Download Unit 4: Polynomial Interpolation

Unit 4: Polynomial Interpolation
(see course book p. 180)
We denote (as above) by Pn the linear space (vector space) of all polynomials
of (max-) degree n.
Definition. [–] Let (xi, yi), i = 0 : n be n + 1 pairs of real numbers
(typically measurement data)
A polynomial p ∈ Pn interpolates these data points if
p(xk ) = yk k = 0 : n
holds.
We assume in the sequel that the xi are distinct.
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4.1: Polynomial Interpolation
An Interpolation Polynomial
5
4
polynomial (6th degree)
3
2
1
0
measurements
-1
-2
0
1
2
3
4
5
6
How do we determine such a polynomial?
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4.2: Vandermonde Approach
Ansats: p(x) = anxn + an−1xn−1 + · · · a1x + a0
Interpolation conditions
p(xj ) = anxni + an−1xn−1
+ · · · a1xi + a0 = yi
i
In matrix form
 n
x0 xn−1
0
n
x xn−1
1
 1

xnn xn−1
n
···
···
···
···
x0
x1
xn
0≤i≤n

  
1
an
y0

  y1 
1
 an−1
  
  ..  =  .. 
a0
yn
1
or V a = y.
V is called a Vandermonde matrix.
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4.3 Vandermonde approach in MATLAB
polyfit sets up V and solves for a (the coefficients)
Alternatively vander sets up V and a = V \y solves for a.
polyval evaluates the polynomial for given x values.
n + 1 points determine a polynomial of (max-)degree n.
Obs! n is input to polyfit .
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4.4 Vandermonde approach in MATLAB
Essential steps to generate and plot an interpolation polynomial:
• Computing the coefficients (polyfit, vander etc)
• Generating x-values for ploting, e.g.
xval=[0:0.1:100] or xval=linspace(0,100,1000)
• Evaluating the polynomial, e.g. yval=polyval(coeff,xval)
• Plotting, e.g. plot(xval,yval)
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4.5 Lagrange Polynomials
We take now another approach to compute the interpolation polynomial
Definition. [–] The polynomials Lni ∈ P n with the property
Lnk(xi) =
0 if
1 if
k=
6 i
k=i
are called Lagrange polynomials.
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4.6 Lagrange Polynomials (Cont.)
It is easy to check, that
n
Y
Lnk(x) =
i=0
i 6= k
(x − xi)
(xk − xi)
The interpolation polynomial p can be written as
p(x) =
n
X
yk Lnk(x)
k=0
Check that it indeed fulfills the interpolation conditions!
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4.7 Lagrange Polynomials: Example
Lagrange polynomials of degree 3:
An Interpolation Polynomial
5
4
polynomial (6th degree)
1
3
2
1
00
measurements
-1
-2
0
1
0.33
2
3
0.66
4
5
61
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4.8 The vector space P n
We have two ways to express a polynomial
Monomial representation p(x) =
Lagrange representation p(x) =
Pn
k
k=0 ak x
Pn
n
k=0 yk Lk (x)
They describe the same polynomial (as the interpolation polynomial is
unique).
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4.9 The vector space P n (Cont.)
We introduced two bases in P n:
Monomial basis {1, x, x2, · · · , xn}, coordinates ak , k = 0 : n
Lagrange basis {Ln0 (x), Ln1 (x), · · · , Lnn(x)}, coordinates yk , k = 0 : n
It is easy to show, that these really are bases (linear independent elements).
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4.10 Inner Product Space
Definition. [9.1] Let V be a linear space and (·, ·) : V × V → R a map
with properties
• (v, v) ≥ 0 and (v, v) = 0 ⇔ v = 0,
• (αv, w) = α(v, w) for α ∈ R,
• (v, w) = (w, v)
• (v + w, u) = (v, u) + (v, u)
then V is called an inner product space and (·, ·) an inner product.
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4.11 Inner Product Space (Examples)
• Rn is an inner product space with the inner product
n
X
(v, u) =
viui = v Tu
see scalar product
i=0
• P n is an inner product space with the inner product
n
X
(p, q)xi =
p(xi)q(xi)
pointwise inner product
i=0
• P n is an inner product space with the inner product
Z b
L2-inner product
(p, q)2 =
p(x)q(x)dx
a
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4.12 Inner Product Space - Orthogonality
Definition. [9.2] Let V be an inner product space and let two elements
p, q ∈ V have the property (p, q) = 0, then they are called orthogonal. One
writes p⊥q or p = q ⊥.
Lagrange polynomials
Pn form an orthogonal basis with respect to the inner
product (p, q) = i=0 p(xi)q(xi).
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