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In: Advanced Math

Numerical Analysis: Perform the steps of the secant method to approximate the first three positive roots...

Numerical Analysis: Perform the steps of the secant method to approximate the first three positive roots of the transcendent equation x = sin (πx).

Solutions

Expert Solution

Iteration scheme under secant method is:

Starting with for which (so we definitely have a positive root here) we get the following table of iterates:

n x_n f(x_n)
-1 2.000000000 2.000000000
0 0.500000000 -0.500000000
1 0.800000000 0.212214748
2 0.710610635 -0.078367145
3 0.734718088 -0.005508773
4 0.736540836 0.000176232
5 0.736484332 -0.000000363

Note that this equation has exactly one positive real root as can be seen from the graph below:

So the only positive root is (approximately)

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