Reference no: EM133168949

CSCI 4650 Numerical Analysis - University of Colorado

Assignment 1:

Question 1. What is the smallest number of multiplications and additions needed to evaluate P (x) = 4x⁶ + 2x² + 2x + 6 at x = 1/3.

Question 2. What are the most number of multiplications and additions needed to evaluate a degree n polynomial, a₀ + a₁x + a₂x² + .... + + a_nxⁿ using nested multiplication.

Question 3. Convert 110 .110 from binary to decimal.

Question 4. Covert 7 .2 from decimal to binary.

Question 5. Create a function in Python which takes a positive integer as an input and outputs its binary repre- sentation.

Question 6. A long double floating point number has 1 bit for a sign, 15 bits for an exponent, and 64 bits for the mantissa. What is ∈_mach? Explain your reasoning.

Question 7. Consider the function

f (x) = 1 - (1 - x)³/x

(a) Give an alternate form of f (x) which avoids the issue of subtraction by nearly equal numbers when x is close to 0.

(b) Create a table of the values that Python outputs for f (x) and your alternate form from (a) for x = 10-k for k = 0, 1, 2, . . . , 10.

Question 8. Consider the function

f (x) = ln(x)

(a) Find the Taylor polynomial of degree 4 around the point x = 1.

(b) Use your answer from (a) to estimate ln(1.01) and use the remainder theorem to say how large your error could be between the estimate and true value. Finally use Python to calculate the error.

Extra Credit: Create a function in Python which takes two numbers, the numerator and denominator of a fraction, as input and outputs the exact binary representation of the number. Determine some readable way for dealing with repeating values.