COT 4810 Homework #5 (10/7-10/9), due 10/21/03

Spline Curves (Cht. 18)

1) Why are quadratic functions not good choices for parametric spline curves?

Although they can fit the point data rather well, they do not have enough flexibility to maintain the "smoothness" of the curve. In particular, the slope at each control point should be the same whether calculated from the left or the right. It's very difficult to do this with quadratic functions.

Mathematical Research (Cht. 9)

2) What is the significance of fractals to computer scientists?

They are an example of how mathematical work can be used in a way that it was never originally intended to be used. It's possible that fractals could be used, or be part of solutions to a varied group of problems. The same holds true for other mathematical discoveries, (such as Euler's theorem a(n) 1 mod n, for all integers n > 1, where gcd(a,n) = 1.)

The Newton-Rhapson Method (Cht. 21)

3) Do four iterations of the Newton-Rhapson method on the function f(x) = x3 - 2. Start with your initial guess for the root as x=1. Fill in the table below showing each improved guess. How far off (what percentage) is your guess for the root from the actual root as calculated by a calculator?

x / f(x)
1 / -1
1.333333 / .370370
1.263888 / .018955
1.259933 / 5.925929x10-5
1.259921 / -2.3492919x10-6

This final answer is .0000391% off from the calculated answer, approximately.

Number Systems for Computing (Cht. 42)

4) Convert 20022121221 from ternary to balanced ternary notation.

1. Add 1 to each digit propogating carries: 101211010102

2. Subtract one from each digit: b0100b0b0b1

(I am using a b to designated -1 or "1 bar.")

Turing Machines and Universal Turing Machine (Chts. 31, 51)

5) Design a turing machine that takes its input (which is a string of 0s and 1s), flips every bit and then halts. Give the formal specification for your machine.

Q = {start, A, halt}

 = {0,1}

 = {0, 1, B} B=blank character

 / 0 / 1 / B
start / (1, R, A) / (0, R, A) / (B, R, halt)
A / (1, R, A) / (0, R, A) / (B, R, halt)

Cat Scanning

6) How is a cat scan better than an X-ray?

A can scan uses many 2D slices to recreate a 3D image of the area of the body that is scanned whereas a X-ray only takes one 2D slice so that the depth of parts of the body are difficult to gauge accurately.