C3 Mei - Numerical Methods to solve equations

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Prital Upadhayay

C3 Coursework

In this coursework, I will use numerical methods to solve the following equation, as I cannot solve it algebraically. I can only obtain an approximation of the solution as it is impossible or hard to find the exact value of the function.

Decimal Search

The graph below is the function I will use decimal search in order to find an approximation of one of the roots.

The table below shows decimal search. Each boundary is tested for sign change which indicates that a root exists between them. The x where the sign change occurs in now the new boundaries and tested for sign change again.

This method is repeated until an approximation of the root is found to a suitable number of decimal places.

        [0, 1]                   [0.6, 0.7]                      [0.66, 0.67]                 [0.662, 0.663]

Root intervals [0.6621, 0.6622]                                          
I know that the root is 0.662 to 3 decimal places


In order to test the boundaries, I put them back into the equation:

(0.6621)4 – 5(0.6621)2 +2 = 0.00292

(0.6622)4 – 5(0.6622)2 + 2 = -0.0025

As the values have different signs, it further indicates that the root is close to 0.662 to 3 decimal places.

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The decimal search will generally fail when two roots are close together or when there is a double root present.

The function   has a repeated root; therefore the decimal search will fail to find the root.

The two diagrams above show that there is a root present, however, as there is a double root, decimal search has failed.

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