Interpolation
Assoc. Prof. Dr. Siridech Boonsang
Electrical Engineering
xi
x
xi+1
Spline Interpolation
• Polynomials are the most common choice of
interpolants.
• There are cases where polynomials can lead to
erroneous results because of round off error and
overshoot.
• Alternative approach is to apply lower-order
polynomials to subsets of data points. Such
connecting polynomials are called spline functions.
Spline provides a superior approximation of the behavior
of functions that have local, abrupt changes (d).
Linear Spline
The first order splines for a group of ordered data
points can be defined as a set of linear functions:
f ( x)  f ( x0 )  m0 ( x  x0 )
f ( x)  f ( x1 )  m1 ( x  x1 )

x0  x  x1
x1  x  x2
f ( x)  f ( xn1 )  mn1 ( x  xn1 ) xn1  x  xn
f ( xi 1 )  f ( xi )
mi 
xi 1  xi
Linear spline - Example
Fit the following data with first order splines. Evaluate the
function at x = 5.
x
f(x)
3.0
4.5
7.0
9.0
2.5
1.0
2.5
0.5
2.5  1
m
 0.6
7  4.5
f (5)  f (4.5)  m(5  4.5)
 1.0  0.6  0.5
 1.3
Parametric Continuity Conditions
•
Zero order parametric (C0): Simply the curves meet,
C1(1) = C2(0) .
•
First order parametric (C1): The first parametric
derivations for two successive curve sections are equal at
their joining point, C´1(1)= C´2(0)
Parametric Continuity Conditions
•
Second order parametric (C2): Both the first and second
parametric derivatives of the two curve sections are the
same at the intersection, C˝1(1)= C˝2(0)
Linear Spline
• The main disadvantage of linear spline is that they are not
smooth. The data points where 2 splines meets called (a knot),
the changes abruptly.
• The first derivative of the function is discontinuous at these
points.
• Using higher order polynomial splines ensure smoothness at
the knots by equating derivatives at these points.
Quadric Splines
• Objective: to derive a second order polynomial for each
2
interval between data points. f i ( x)  ai x  bi x  ci
• Terms: Interior knots and end points
For n+1 data points:
• i = (0, 1, 2, …n),
• n intervals,
• 3n unknown
constants (a’s, b’s and
c’s)
Quadric Splines
• The function values of adjacent polynomial must be
equal at the interior knots 2(n-1).
ai 1 xi 1  bi 1 xi 1  ci 1  f i ( xi 1 ) i  2, 3, 4,..., n
2
ai xi 1  bi xi 1  ci  f i ( xi 1 )
2
i  2, 3, 4,..., n
• The first and last functions must pass through the end
points (2).
a1 x0  b1 x0  c1  f ( x0 )
2
an xn  bn xn  cn  f ( xn )
2
Quadric Splines
• The first derivatives at the interior knots must be equal
(n-1).
f i ( x)  2ai x  bi
'
2ai 1 xi 1  bi 1  2ai xi 1  bi
• Assume that the second derivate is zero at the first point
(1)
a1  0
(The first two points will be connected by a straight line)
Quadric Splines - Example
Fit the following data with quadratic splines. Estimate
the value at x = 5.
x
f(x)
3.0
4.5
7.0
9.0
2.5
1.0
2.5
0.5
Solutions:
There are 3 intervals (n=3), 9 unknowns.
Quadric Splines - Example
1. Equal interior points:
 For first interior point (4.5, 1.0)
The 1st equation:
x12 a1  x1b1  c1  f ( x1 )
(4.5) 2 a1  4.5b1  c1  f (4.5)
The 2nd equation:
20.25 a1  4.5 b1  c1  1.0
x12 a2  x1b2  c2  f ( x1 )
(4.5) 2 a2  4.5b2  c2  f (4.5)
20.25a2  4.5b2  c2  1.0
Quadric Splines - Example
 For
second interior point (7.0, 2.5)
The 3rd equation:
x22 a2  x2b2  c2  f ( x2 )
(7) 2 a2  7b2  c2  f (7)
49a2  7b2  c2  2.5
The 4th equation:
x22 a3  x2b3  c3  f ( x2 )
(7) a3  7b3  c3  f (7)
2
49a3  7b3  c3  2.5
Quadric Splines - Example
 First and last functions pass the end points
For the start point (3.0, 2.5)
x a1  x0 b1  c1  f ( x0 )
2
0
9a1  3b1  c1  2.5
For the end point (9, 0.5)
x a  x3b3  c3  f ( x3 )
2
3 1
81a3  9b3  c3  0.5
Quadric Splines - Example
Equal derivatives at the interior knots.
For first interior point (4.5, 1.0)
2 x1 a1  b1  2 x1 a2  b2
9a1  b1  9a2  b2
For second interior point (7.0, 2.5)
2 x2 a2  b2  2 x3 a3  b3
14a2  b2  14a3  b3
Second derivative at the first point is 0
f '' ( x0 )  a1  0
Quadric Splines - Example
4.5 1
0 0

0 0

0 0
3 1

0 0
1 0

0 0
0
0
0
0
0
20.25 4.5 1
0
0
49
7
1
0
0
0
0
0
49
7
0
0
0
0
0
0
0
0
81
9
0
0
 9 1
0
14
0
1
 14  1
0  b1  1 
c   

0   1  1 
0   a 2   2 .5 
   
1  b2  2.5



 2 .5 
0  c2
   
1  a3  0.5
0  b3  0 
   
0  c3  0 
Quadric Splines - Example
Solving these 8 equations with 8 unknowns
a1  0,
b1  1,
c1  5.5
a2  0.64, b2  6.76, c2  18.46
a3  1.6, b3  24.6,
c3  91.3
f1 ( x)   x  5.5,
3.0  x  4.5
f 2 ( x)  0.46 x 2  6.76 x  18.46,
4.5  x  7.0
f3 ( x)  1.6 x 2  24.6 x  91.3,
7.0  x  9.0
Cubic Splines
Objective: to derive a third order polynomial for
each interval between data points.
Terms: Interior knots and end points
f i ( x)  ai x  bi x  ci x  d i
3
2
For n+1 data points:
• i = (0, 1, 2, …n),
• n intervals,
• 4n unknown constants (a’s, b’s ,c’s and d’s)
Cubic Splines
• The function values must be equal at the interior knots (2n-2).
• The first and last functions must pass through the end points (2).
• The first derivatives at the interior knots must be equal (n-1).
• The second derivatives at the interior knots must be equal (n-1).
• The second derivatives at the end knots are zero (2), (the 2nd derivative
function becomes a straight line at the end points)
Alternative technique to get Cubic Splines
• The second derivative within each interval [xi-1, xi ] is a straight line. (the 2nd
derivatives can be represented by first order Lagrange interpolating polynomials.
x  xi
x  xi 1
''
f i ( x)  f i ( xi 1 )
 f i ( xi )
xi 1  xi
xi  xi 1
''
''
A straight line
connecting the first
knot f’’(xi-1) and the
second knot f’’(xi)
The second derivative at any point x within the interval
Cubic Splines
• The last equation can be integrated twice
2 unknown constants of integration can be evaluated by
applying the boundary conditions:
1. f(x) = f (xi-1) at xi-1
2. f(x) = f (xi) at xi
''
f i ( x) 
''
f i ( xi 1 )
xi  x 3  f i ( xi ) x  xi 1 3
6xi  xi 1 
6xi  xi 1 
 f i ( xi 1 ) f i ( xi 1 )xi  xi 1 


 xi  x 
6
 xi  xi 1

''
''
 f i ( xi )
f i ( xi )xi  xi 1 


 x  xi 1 
6
 xi  xi 1

Unknowns:
f ' ' ( xi )
f ' ' ( xi 1 )
i = 0, 1,…, n
Cubic Splines
•For each interior point xi (n-1):
f i1 ( xi )  f i ( xi )
'
'
( xi  xi 1 ) f '' ( xi 1 )  2( xi 1  xi 1 ) f '' ( xi )
 ( xi 1  xi ) f '' ( xi 1 ) 
6
 f ( xi 1 )  f ( xi )
xi 1  xi

6
 f ( xi 1 )  f ( xi )
xi  xi 1
This equation result with n-1 unknown second
derivatives where, for boundary points:
f˝(xo) = f˝(xn) = 0
Cubic Splines - Example
Fit the following data with cubic splines
Use the results to estimate the value at x=5.
x
f(x)
3.0
2.5
4.5
1.0
7.0
2.5
9.0
0.5
Solution:
Natural Spline:
f '' ( x0 )  f '' (3)  0,
f '' ( x3 )  f '' (9)  0
Cubic Splines - Example
For 1st interior point (x1 = 4.5)
x
f(x)
3.0
2.5
4.5
1.0
7.0
2.5
9.0
0.5
- xi  xi 1  x1  x0  4.5  3.0  1.5
- xi 1  xi 1  x2  x0  7  3.0  4
- xi 1  xi  x2  x1  7  4.5  2.5
Apply the following equation:
( xi  xi 1 ) f '' ( xi 1 )  2( xi 1  xi 1 ) f '' ( xi )  ( xi 1  xi ) f '' ( xi 1 )
6
6
 f ( xi 1 )  f ( xi ) 
 f ( xi 1 )  f ( xi )

xi 1  xi
xi  xi 1
Cubic Splines - Example
6
6
1.5 f (3)  2  4 f (4.5)  2.5 f (7) 
(2.5  1) 
(2.5  1)
2.5
1.5
''
f
(3)  0
Since
''
''
''
8 f '' (4.5)  2.5 f '' (7)  9.6 .............. (eq.1)
 For 2nd interior point (x2 = 7 )
x
3.0
4.5
7.0
9.0
f(x)
2.5
1.0
2.5
0.5
xi  xi 1  x2  x1  7  4.5  2.5
xi 1  xi 1  x3  x1  9  4.5  4.5
xi 1  xi  x3  x2  9  7  2
Cubic Splines - Example
Apply the following equation:
( xi  xi 1 ) f '' ( xi 1 )  2( xi 1  xi 1 ) f '' ( xi )  ( xi 1  xi ) f '' ( xi 1 )
6
6
 f ( xi 1 )  f ( xi ) 
 f ( xi 1 )  f ( xi )

xi 1  xi
xi  xi 1
2.5 f '' (4.5)  2  4.5 f '' (7)  2 f '' (9) 
Since
6
6
(0.5  2.5) 
(1  2.5)
2
2.5
f '' (9)  0
2.5 f '' (4.5)  9 f '' (7)  9.6 ............. (equ 2)
Cubic Splines - Example
Solve the two equations:
8 f i '' (4.5)  2.5 f i '' (7)  9.6 

''
''
yeild
f
(
4
.
5
)

1
.
67909
,
f
(7)  1.53308

''
''
2.5 f i (4.5)  9 f i (7)  9.6

The first interval (i=1), apply for the equation:
''
''
f (x )
f i ( xi )
3
x  xi 1 3
f i ( x)  i i 1  xi  x  
6 xi  xi 1 
6 xi  xi 1 
''
 f i ( xi 1 ) f i '' ( xi 1 ) xi  xi 1  
 f i ( xi )
f i ( xi ) xi  xi 1  



  xi  x   
  x  xi 1 
6
6
 xi  xi 1

 xi  xi 1

f1 ( x)  0 ( xi  3)3 
1.67909
 2.5 0(1.5) 
 1 1.67909(1.5) 


( x  3)3  

4
.
5

x


( x  3)



6(1.5)
1
.
5
6
1
.
5
6




f1 ( x)  0.186566 ( x  3)3  1.6667 (4.5  x)  0.24689( x  3)
Cubic Splines - Example
The 2nd interval (i =2), apply for the equation:
f 2 ( x) 
1.67909
 1.53308
 1  1.67909(2.5) 
7  x 
(7  x ) 3 
( x  4.5)3  


6(2.5)
6(2.5)
6
 2.5

 2.5  1.53308(2.5) 


( x  4.5)

6
 2.5

f 2 ( x)  0.111939(7  x)3  0.102205 ( x  4.5)3  0.29962(7  x)  1.638783 ( x  4.5)
The 3rd interval (i =3),
f 3 ( x)  0.127757(9  x)3  1.761027 (9  x)  0.25 ( x  7)
For x = 5:
f 2 ( x)  f 2 (5)  1.102886