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Unit 5: Polynomial Interpolation

We denote (as above) by Pn the linear space (vector space) of all polynomialsof (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.

C. Fuhrer/ A. Sopasakis: FMN050/FMNF01-2015

81

5.1: Polynomial Interpolation

0 1 2 3 4 5 6-2

-1

0

1

2

3

4

5An Interpolation Polynomial

measurements

polynomial (6th degree)

How do we determine such a polynomial?C. Fuhrer/ A. Sopasakis: FMN050/FMNF01-2015

82

5.2: Vandermonde Approach

Ansats: p(x) = anxn + an−1x

n−1 + · · · a1x+ a0

Interpolation conditionsp(xj) = anx

ni + an−1x

n−1i + · · · a1xi + a0 = yi 0 ≤ i ≤ n

In matrix formxn0 xn−1

0 · · · x0 1xn1 xn−1

1 · · · x1 1· · ·

xnn xn−1n · · · xn 1

anan−1

...a0

=

y0

y1...yn

or V a = y.

V is called a Vandermonde matrix.C. Fuhrer/ A. Sopasakis: FMN050/FMNF01-2015

83

5.3 Vandermonde approach in MATLAB

polyfit sets up V and solves for a (the coefficients)

Alternatively vander sets up V and a = solve(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 .


84

5.4 Vandermonde approach in MATLAB or Python

Essential steps to generate and plot an interpolation polynomial:

• Computing the coefficients (polyfit, vander etc)

• Generating x-values for ploting, e.g.xval=linspace(0,100,1000)

• Evaluating the polynomial, e.g. yval=polyval(coeff,xval)

• Plotting, e.g. plot(xval,yval).


85

5.5 Lagrange Polynomials

We take now another approach to compute the interpolation polynomial

Definition. [–] The polynomials Li ∈ Pn with the property

Li(xj) =

{0 if i 6= j1 if i = j

are called Lagrange polynomials.


86

5.6 Lagrange Polynomials (Cont.)

It is easy to check, that

Li(x) =

n∏j = 0i 6= j

(x− xj)(xi − xj)

The interpolation polynomial p can be written as

p(x) =

n∑i=0

yiLi(x)

Check that it indeed fulfils the interpolation conditions!C. Fuhrer/ A. Sopasakis: FMN050/FMNF01-2015

87

5.7 Lagrange Polynomials: Example

Lagrange polynomials of degree 3:

0 1 2 3 4 5 6-2

-1

0

1

2

3

4

5An Interpolation Polynomial

measurements

polynomial (6th degree)

0 0.33 0.66 1

0

1


88

5.8 Lagrange Polynomials in Python

def lagrange(x,i,xm):

"""

Evaluates the i-th Lagrange polynomial at x

based on grid data xm

"""

n=len(xm)-1

y=1.

for j in range(n+1):

if i!=j:

y*=(x-xm[j])/(xm[i]-xm[j])

return y

and an interpolation functiondef interpolation(x,xm ,ym):

n=len(xm)-1

lagrpoly=array([lagrange(x,i,xm) for i in range(n+1)])

y = dot(ym ,lagrpoly)

return y


89

5.9 Example (cont.)

Interpolation data and a plot

xm = array([1,2,3,4,5,6])

ym = array([-3,0,-1,2,1,4])

xplot = \

linspace(0.9,6.1,100)

yplot = \

interpolation(xplot ,xm,ym)


90

5.10 The vector space Pn

We have two ways to express a polynomial

Monomial representation p(x) =∑nk=0 akx

k

Lagrange representation p(x) =∑nk=0 ykL

nk(x)

They describe the same polynomial (as the interpolation polynomial isunique).


91

5.11 The vector space Pn (Cont.)

We introduced two bases in Pn:

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).


92

Download - Unit 5: Polynomial Interpolation - Matematikcentrum | … · 2015-04-12 · Unit 5: Polynomial Interpolation We denote (as above) by P ... 5.3 Vandermonde approach in MATLAB ... 5.5

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