MATH 609-602
Homework #5
Polynomial interpolation of functions of one variable
(1) Let f(x) = xnand f[x0, x1, ..., xn] be the divided difference of order nusing
the points x0< x1< ... < xn. Prove that:
(a) (10 pts) f[x0, x1, ..., xn] = 1;
(b) (10 pts) f[x0, x1, ..., xn−1] = x0+x1+... +xn−1.
(2) (10 pts) If f[x0, x1, . . . , xn] denotes the divided difference of order nprove the
Leibnitz formula
(fg)[x0, x1, . . . , xn] =
n
X
k=0
f[x0, x1, . . . , xk]g[xk, xk+1, . . . , xn].
(3) (10 pts) Find the Lagrange and backward Newton divided difference interpo-
lating polynomials for the data (0,1),(0.5,2),(1,3),(1.5,4).
(4) (20 pts) Let
Ln,k(x) = (x−x0)...(x−xk−1)(x−xk+1)...(x−xn)
(xk−x0)...(xk−xk−1)(xk−xk+1)...(xk−xn).
Show that for any xthe following relations are valid:
(a) Pn
k=0 Ln,k(x) = 1;
(b) Pn
k=0 xm
kLn,k(x) = xmfor m= 1, ..., n.
(5) (20 pts) Estimate the interpolation error of cos xin the interval (0,0.4) by a
polynomial of degree 2 using the interpolation nodes x0= 0, x1= 0.2, x2=
0.4.
(6) (20 pts) Pn
k=0(x−xk)mLn,k (x) = 0 for m= 1, ..., n.
In your free time and for your amusement:
Show that if ω(x) = (x−x0)(x−x1)...(x−xn) then:
(1) Pn
k=0(x−xk)n+1Ln,k (x) = (−1)nω(x)
(2) Pn
k=0(x−xk)n+2Ln,k (x) = (−1)nω(x)Pn
k=0(x−xk)
(3) Pn
k=0 Ln,k(0)xn+1
k= (−1)nx0.x1...xn.
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