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RRoooottss ooff aa NNoonnlliinneeaarr 
EEqquuaattiioonn 
NNeewwttoonn--RRaapphhssoonn 
MMeetthhoodd 
1
NNeewwttoonn--RRaapphhssoonn MMeetthhoodd 
x = x - f(x ) 
i 
f (x ) 
2 
i 
i+1 i ¢ 
f(x) 
f(xi) 
f(xi-1) 
q   
[ ( ) ] i i x f x , 
xi+2 xi+1 xi X
DDeerriivvaattiioonn 
tan(a ) = AB 
AC 
f x f x 
i i x x 
+ - 
1 
'( ) ( ) 
( ) 
i 
= 
i i 
i i f x 
3 
f(x) 
f(xi) 
xi+1 xi 
X 
B 
1 
C a  A '( ) 
i 
x = x - f x +
AAllggoorriitthhmm ffoorr NNeewwttoonn-- 
RRaapphhssoonn MMeetthhoodd 
4
SStteepp 11 
Evaluate f¢(x) symbolically 
5
SStteepp 22 
6 
Calculate the next estimate of the root 
x = x - f(x ) 
i 
f'(x ) 
i 
i+1 i 
Find the absolute relative approximate error 
x 100 
= x - x 
i i 
1 
1 
x 
i 
a 
+ 
Î +
SStteepp 33 
Find if the absolute relative approximate error is greater 
than the pre-specified relative error tolerance. 
If so, go back to step 2, else stop the algorithm. 
Also check if the number of iterations has exceeded the 
maximum number of iterations. 
7
EExxaammppllee 
Solve the given equation using Newton’s method using 3 iterations: 
8 
f ( x) = x3-0.165x2+3.993x10-4
GGrraapphh ooff ffuunnccttiioonn ff((xx)) 
9 
f ( x) = x3-0.165x2+3.993x10-4
IItteerraattiioonn ##11 
x x f x 
0 
= 
= - 
0.02 3.413x10 
10 
( ) 
( ) 
0.08320 
75.96% 
5.4x10 
' 
0.02 
- 
4 
3 
0 
1 
0 
1 0 
= 
Î = 
- 
= - 
- 
a 
x 
f x 
x
IItteerraattiioonn ##22 
0.08320 
x x f x 
1 
= 
= - - 
0.08320 1.670x10 
11 
( ) 
( ) 
0.05824 
42.86% 
- 
6.689x10 
' 
4 
3 
1 
2 
1 
2 1 
= 
Î = 
- 
= - 
- 
a 
x 
f x 
x
IItteerraattiioonn ##33 
0.05824 
2 
= 
x = x - 
f x 
0.05284 3.717x10 
12 
( ) 
( ) 
0.06235 
6.592% 
- 
9.043x10 
' 
2 
a 
3 
5 
2 
3 2 
= 
Î = 
- 
= - 
- 
f x 
x
AAddvvaannttaaggeess 
CCoonnvveerrggeess ffaasstt,, iiff iitt ccoonnvveerrggeess 
RReeqquuiirreess oonnllyy oonnee gguueessss 
13
DDrraawwbbaacckkss 
14 
10 
5 
0 
f(x) 
-2 -1 0 1 2 
-5 
-10 
-15 
-20 
1 
2 3 
x 
f ( x) = ( x -1)3 = 0 
IInnfflleeccttiioonn PPooiinntt
DDrraawwbbaacckkss 
-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 
f ( x) = x3 - 0.03x2 + 2.4x10-6 = 0 
15 
1.00E-05 
7.50E-06 
5.00E-06 
2.50E-06 
0.00E+00 
-2.50E-06 
-5.00E-06 
-7.50E-06 
-1.00E-05 
x 
f(x) 
0.02 
Division by zero
DDrraawwbbaacckkss 
16 
Root Jumping 
f ( x) = Sin x = 0
17

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