Polynomial interpolation
Polynomial interpolation
Emma Perracchione
Corso di Calcolo Numerico per Ingegneria Meccanica - Matr. PARI (Univ. PD)
Gli esercizi sono presi dal libro: S. De Marchi, D. Poggiali, Exercices of numerical calculus with solutions in Matlab/Octave.
A.A. 2018/2019
Polynomial interpolation Materiale
Materiale
TUTTO IL MATERIALE SI TROVA AL SEGUENTE LINK E VERRA’
AGGIORNATO AD OGNI LEZIONE.
https://www.math.unipd.it/~emma/CN1819.html OPUURE VEDASI
https://elearning.unipd.it/dii/course/view.php?id=1720
Polynomial interpolation Remarks
Introduction
For a given set of points (xi, fi), i = 0, . . . , n, the Lagrange polynomial Πn is the polynomial of degree n that assumes at each value xi the corresponding value fi.
Lagrange interpolation is susceptible to Runge’s phenomenon of large oscillation.
This lecture is a kind of basis for the following ones. Helpfull, for instance, for numerical differentiation and integral approximation.
Polynomial interpolation Remarks
Lagrange
The Lagrange form is given by:
Πn(x ) :=
n
X
i =0
fi`i(x ),
where
`i(x ) := Y
0≤j ≤n j 6=i
x − xj
xi − xj.
When interpolating a given function f by a polynomial of degree n at the nodes x0, . . . , xn we get the error E (x ) = |R(x )|, where
R(x ) = f (x ) − Πn(x ), and
R(x ) = ωn+1(x )fn+1(ξ)
(n + 1)!, x0 < ξ < xn, ωn+1(x ) =
n
Y
i =0
(x − xi).
Polynomial interpolation Exercises
Exercise 1
Exercise
Construct the interpolating Lagrange polynomial on equispaced points of degree n = 5, . . . , 15 for the Runge function:
g (x ) = 1
1 + x2, x ∈ [−5, 5].
On the script Esercizio1, using the command pause(.5), make the plots of the function and of the polynomials.
Hint
The function named Lagrange.m is already in your folder. It computes the Lagrange basis. Then, recall this function on an independent script and complete the first exercise.
Polynomial interpolation Exercises
Exercise 1
function l = lagrange(i,x,xbar)
% i-th elementary Lagrange polynomial of
% the points x evalued on xbar n = length(x); m = length(xbar);
l=prod(repmat(xbar,1,n-1)-repmat(x([1:i-1,i+1:n]),m,1),2)...
/prod(x(i)-x([1:i-1,i+1:n]));
% Denominator of the product in Lagrange form:
% prod(x(i)-x([1:i-1,i+1:n]))
% Numerator of the product in Lagrange form:
% repmat(xbar,1,n-1)-repmat(x([1:i-1,i+1:n]),m,1)
% repmat is needed to create an array for evaluating
% the interpolant.
% For understanding the program write help repmat
Polynomial interpolation Matlab pre-built routines
Polyfit and polyval
The coefficients of the interpolating polynomial can be obtained via the command polyfit.
At first, let us see the Matlab help.
>> help polyfit
polyfit Fit polynomial to data.
P = polyfit(X,Y,N) finds the coefficients of a polynomial P(X) of degree N that fits the data Y best in a
least-squares sense. P is a row vector of length N+1 containing the polynomial coefficients in
descending powers,
P(1)*X^N + P(2)*X^(N-1) +...+ P(N)*X + P(N+1).
... ...
Reference page for polyfit Other functions named polyfit
Polynomial interpolation Matlab pre-built routines
Polyfit and polyval
For evaluating the interpolant, let us see the Matlab help for polyval.
>>help polyval
polyval Evaluate polynomial.
Y = polyval(P,X) returns the value of a polynomial P evaluated at X. P is a vector of length N+1 whose elements are the coefficients of the
polynomial in descending powers.
Y = P(1)*X^N + P(2)*X^(N-1) + ... + P(N)*X + P(N+1) ...
Polynomial interpolation Exercises
Exercise 2
Exercise
Find the coefficients of the polynomial interpolating x = (−2, 1, 3) and y = (−2, 11, 17) via the polyfit command on a script named
Esercizio2 and “on paper”. Evaluate and plot the interpolant for 100 evaluation points.
Hint
Compute the coefficients of the Lagrange polynomial p2(x ) = a2· x2+ a1· x + a0. Then you will obtain
p2(x ) = (−4/15) · x2+ (61/15) · x + (36/5)
≈ −0.26x2+ 4.06x + 7.2.
