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INTERPOLATION(1)
ELM1222 Numerical Analysis
1
Some of the contents are adopted from
Laurene V. Fausett, Applied Numerical Analysis using MATLAB. Prentice Hall Inc., 1999
ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Today’s lecture
• Polynomial Interpolation
• Lagrange Interpolation
• Newton Interpolation
• Difficulties with Polynomial Interpolation
• Hermite Interpolation
• Rational-Function Interpolation
2 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Lagrange Interpolation Polynomials
• Basic concept
• The Lagrange interpolating polynomial is the polynomial of degree n-1
that passes through the n points.
• Using given several point, we can find Lagrange interpolation polynomial.
3 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
General Form of Lagrange
• The general form of the polynomial is
p(x) = L1y1 + L2y2 + … + Lnyn
where the given points are (x1,y1), ….. , (xn,yn).
• The equation of the line passing through two points (x1,y1) and (x2,y2) is
• The equation of the parabola passing through three points (x1,y1), (x2,y2), and
(x3,y3) is
2
12
11
21
2
)(
)(
)(
)()( y
xx
xxy
xx
xxxp
3
2313
212
3212
311
3121
32
))((
))((
))((
))((
))((
))(()( y
xxxx
xxxxy
xxxx
xxxxy
xxxx
xxxxxp
))...()()...((
))...()()...(()(
111
111
nkkkkkk
nkkk
xxxxxxxx
xxxxxxxxxL
4 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Lagrange Interpolation
Example 1 :
(x1,y1)=(-2,4), (x2,y2)=(0,2), (x3,y3)=(2,8)
288
)2(2
4
)2)(2(4
8
)2(
8)02))(2(2(
)0))(2((2
)20))(2(0(
)2))(2((4
)22)(02(
)2)(0()(
2
xxxxxxxx
xxxxxxxp
5 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Lagrange Interpolation
• We can represent the Lagrange polynomial with coefficient ck.
𝑝(𝑥) = 𝑐1𝑁1 + 𝑐2𝑁2 + … + 𝑐𝑛𝑁𝑛
))...()()...(( 111 nkkkkkk
kk
xxxxxxxx
yc
))...()()...(()( 111 nkkk xxxxxxxxxN
6 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Higher Order Interpolation Polynomials
Example 2: Higher order interpolation polynomials
x = [ -2 -1 0 1 2 3 4],
y = [ -15 0 3 0 -3 0 15]
7 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Review and Discussion
• In Lagrange interpolation polynomial, it always go through given points.
• It is less convenient than the Newton form when additional data points may
be added to the problem.
3
2313
212
3212
311
3121
32
))((
))((
))((
))((
))((
))(()( y
xxxx
xxxxy
xxxx
xxxxy
xxxx
xxxxxp
8 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
1-9
Newton form of the equation of a straight line passing through two points
(x1, y1) and (x2, y2) is
Newton form of the equation of a parabola passing through three points
(x1, y1), (x2, y2), and (x3, y3) is
The general form of the polynomial passing through n points
(x1, y1), …,(xn, yn) is
Newton Interpolation Polynomials
)()( 121 xxaaxp
))(()()( 213121 xxxxaxxaaxp
))...((...))(()()( 11213121 nn xxxxaxxxxaxxaaxp
9 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Substituting (x1, y1) into
Substituting (x2, y2) into
Substituting (x3, y3) into
))(()( 213121 xxxxaxxaay
))(()( 213121 xxxxaxxaay
))(()()( 213121 xxxxaxxaaxp
))(()( 213121 xxxxaxxaay
11 ya
12
122
xx
yya
13
12
12
23
23
3xx
xx
yy
xx
yy
a
Consider a parabola equation obtained using three points
10 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Newton Interpolation Polynomials
Example 3: Passing through the points
(x1, y1)=(-2, 4), (x2, y2)=(0, 2), and (x3, y3)=(2, 8).
The equations is
Where the coefficients are
)0))(2(())2(()( 321 xxaxaaxp
411 ya
1)2(0
42
12
122
xx
yya
1)2(2
)2(0
42
02
28
13
12
12
23
23
3
xx
xx
yy
xx
yy
a
2)2()2(4)( 2 xxxxxxp
11 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Newton Interpolation Polynomials
Passing through the points (x1, y1)=(-2, 4), (x2, y2)=(0, 2), and (x3, y3)=(2, 8).
2)2()2(4)( 2 xxxxxxp
))(()()( 213121 xxxxaxxaaxp
12 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Newton Interpolation Polynomials
Additional Data Points
Example 4: adding the points (x4, y4) = (-1, -1) and (x5, y5) = (1, 1) to the
previous data
)1)(2)(2()2)(2()2()2(4)( xxxxxxxxxxxp
13 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Example 5: Consider again the data
with Lagrange form.
x = [ -2 -1 0 1 2 3 4 ],
y = [ -15 0 3 0 -3 0 15]
Higher Order Interpolation Polynomials
)1)(2()1)(2(6)2(1515)( xxxxxxxp
Do it again with Newton form.
the polynomial is cubic
14 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Higher Order Interpolation Polynomials
• Example 6: If the y values are modified slightly, the divided-difference table
shows the small contribution from the higher degree terms:
)3)(1)(1)(2(0007.0
)2)(1)(1)(2(0042.0)1)(1)(2(0167.0
)1)(2(0333.1)1)(2(95.5)2(5.1414)(
xxxxx
xxxxxxxxx
xxxxxxxN
15 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Example 7: The data
x = [ -2 -1.5 -1 -0.5 0 – 0.5 1 1.5 2]
y = [ 0 0 0 0.87 1 0.87 0 0 0]
illustrate the difficulty with using higher order polynomials to interpolate a
moderately large number of points.
Difficulties: Humped and Flat Data
16 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Example 8: The data
x = [ 0.00 0.20 0.80 1.00 1.20 1.90 2.00 2.10 2.95 3.00]
y = [ 0.01 0.22 0.76 1.03 1.18 1.94 2.01 2.08 2.90 2.95]
Not good with noisy straight line.
Difficulties: Noisy Straight Line
Newton polynomial coefficients
17 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Difficulties: Runge Function
Example 9:
The function
is an example of the fact that polynomial interpolation does not produce a good
approximation for some functions
using more function values (at evenly spaced x values) does not necessarily
improve the situation.
x = [ -1 -0.5 0.0 0.5 1.0 ]
y = [0.0385 0.1379 1.0000 0.1379 0.0385]
2251
1)(
xxf
18 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
1-19
Example 10:
x = [-1.000 -0.750 -0.500 -0.250 0.000 0.250 0.500 0.750 1.000 ]
y = [0.0385 0.0664 0.138 0.3902 1.000 0.3902 0.138 0.0664 0.0385]
The interpolation polynomial overshoots the true polynomial much more
severely than the polynomial formed by using only five points.
Difficulties: Runge Function
19 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Hermite Interpolation
Rational-Function Interpolation
20
Some of the contents are adopted from
Laurene V. Fausett, Applied Numerical Analysis using MATLAB. Prentice Hall Inc., 1999
ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Hermite Interpolation
• Hermite interpolation allows us to find a ploynomial that matched both
function value and some of the derivative values
21 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Hermite Interpolation Example 11:
22 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Difficult Data
• As with lower order polynomial interpolation, trying to interpolate in humped
and flat regions brings overshootings.
Example 12:
23 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Rational-Function Interpolation • Why use rational-function interpolation?
• Some functions are not well approximated by polynomials.(runge-function)
• but are well approximated by rational functions, that is quotients of polynomials.
24 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Bulirsch-Stoer algorithm
• Bulirsch-Stoer algorithm
• The approach is recursive,
• Given a set of m+1 data points (x1,y1), … , (xm+1, ym+1), we seek an interpolation function of the form
25 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
11)1)...(1())...(1(
)1...())...(1(
)1...())...(1(
))...(1())...(1(
miimii
miimii
mi
i
miimii
miimiii
ii
RR
RR
xx
xx
RR
RR
yRBulirsch-Stoer pattern
Bulirsch-Stoer algorithm
• Bulirsch-Stoer method for three data points
26 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
• Bulirsch-Stoer method for five data points Third stage
114
34
4
3
34434
R
RR
xx
xx
RRRR
115
45
5
4
45545
R
RR
xx
xx
RRRR
11334
2334
4
2
233434234
RR
RR
xx
xx
RRRR
11445
3445
5
3
344545345
RR
RR
xx
xx
RRRR
R5=y5 x5 y5
R4=y4 x4 y4
R3=y3 x3 y3
R2= y2 x2 y2
R1= y1 x1 y1
Second stage First stage data
112
12
2
1
12212
R
RR
xx
xx
RRRR
113
23
3
2
23323
R
RR
xx
xx
RRRR
11223
1223
3
1
122323123
RR
RR
xx
xx
RRRR
Bulirsch-Stoer algorithm
27 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Fifth stage Forth stage
1123234
123234
4
1
1232342341234
RR
RR
xx
xx
RRRR
1134345
234345
5
2
2343453452345
RR
RR
xx
xx
RRRR
112342345
12342345
5
1
12342345234512345
RR
RR
xx
xx
RRRR
Bulirsch-Stoer algorithm
28 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Rational-function interpolation
data points:
x = [-1 -0.5 0.0 0.5 1.0]
y = [0.0385 0.1379 1.0000 0.1379 0.0385]
29 ELM1222 Numerical Analysis | Dr Muharrem Mercimek
Example 13: