Interpolation Generalized

Generalized Approach to Interpolation
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Generalized Approach to
Interpolation
Mohammad Tawfik

Mohammad Tawfik
#WikiCourses
Generalization
• Lets represent the relation between the
dependent and independent variables as
follows:
     axhxy
n
i
i
i xa  0
    
   
T
nn
nn
aaaaaa
xxxxxh
1210
12
...
...1





Mohammad Tawfik
#WikiCourses
Generalization (cont’d)
• That is valid if you have (n+1) data points
through which you want to interpolate an
nth order polynomial.

Mohammad Tawfik
#WikiCourses
• Now, let’s use those points:
        ahaxhxyy
n
i
i
i xa 11
0
11 1
 
        ahaxh
n
xyy nn
n
i
i
inn xa 11
0
11 1 

 

 


Mohammad Tawfik
#WikiCourses
• Or simply:
 
 
 
 
    aTa
h
h
h
y
n














1
2
1

1
,

 j
iji xTwhere

Mohammad Tawfik
#WikiCourses
Now, we may …
1. Solve the above equations
simultaneously to obtain the values for {a}
and explicitly write the relation as a
polynomial in x:
       yTxhxy
1


Mohammad Tawfik
#WikiCourses
Now, we may …
2. Just invert the transformation matrix [T] and
substitute in the original relation to obtain a
summation of weighted functions
In this case, each function N(x) is an nth order
polynomial! (Recall Lagrange interpolation)
           
 





1
1
1
n
j
jj yxN
yxNyTxhxy

Mohammad Tawfik
#WikiCourses
• Where
  


n
i
ji
i
j TxxN
0
1
,1.

Mohammad Tawfik
#WikiCourses
Interpolation Involving
Derivatives!

Mohammad Tawfik
#WikiCourses
Interpolation Involving Derivatives!
• In many practical cases, we may
encounter data involving the first or
second derivatives of the function!
• However, we still need to have (n+1)
pieces of information to be able to
interpolate nth order polynomial
• We will follow the same approach, but with
a simple modification

Mohammad Tawfik
#WikiCourses
Given the data …
• Assume that you are given the data of the
function value and the slope at m-points.
Where 2*m=n+1 (the order of the
polynomial)

Mohammad Tawfik
#WikiCourses
Following the previous procedure …
• Lets represent the relation between the
dependent and independent variables as
follows:
     axhxy
n
i
i
i xa  0
    
   
T
nn
nn
aaaaaa
xxxxxh
1210
12
...
...1





Mohammad Tawfik
#WikiCourses
The twist …
• Now, let’s use that data:
        
      aha
dx
xdh
dx
dy
y
but
ahaxhxyy
xxxx
n
i
i
i xa
'
1
'
1
11
0
11
11
1












Mohammad Tawfik
#WikiCourses
Which gives us …
 
 
 
 
    aTa
h
h
h
h
y
y
y
y
m
m
m
m


































'
'
1
1
'
'
1
1


Mohammad Tawfik
#WikiCourses
Now …
• We can solve for the parameters {a}
OR
• Obtain the N(x) functions as before.

Mohammad Tawfik
#WikiCourses
Interpolation with more than
one variable!

Mohammad Tawfik
#WikiCourses
We will be using the same
approach …
• If you have a function f(x,y) measured at
(n+1) points in the x-direction, and at
(m+1) points in the y-directions, you may
be able to write:
• The follow the same procedure described
before
       1),1)*(1()1)*(1(,1
0 0
, ,, 
 
  mnmn
n
i
m
j
ji
ji ayxhyxayxf

Interpolation Generalized

Recommended

Recommended

More Related Content

What's hot

What's hot (9)

Viewers also liked

Viewers also liked (20)

More from Mohammad Tawfik

More from Mohammad Tawfik (20)

Recently uploaded

Recently uploaded (20)

Interpolation Generalized