x values.

This method can fail to find roots in some cases. For example if the equation has a repeated root as shown below for the equation x³−0.96x²−5.6471x+7.48225=0.

As the root between 1 and 2 is repeated, the line of the graph never crosses the x axis so the function does not ever take a negative value between 1 and 2. As a result of this no change of sign will be found using the method above and so the root will not be found.

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Newton-Raphson Method

The Newton-Raphson method takes a value of x close to the root which is to be found and then uses and iterative formula to generate a value which is closer to the root. This process is then repeated on the new x values until they converge on the root to the required level of accuracy. Taking x0 as the first guess at the root, the tangent to the curve at (x0,f(x0)) crosses the x axis at x1, the second guess. This gives the iterative formula xn+1=xn-f(xn)/f'(xn)

This method will be used to find the roots of ...

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