First Semester in Numerical Analysis with Julia, 2020a

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CHAPTER 1. INTRODUCTION 47 b) Compute the x-intercept using each formula when (x 1 ,y 1 )=(1.02, 3.32) and (x 2 ,y 2 )= (1.31, 4.31). Use three-digit rounding arithmetic. c) Use Julia (or a calculator) to compute the x-intercept using the full-precision of the device (you can use either one of the formulas). Using this result, compute the relative and absolute errors of the answers you gave in part (b). Discuss which formula is better and why. Exercise 1.3-3: Write two functions in Julia to compute the binomial coefficient ( ) m k using the following formulas: a) ( ) m k = m! (m! is factorial(m) in Julia.) k!(m−k)! b) ( ) m k =( m m−1 m−k+1 )( ) × ... × ( ) k k−1 1 Then, experiment with various values for m, k to see which formula causes overflow first. Exercise 1.3-4: Polynomials can be evaluated in a nested form (also called Horner’s method) that has two advantages: the nested form has significantly less computations, and it can reduce roundoff error. For p(x) =a 0 + a 1 x + a 2 x 2 + ... + a n−1 x n−1 + a n x n its nested form is p(x) =a 0 + x(a 1 + x(a 2 + ... + x(a n−1 + x(a n ))...)). Consider the polynomial p(x) =x 2 +1.1x − 2.8. a) Compute p(3.5) using three-digit rounding, and three-digit chopping arithmetic. What are the absolute errors? (Note that the exact value of p(3.5) is 13.3.) b) Write x 2 +1.1x − 2.8 in nested form by these simple steps: x 2 +1.1x − 2.8 =(x 2 +1.1x) − 2.8 =(x +1.1)x − 2.8. Then compute p(3.5) using three-digit rounding and chopping using the nested form. What are the absolute errors? Compare the errors with the ones you found in (a).
CHAPTER 1. INTRODUCTION 48 Exercise 1.3-5: Consider the polynomial written in standard form: 5x 4 +3x 3 +4x 2 + 7x − 5. a) Write the polynomial in its nested form. (See the previous problem.) b) How many multiplications does the nested form require when we evaluate the polynomial at a real number? How many multiplications does the standard form require? Can you generalize your answer to any nth degree polynomial?
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Chapter 5 Approximation Theory 5.1
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Index Absolute error, 38 Beasley-Sp
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INDEX 220 Chebyshev, 203 Julia code
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First Semester in Numerical Analysis with Julia, 2020a

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