Name: Victoria Fung
C3
Numerical Methods
OCR Advanced GCE Unit 4753/02
Methods for Advanced Mathematics Coursework
I’m going to investigate the solution of equations using three different numerical
methods. Numerical methods are used for solving equations.
The first one is a systematic search for the change of sign using a decimal search.
The second one is using the Newton Raphson method and finally the third method
is using the rearranging the formula so instead of having f(x)=0, I’d rearrange it into
the form of x=g(x).
x³–9x+5=0
Page 1 of 7
Name: Victoria Fung
C3
Numerical Methods
Change of sign method:
Decimal Search:
The root lies within unit interval (-4,-3)
F(-4)= -4³–(9x-4)+5=-23
F(-3)= -3³–(9x-3)+5=5
Change in sign means………
x
F(x)
-4
-23
-3.9
-19.22
-3.8
-15.67
-3.7
-12.35
-3.6
-9.256
-3.5
-6.375
-3.4
-3.704
-3.3
-1.237
-3.2
1.032
So the root lies within the interval (-3.3,-3.2) (1dp)
X
-3.29
-3.28
-3.27
-3.26
-3.25
-3.24
-3.23
-3.22
Page 2 of 7
-3.21
Name: Victoria Fung
C3
Numerical Methods
F(x)
-1.00129
-0.76755
-0.53578
-0.30598
-0.07813
0.147776
0.371733
0.593752
0.813839
-3.245
-3.244
-3.243
-3.242
0.08021
0.102752
So the root lies within the interval (-3.25, -3.24) (2dp)
X
-3.25
-3.249
-3.248
-3.247
-3.246
F(x)
-0.07813
-0.05545
-0.03279
-0.01015
0.012469
0.035069 0.057649
The roots lie between the interval -3.247 and -3.246 (3dp)
x
F(x)
-3.247
-0.01015
-3.2469
-0.00789
-3.2468
-0.00562
-3.2467
-0.00336
-3.2466
-0.0011
-3.2465
0.001162
-3.2464
-3.2463
-3.2462
-3.2461
-3.246
0.003424 0.005685 0.007947 0.010208 0.012469
X=-3.247 (3dp)
or
x= 3.246 55
With maximum error bound ±0.000 05
But a disadvantage of using the decimal search is that ………………………
Page 3 of 7
Name: Victoria Fung
C3
Numerical Methods
Newton-Raphson method:
x4+x-9=0
The roots
f(x)=x4+x-9
f’(x)=4x3+1
Page 4 of 7
Name: Victoria Fung
C3
Numerical Methods
xr+1=xr - f(xr)
f’(xr)
xr+1=xr - x4+x-9
4x3+1
xn
f(xn)
f'(xn)
x n+1
1
2.4
1.927895
1.700591
1.649107
1.646727
1.646722
1.646722
-7
26.5776
6.74235427
1.064316798
0.045073275
9.23294E-05
3.89861E-10
0
5
56.296
29.6622579
20.6725121
18.9393292
18.8617748
18.8616155
18.8616155
2.4
1.927895
1.700591
1.649107
1.646727
1.646722
1.646722
1.646722
Page 5 of 7
Name: Victoria Fung
C3
Numerical Methods
Page 6 of 7
Name: Victoria Fung
C3
Numerical Methods
x1
x2
x3
x4
x5
x6
2
1.727273
1.651931
1.646745
1.646722
1.646722
Page 7 of 7