1. Use the Newton forward-difference formula to construct interpolating polynomials ofdegree one, two, and three for the following data. Approximate the specified value using each of the polynomials.
i. f(0.43) if f(0) = 1, f(0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169
ii. f(0.18) if f(0.1) = −0.29004986, f(0.2) = −0.56079734, f(0.3) = −0.81401972, f(0.4) = −1.0526302
2. Use the Newton backward-difference formula to construct interpolating polynomials ofdegree one, two, and three for the following data. Approximate the specified value using each of the polynomials.
i. f(−1/3) if f(−0.75) = −0.07181250, f(−0.5) = −0.02475000, f(−0.25) = 0.33493750,
f(0) = 1.10100000
ii. f(0.25) if f(0.1) = −0.62049958 , f(0.2) = −0.28398668 , f(0.3) = 0.00660095, f(0.4) = 0.24842440
3. A fourth-degree polynomial P(x) satisfies ∆4P(0) = 24, ∆3P(0) = 6, and ∆2P(0) = 0, where ∆P(x) = P(x + 1) − P(x). Compute ∆2P(10). 4. The following data are part of a table for .
Calculate g(0.25) as accurately as possible
i. by forward difference interpolating directly in this table,
ii. by first tabulating xg(x) and then forward difference interpolating in that table,
iii. explain the difference between the results in (i) and (ii) respectively.
5. i. Show that the cubic polynomials
P(x) = 3 − 2(x + 1) + 0(x + 1)(x) + (x + 1)(x)(x − 1)
and
Q(x) = −1 + 4(x + 2) − 3(x + 2)(x + 1) + (x + 2)(x + 1)(x)
both interpolate the data
f(−2) = −1,f(−1) = 3,f(0) = 1,f(1) = −1,f(2) = 3
ii. Why does part (i) not violate the uniqueness property of interpolating polynomials?
6. The following data are given for a polynomial P(x) of unknown degree.
P(0) = 4,P(1) = 9,P(2) = 15,P(3) = 18
Determine the coefficient of x3 in P(x), if all fourth-order forward differences are 1.
7.
For a function f , the Newton divided-difference formula gives the interpolating polynomial
,
on the nodes x0 = 0,x1 = 0.25,x2 = 0.5, and x3 = 0.75. Find f(0.75).
