A numerical method is used to find an approximate solution to an initial value p

Question

A numerical method is used to find an approximate solution to an initial value problem

using various different step-sizes. The following estimates for r(1) are obtained: number of steps approximation of x(1 6.229247941 6.231035466 6.231171834 6.231181291 6.231181913 2 16 32 (a) Estimate r(1) accurate to 6 decimal places. Explain how you got your answer. (b) Estimate the errors in the approximation of r(1) using 4 and 8 steps (c) Calculate the effective order of the method at stepsize h 0.125. (d) What numerical method do you think might have been used? Give a reason for your answer.

Explanation / Answer

a. x(1)=6.231181 (correct upto 6 decimal places)

For checking this you have to check that whatever value you will get in last step for how many decimal places that value is same with your previous value.suppose in the given problem in 1st step you got 6.208333333,in 2nd step you got 6.229247941 that means that value is stable for 1 decimal place.In next step you have to check for that value is stable for 2 decimal places and so on.

b.In 4th step,x(1)=6.231035466, in 8th step, x(1)=6.231171834

Error= value of x(1) in 8th step - value of x(1) in 4th step

=6.231171834-6.231035466=0.000136368

c. Step size h=0.125=125/1000=1/8

Therefore, number of steps n=1/h=8

In 8th step, x(1)=6.231171834 this value is same with the previous step till 3 decimal places.So,effective order will be 3.

d. Here ,Euler's method is used for solving the given initial value problem.

We used Euler's method only when we are estimating one numerical value using different step sizes.Here also we are doing same thing.

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