Homework #6
1. Show that for any z,
f[x0, x1, . . . , xn, z] = f(n+1)(ξ(z))
(n+ 1)!
by consider the error formula of the Lagrange interpolatory polynomial passing through
the n+ 1 data points at x0, x1, . . . , xnand the Newton interpolatory divided difference
form of the Lagrange interpolatory polynomial with the additional data point at z,
both evaluated at z.
2. Consider the Hermite interpolating polynomial
H(x) =
n
X
j=0
f(xj)Hn,j(x) +
n
X
j=0
f0(xj)ˆ
Hn,j(x).
Use the forms of Hn,j and ˆ
Hn,j to show:
(a) Hn,j(xk) = 0 for all k6=jand Hn,j(xj) = 1.
(b) ˆ
Hn,j(xk) = 0 for all k.
(c) H0
n,j(xk) = 0 for all k.
(d) ˆ
H0
n,j(xk) = 0 for all k6=jand ˆ
H0
n,j(xj) = 1.
3. Consider x0=−1, x1= 1 and f(x0) = 0, f (x1) = 2 and f0(x0) = 1, f 0(x1) = −1.
(a) Find the Hermite interpolating polynomial using divided differences.
(b) Add the information x2= 0 and f(x2) = 1 and f0(x2) = 0 and find the resulting
Hermite interpolating polynomial.
(c) Approximate f(1/2) and f0(1/2) using the results of part (b).
4. Consider x0= 0, x1= 1, x2= 2 and f(x0)=0, f (x1)=1, f(x2) = 2 and f0(x0) =
0, f 0(x1) = 1, f0(x2) = 2. Find the Hermite interpolating polynomial that interpolates
this data and use it to approximate f(3/2).
5. (a) Verify that H(x) = x3+ 3x2−x−2 is the Hermite interpolating polynomial for
the data x0= 0, f(x0) = −2, f0(x0) = −1 and x1= 1, f(x1) = 1, f0(x1) = 8.
(b) Add the data x2= 2, f(x2) = 0, f0(x2) = 0 and find the resulting Hermite
interpolating polynomial.
6. Consider the underlying function f(x) = sin xand nodes x0, x1, . . . , xn∈[0,2π] satis-
fying xj= 2πj/n.
(a) In the case n= 10, use error bounds to bound the absolute error between f(1)
and H(1), where H(x) is the piecewise cubic Hermite interpolating polynomial.
(b) Using error bounds, determine nsuch that the absolute error between f(x) and
the piecewise cubic Hermite interpolating polynomial H(x) will be less than 10−10 .
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