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Solving the equation of 0 = 3x^5 - 3x + 1 using different methods

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Introduction

Solving the equation of   0 = 3x^5 - 3x + 1 using different methods. This is to compare all three methods to find out which is best. To be fair, the same root will be calculated by all three methods.

Method:Decimal Search/Change of sign

        The equation, 0 = 3x^5 – 3x + 1, will be solved using the method Decimal Search/Change of Sign.

image00.png

I have

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Middle

x

y

0.33

0.021740618

0.331

0.018919587

0.332

0.016100733

0.333

0.013284074

0.334

0.010469631

0.335

0.007657423

0.336

0.004847471

0.337

0.002039795

0.338

-0.000765585

0.339

-0.003568648

0.34

-0.006369373

x

y

0.3377

7.57868E-05

0.33771

4.77377E-05

0.33772

1.96889E-05

0.33773

-8.35975E-06

0.33774

-3.64081E-05

0.33775

-6.44563E-05

0.33776

-9.25042E-05

0.33777

-0.000120552

0.33778

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Conclusion

The Rearrangement method would, in my opinion, be the quickest. This is because the equation just has to be rearranged, and then the numbers substituted into the equation. It also has only a few calculations (5), normally slightly more than the Newton Raphson method.

Overall, I think, without the aid of a computer, the Rearrangement method would be the quickest and most efficient way of calculating the value of the function. The Newton Raphson method would be second as there are less calculations to do, even though the method is harder. Decimal Search/Change of Sign would come last as there are too many calculations to do.

However, if a computer was available, any method would be suitable, if the correct software was available. This is because they all take roughly the same amount of time.

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