Solving Equations by Numerical Methods
Introduction
In this coursework I am going to investigate the three different numerical methods used to solve equations. These include the change of sign method, Newton Raphson method and the Rearranging method. To carry out the investigation I will explain how each method works, using an example of a working equation in each case. I will also show when each of the methods will not work with other equations. I will then compare all three of the methods.
Change of sign method
Decimal search and Interval bisection are both ways of finding an interval where there is a change of sign. The change of sign can be found on a graph when the line crosses the x-axis. Wherever there is a change of sign there will be a root. These two methods find the interval where the root lies.
Decimal Search
To find the roots using decimal search, the y-values must be found, using the values of X from 0.1, 0.2, 0.3, all the way up to 1. When a change of sign occurs, this means there is a root lying here. Once the root interval to 1 decimal place has been found, it must then be found to 2 and 3 decimal places and so on to the required number of places.
Y=x3-5x+1 can be solved using decimal search.
The table shows that the routes can be found.
X
Y
X
Y
X
Y
0
2
-1
-2
3
0.1
0.501
2.1
-0.239
-2.1
2.239
0.2
0.008
2.2
0.648
-2.2
.352
0.3
-0.473
2.11
-0.15607
-2.3
0.333
0.21
-0.04074
2.12
-0.07187
-2.4
-0.824
0.201
0.003121
2.13
0.013597
-2.31
0.223609
0.202
-0.00176
2.121
-0.06338
-2.32
0.112832
0.2011
0.002633
2.122
-0.05488
-2.33
0.000663
0.2012
0.002145
2.123
-0.04637
-2.34
-0.1129
0.2013
0.001657
2.124
-0.03784
-2.331
-0.01063
0.2014
0.001169
2.125
-0.0293
-2.3301
-0.00047
0.2015
0.000681
2.126
-0.02074
-2.33001
0.00055
0.2016
0.000194
2.127
-0.01218
Root= -2.33005
0.2017
-0.00029
2.128
-0.0036
Error= +/- 0.00005
0.20161
0.000145
2.129
0.004993
0.20162
9.6E-05
2.1281
-0.00274
0.20163
4.72E-05
2.1282
-0.00188
0.20164
-1.6E-06
2.1283
-0.00102
Root= 0.201635
2.1284
-0.00016
Error= +/- 0.000005
2.1285
0.000695
Root= 2.12845
Error= +/-0.00005
2x4+x3+3x2 cannot be solved using decimal search.
X
Y
X
Y
0
0
-0.1
0.0292
0.1
0.0312
-0.2
0.1152
0.2
0.1312
-0.3
0.2592
0.3
0.3132
-0.4
0.4672
0.4
0.5952
-0.5
0.75
0.5
-0.6
.1232
0.6
.5552
-0.7
.6072
0.7
2.2932
-0.8
2.2272
0.8
3.2512
0.9
4.4712
0.9
4.4712
-1
4
6
The graph shows below that the line only touches the x-axis and so there is no change of sign. This means the decimal search method cannot be used to find the route of the equation.
Introduction
In this coursework I am going to investigate the three different numerical methods used to solve equations. These include the change of sign method, Newton Raphson method and the Rearranging method. To carry out the investigation I will explain how each method works, using an example of a working equation in each case. I will also show when each of the methods will not work with other equations. I will then compare all three of the methods.
Change of sign method
Decimal search and Interval bisection are both ways of finding an interval where there is a change of sign. The change of sign can be found on a graph when the line crosses the x-axis. Wherever there is a change of sign there will be a root. These two methods find the interval where the root lies.
Decimal Search
To find the roots using decimal search, the y-values must be found, using the values of X from 0.1, 0.2, 0.3, all the way up to 1. When a change of sign occurs, this means there is a root lying here. Once the root interval to 1 decimal place has been found, it must then be found to 2 and 3 decimal places and so on to the required number of places.
Y=x3-5x+1 can be solved using decimal search.
The table shows that the routes can be found.
X
Y
X
Y
X
Y
0
2
-1
-2
3
0.1
0.501
2.1
-0.239
-2.1
2.239
0.2
0.008
2.2
0.648
-2.2
.352
0.3
-0.473
2.11
-0.15607
-2.3
0.333
0.21
-0.04074
2.12
-0.07187
-2.4
-0.824
0.201
0.003121
2.13
0.013597
-2.31
0.223609
0.202
-0.00176
2.121
-0.06338
-2.32
0.112832
0.2011
0.002633
2.122
-0.05488
-2.33
0.000663
0.2012
0.002145
2.123
-0.04637
-2.34
-0.1129
0.2013
0.001657
2.124
-0.03784
-2.331
-0.01063
0.2014
0.001169
2.125
-0.0293
-2.3301
-0.00047
0.2015
0.000681
2.126
-0.02074
-2.33001
0.00055
0.2016
0.000194
2.127
-0.01218
Root= -2.33005
0.2017
-0.00029
2.128
-0.0036
Error= +/- 0.00005
0.20161
0.000145
2.129
0.004993
0.20162
9.6E-05
2.1281
-0.00274
0.20163
4.72E-05
2.1282
-0.00188
0.20164
-1.6E-06
2.1283
-0.00102
Root= 0.201635
2.1284
-0.00016
Error= +/- 0.000005
2.1285
0.000695
Root= 2.12845
Error= +/-0.00005
2x4+x3+3x2 cannot be solved using decimal search.
X
Y
X
Y
0
0
-0.1
0.0292
0.1
0.0312
-0.2
0.1152
0.2
0.1312
-0.3
0.2592
0.3
0.3132
-0.4
0.4672
0.4
0.5952
-0.5
0.75
0.5
-0.6
.1232
0.6
.5552
-0.7
.6072
0.7
2.2932
-0.8
2.2272
0.8
3.2512
0.9
4.4712
0.9
4.4712
-1
4
6
The graph shows below that the line only touches the x-axis and so there is no change of sign. This means the decimal search method cannot be used to find the route of the equation.