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# The decimal search

Extracts from this document...

Introduction

The Decimal Search method

I would like to solve the equation x³ + 2x² − 0.5 = 0. Let f(x) = x³ + 2x² − 0.5

The graph below shows the equation y = x³ + 2x² − 0.5. As you can see, there are three roots/solutions when f(x) = 0.

The equation x³ + 2x² − 0.5 = 0 cannot be solved algebraically so I will use the decimal search method.

This method looks for a change in sign of the value of f(x). The following table gives values of x and f(x).

 x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 f(x) -0.5 0.625 0.5 -0.125 -0.5 0.125 2.5 7.375 15.5

As the curve crosses the x axis, the value of the function f(x) changes sign, for example:

Middle

0.1225

0.1480

The interval is reduced to 0.001, the root is between [-1.854,-1.855].

 x -1.86 -1.859 -1.858 -1.857 -1.856 -1.855 -1.854 -1.853 -1.852 -1.851 -1.85 f(x) -0.0157 -0.0127 -0.0098 -0.0069 -0.004 -0.0011 0.0018 0.0047 0.0076 0.0105 0.0134

The interval is reduced to 0.0001, the root is between [-1.8546,-1.8547].

 x -1.855 -1.8549 -1.8548 -1.8547 -1.8546 -1.8545 -1.8544 -1.8543 -1.8542 -1.8541 -1.854 f(x) -0.0011 -0.00076 -0.00047 -0.00018 0.00011 0.0004 0.00069 0.00098 0.0013 0.0016 0.0018

Conclusion

- 1.854640

-1.854639

-1.854638

-1.854637

-1.854636

-1.854635

-1.854634

-1.854633

-1.854632

-1.854631

-1.854630

f(x)

-6.7E-06

-3.8E-06

-9.3E-07

2.0E-06

4.9E-06

7.8E-06

1.1E-05

1.4E-05

1.6E-05

1.9E-05

2.2E-05

In error bounds x= 1.8546375 ± 0.0000005.

Where decimal search may fail

Let us solve the equation y = 4x³ + 5.84x² − 4.7104x − 7.20896.

f (x) = 4x³ + 5.84x² − 4.7104x − 7.20896

The following table shows the values of f(x) where is x between [-3, 3].

 x -3 -2 -1 0 1 2 3 f(x) -48.5178 -6.42816 -0.65856 -7.20896 -2.07936 38.7302 139.22

The table confirms there is a root between [1, 2], as you can see from the graph of y = 4x³ + 5.84x² − 4.7104x − 7.20896 below, there is a root between [1, 2], but also a repeated root between [-1, -2]. The decimal search method fails to identify the repeated root between [-1, -2].

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