L1 NR Method.ppt

Numerical methods

1. Newton-Raphson Method
Numerical
Methods
f(x)
f(xi)
f(xi-1)
xi+2 xi+1 xi
X


 
ii
xfx
,
)(xf
)f(x
- = xx
i
i
ii

1

2. Step 1:
Step 2:
Step 3:
Algorithm for Newton-Raphson Method
Step 4:
Department of Mathematics

3. The floating ball has a specific gravity of 0.6 and has a radius of 5.5 cm.
Find the depth to which the ball is submerged when floating in water. The
equation that gives the depth xin meters to which the ball is submerged
under water is given by
http://numericalmethods.eng.usf.edu
Newton-Raphson Method
423
1099331650
-
.+x.-xxf 
UsetheNewton’smethodoffindingrootsof
equationstofind
a)thedepth‘x’towhichtheballis
submergedunderwater.Conductthree
iterationstoestimatetherootofthe
aboveequation.
Department of Mathematics

4. http://numericalmethods.eng.usf.edu
Newton-Raphson Method
Department of Mathematics
Graph of the function f(x)
423
1099331650
-
.+x.-xxf  
 x-xxf
.+x.-xxf
-
33.03'
1099331650
2
423

 

 
 
 
06242.0
01242.00.05
109
10118.1
0.05
05.033.005.03
10.993305.0165.005.0
05.0
'
3
4
2
423
0
0
01












xf
xf
xx
Iteration 1: The estimate of the root is

5. http://numericalmethods.eng.usf.edu
Newton-Raphson Method
Department of Mathematics

   
   
 
06238.0
104646.406242.0
1090973.8
1097781.3
06242.0
06242.033.006242.03
10.993306242.0165.006242.0
06242.0
'
5
3
7
2
423
1
1
12













xf
xf
xx
Iteration 2

   
   
 
06238.0
109822.406238.0
1091171.8
1044.4
06238.0
06238.033.006238.03
10.993306238.0165.006238.0
06238.0
'
9
3
11
2
423
2
2
23













xf
xf
xx
Iteration 3

6. http://numericalmethods.eng.usf.edu
Newton-Raphson Method
Department of Mathematics
Estimate of the root
in first iteration
Estimate of the root
in second iteration
Estimate of the root
in third iteration