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ECIV 301Programming & Graphics
Numerical Methods for Engineers
Lecture 23CURVE FITTING Chapter 18
Function Interpolation and Approximation
Lagrange Interpolating Polynomials
• Reformulation of Newton’s Polynomials
• Avoid Calculation of Divided Differences
n
iiin xfxLxf
0
)()(x f(x)xo f(xo )
x1 f(x1 )
x2 f(x2 )
… …
xn f(xn)
n
ijj ji
ji xx
xxxL
0
)(
Lagrange Interpolating PolynomialCardinal Functions: Product of n-1 linear factors
ni
n
ii
i
ii
i
iii xx
xx
xx
xx
xx
xx
xx
xx
xx
xxxL
1
1
1
1
2
2
1
1
Skip xi
Property:
ji if 1
ji if 0ijji xL
Other Methods
nn xaxaxaaoxf 2
21)(
Direct Evaluation
n+1 coefficients
n210
n210
y y yy yf(x)
x x x x x
n+1 Data Points
Interpolating Polynomial should represent them exactly
Other Methods
nn xaxaxaaoxf 2
21)(
Direct Evaluation
n210
n210
y y yy yf(x)
x x x x x
nn xaxaxaaoy 0
202010
nn xaxaxaaoy 1
212111
nnnnnn xaxaxaaoy 2
21
Other Methods
n
1
0
2
1211
0200
n
1
0
a
a
a
y
y
y
nnnn
n
n
xxx
xxx
xxx
Solve Using any of the methods we have learned
Other Methods
•Not the most efficient method
•Ill-conditioned matrix (nearly singular)
•If n is large highly inaccurate coefficients
•Limit to lower order polynomials
Inverse Interpolation
n321
n321
y y y yy f(x)
x x x xx
-0.01
-0.005
0
0.005
0.01
0.015
0.92 0.925 0.93 0.935 0.94 0.945
Xr=?Xr=?
Yr=GivenYr=Given
Inverse Interpolation
-0.01
-0.005
0
0.005
0.01
0.015
0.92 0.925 0.93 0.935 0.94 0.945
Xr=?Xr=?
Switch x and y and then interpolate?
Not a Good Idea!
n321
n321
x x x x x
y y y y f(x)
Yr=GivenYr=Given
Inverse Interpolation
-0.01
-0.005
0
0.005
0.01
0.015
0.92 0.925 0.93 0.935 0.94 0.945
Fit and nth order polynomial to x, f(x) data
Solve Equation 0 Yrxf rn
Xr=?Xr=?
Yr=GivenYr=Given
Errors in Polynomial Interpolation
n321
n321
y y y yy f(x)
x x x xx
-0.01
-0.005
0
0.005
0.01
0.015
0.92 0.925 0.93 0.935 0.94 0.945
It is expected that as number of nodes increases, error decreases, HOWEVER….
n
iiin xfxLxf
11
At all interpolation nodes xi Error=0At all intermediate points
Error: f(x)-fn-1(x)
f(x)
Errors in Polynomial Interpolation
Beware of Oscillations….
For Example:Consider f(x)=(1+x2)-1 evaluated at 9 points in [-5,5]And corresponding p8(x) Lagrange Interpolating Polynomial
P8(x)f(x)
E.G Quadratic Splines
• Function Values at adjacent polynomials are equal at interior nodes
11112
11 iiiiii xfcxbxa
112
1 iiiiii xfcxbxa
ni 2
conditions )1(2 n
E.G Quadratic Splines• First and Last Functions pass through end
points
011201 xfcxbxa i
nnnnnn xfcxbxa 2
conditions )1(2 n
conditions 2
conditions 2n
ni 2
E.G Quadratic Splines• First Derivatives at Interior nodes are equal
baxxf 20
ni 2
conditions )1(2 n
conditions 2
conditions 13 n
iii
iii
bxa
bxa
1
111
2
2
conditions 1-n
E.G Quadratic Splines• Assume Second Derivative @ First Point=0
02 10 axf
conditions )1(2 n
conditions 2
conditions 3n
conditions 1-nconditions 1
E.G Quadratic Splines• Assume Second Derivative @ First Point=0
conditions 3n
tscoefficien edundetermin 3n
Solve 3nx3n system of Equations
baC
ix on based )( and
)( on based
xf
xf