Numerical Analysis Problem?

[d_u_guy]

Given the polynomial f(z)= z^3 - 7z^2 + 14z - 6

a. use Horner's algorithm to find p(4).

b. Use repeated Horner's algorithm to find the Taylor expansion of p(z) about the point z0 = 1. Compute the Taylor expansion without using Horner's method and compare the result.

c. Loate an approximate root z3 using Newton's and Horner's method when z0 = 2.

[Helmut]

p(z) = ((z - 7)z + 14)z - 6

4 - 7 = 3
3*4 + 14 = 26
26*4 - 6 = 98

p(4) = 98

b.
f(z) = z^3 - 7z^2 + 14z - 6
f'(z) = 3z^2 - 14z + 14
f''(z) = 6z - 14
f'''(z) = 6

1 - 7 = - 6
- 6*1 + 14 = 8
8*1 - 6 = 2
f(1) = 2

3*1-14 = - 11
- 11*1 + 14 = 3

f'(1) = 3
f''(1) = 6*1-14 = - 8
f'''(1) = 6

f(z) = 2 + 3(z - 1) - 4(z - 1)^2 + (z - 1)^3

f(1) = 2 + 3*0 - 4*0 + 0 = 2

c.
2 - 7 = - 5
- 5*2 + 14 = 4
4*2 - 6 = 2
f(2) = 2

x = x0 + (z^3 - 7z^2 + 14z - 6 - 0)/(3z^2 - 14z + 14)

3*2 - 14 = - 8
-8*2 + 14 = 2

x = 2 + 2/2 = 3
3 - 7 = - 4
- 4*3 + 14 = 2
2*3 - 6 = 0

root = 3