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The decimal search

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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.

image11.png

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

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

f(x)

-0.5

0.625

0.5

-0.125

-0.5

0.125

2.5

7.375

15.5

image00.png

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

image01.png

...read more.

Middle

0.1225

0.1480

image13.png

image05.png

image06.png

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

x

-1.860

-1.859

-1.858

-1.857

-1.856

-1.855

-1.854

-1.853

-1.852

-1.851

-1.850

f(x)

-0.0157

-0.0127

-0.0098

-0.0069

-0.0040

-0.0011

0.0018

0.0047

0.0076

0.0105

0.0134

image07.png

image14.png


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

x

-1.8550

-1.8549

-1.8548

-1.8547

-1.8546

-1.8545

-1.8544

-1.8543

-1.8542

-1.8541

-1.8540

f(x)

-1.1E-03

-7.6E-04

-4.7E-04

-1.8E-04

1.1E-04

4.0E-04

6.9E-04

9.8E-04

1.3E-03

1.6E-03

1.8E-03

image08.png

image15.png

...read more.

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

image02.png

image17.png

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)

-48.51776

-6.42816

-0.65856

-7.20896

-2.07936

38.73024

139.2198

image03.png

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].

image18.png

...read more.

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