Homework 2 Foundations of Computational Math 1 Fall 2018

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1 Homework 2 Foundations of Computational Math 1 Fall 2018 Note that Problems 2 and 8 have coding in them. Problem 2 is a simple task while Problem 8 is very involved (and has in fact been given as a programming assignment in the past in this class). This Homework is not a progamming assignment and therefore you need not turn in anything. It is strongly recommended however that you do the coding requested especially for Problem 8 since it reinforces key computational, error and theoretical concepts as well as the consistency between theory and computation. Problem 2.1 Suppose the n-bit 2 s complement representation is used to encode a range of integers, 2 n 1 x 2 n a. If x 0 then x is represented by bit pattern obtained by complementing all of the bits in the binary encoding of x, adding 1 and ignoring all bits in the result beyond the n-th place, i.e., the bit with weight 2 n 1. This procedure is also used when x < 0 to recover the encoding of x 0. What is the relationship between the binary encoding of 2 n 1 x 2 n 1 1 and the binary encoding of x in terms of the number of bits n? 2.1.b. Show that simple addition modulo 2 n on the encoded patterns is identical to integer addition (subtraction) for 2 n 1 x, y 2 n 1 1. You may ignore results that are out of range, i.e., overflow. 2.1.c. Show how overflow in addition (subtraction) can be detected efficiently. 2.1.d. Multiplying an unsigned binary number by 2 or 1/2 corresponds to shifting the binary representation left and right respectively (a so-called logical shift). Show how multiplying signed integers encoded via 2 s complement representation by 2 or 1/2 can be done via a shifting operation (an arithmetic shift). Problem 2.2 Consider the following numbers:

2 2.2.a. Express the numbers as floating point numbers with β = 10 and t = 4 using rounding to even and using chopping. 2.2.b. Express the numbers as floating point numbers with in single precision IEEE format using rounding to even. It is strongly recommended that you implement a program to do this rather than computing the representation manually. 2.2.c. Calculate the relative error for each number and verify it satisfies the bounds implied by the floating point system used. Problem a. Suppose x R and y R with x < y. Is it always true that fl(x) < fl(y) in any standard model floating point system? 2.3.b. Suppose x, y and z are floating point numbers in a standard model floating point arithmetic system. Is floating point arithmetic associative, i.e., is it true that (x op (y op z)) = ((x op y) op z)? 2.3.c. Is floating point arithmetic distributive, i.e., is it true that fl(fl(x + z) y) = fl(fl(fl(x y) + fl(y z)))? 2.3.d. Suppose x and y are two floating point numbers in a system F(β, t, L, U) with opposite signs. How close do x and y have to be in magnitude in order for the result of the floating point computation to be exact? (x + y) Problem 2.4 Consider the function f(x) = x 1.01 x 2.4.a. Find the absolute condition number for f(x). 2.4.b. Find the relative condition number for f(x). 2.4.c. Evaluate the condition numbers around x = 1. 2

3 2.4.d. Check the predictions of the condition numbers by examining the relative error and the absolute error err rel = f(x 1) f(x 0 ) f(x 0 ) err abs = f(x 1 ) f(x 0 ) with x 0 = 1, x 1 = x 0 (1 + δ) and δ small. Problem 2.5 Let f(ξ 1, ξ 2,..., ξ k ) be a function of k real parameters ξ i, 1 i k. Recall, the relative condition number of f with respect to ξ 1 can be expressed κ rel = max(1, c(ξ 1, ξ 2,..., ξ k )) where 0 c(ξ 1, ξ 2,..., ξ k ) is a value that indicates the sensitivity of f to small relative perturbations to ξ 1 as a function of the parameters ξ i, 1 i k. If c(ξ 1, ξ 2,..., ξ k ) 1 then f is considered well-conditioned. Additionally, however, when c < 1 its value gives important information. The smaller c is the less sensitive f is to a relative perturbations in ξ 1. Let n 2 be an integer and β > 0. Consider the polynomial equation p(x) = x n + x n 1 β = a. Show that the equation has exactly one positive root ρ(β). 2.5.b. Derive a formula for c(β, n) that indicates the sensitivity of ρ(β) to small relative perurturbations to β. 2.5.c. Derive a upper bound on c(β, n). 2.5.d. Comment on the conditioning of ρ(β) with respect to β. Problem 2.6 The evaluation of ( x ) f(x) = x + 1 x encounters cancellation for x 0. Rewrite the formula for f(x) to give an algorithm for its evaluation that avoids cancellation. 3

4 Problem a Suppose that x and y are two floating point numbers in a system that supports gradual underflow and satisfies the standard model. Show that if y/2 x 2y then 2.7.b fl(x y) = x y Suppose a triangle has sides with lengths a b c. Heron s formula for its area is A = s(s a)(s b)(s c), Kahan has suggested the following formula s = a + b + c 2 A = 1 4 (a + (b + c))(c (a b))(c + (a b))(a + (b c)) (i) What happens with the Heron s formula with needle-shaped triangles? (ii) Give an informal proof that Kahan s formula is reliable numerically. You may consult the literature of course. (iii) Compare the accuracy of the two formulae in single-precision for several examples to illustrate your points. Problem 2.8 Consider a polynomial of a single variable x written in terms of monomials p n (x) = α 0 + α 1 x + + α n x n p n (x) can be evaluated using Horner s rule given by the procedure c n = α n for i = n 1 : 1 : 0 c i = xc i+1 + α i end p n (x) = c 0 If the roots of the polynomial are known we can use a recurrence based on p n (x) = α n (x ρ 1 ) (x ρ n ) (1) given by: 4

5 d 0 = α n for i = 1 : n d i = d i 1 (x ρ i ) end p n (x) = d n This algorithm can be shown to compute p n (x) to high relative accuracy (Higham 2002 Accuracy and Stability of Numerical Algorithms, Second Edition). Specifically, d n = p n (x)(1 + µ), µ γ 2n+1 where γ k = ku/(1 ku) and u is the unit roundoff of the floating point system used. 2.8.a An error analysis of Horner s rule shows that the computed value of the polynomial satisfies ĉ 0 = (1 + θ 1 )α 0 + (1 + θ 3 )α 1 x + + (1 + θ 2n 1 )α n 1 x n 1 + (1 + θ 2n )α n x n (2) where θ k γ k (Higham 2002 Accuracy and Stability of Numerical Algorithms, Second Edition). The pattern on the subscript is odd numbers, i.e., increment of 2, until the last which is even, i.e., last increment is 1. Let p n (x) = α 0 + α 1 x + + α n x n. 1. Show that p n (x) ĉ 0 p n (x) γ 2n p n ( x ) p n (x) (3) and therefore κ rel = p( x ) p(x) is a relative condition number for perturbations to the coefficients bounded by γ 2n. 2. Is Equation(2) and the associated bound a backward stability bound? 3. Find examples of conditions on p n (x) and x that guarantee that p n (x) is perfectly conditioned with respect to perturbations to the coefficients of the monomial form of p n (x). 4. When would you expect the value of p n (x) to be relatively very sensitive with respect to perturbations to the coefficients? 5

6 2.8.b Equation (3) also yields an a priori bound on the forward error p n (x) ĉ 0 that can be computed along with evaluating p n (x) with Horner s rule. Write a code that evaluates p n (x) and the forward error bound using Horner s rule as well as the product form (1). Your code should allow the selection of single or double precision evaluation of each of these. 1. Apply the code to the polynomial p 9 (x) = (x 2) 9 = x 9 18x x 7 672x x x x x x 512 to evaluate p 9 (x) via Horner s rule and the a priori bound on forward error at several hundred points in the interval [1.91, 2.1] in single precision. Use p 9 (x) evaluated using the product form in double precision as exact for the purposes of this exercise. Plot the product form values across the interval and use the forward error bound to plot curves above and below the exact product form curve to show where the computed values must lie. (Recall, for IEEE single precision u and for IEEE double precision u ) Also plot the values of p 9 (x) computed with Horner s rule and verify the correctness of the a priori bounding curves. Comment on the tightness of the bounds and the computed values of p 9 (x) and quantify your conclusions, i.e., do not simply appeal to the picture. 2. Repeat the previous procedure with p 9 (x) evaluated using the product form in single precision as the exact value of p 9 (x). Are the conclusions significantly changed? 3. Is the fact that the a priori error bound is evaluated in single precision significant? Would you expect evaluating it in double precision to be a good idea? 2.8.c The computed value on step i of Horner s rule satisfies (1 + ɛ i )ĉ i = xĉ i+1 (1 + δ i ) + α i, δ i u, ɛ i u Define ĉ i = c i + e i with e n = 0 and c i the exact value of the parameter in Horner s rule evaluated in exact arithmetic. Show that e i = xe i+1 + xĉ i+1 δ i ɛ i ĉ i e i uβ i β i = x β i+1 + x ĉ i+1 + ĉ i, β n = 0 6

7 and therefore we have the bound p n (x) ĉ 0 uβ 0 This bound is called a running error bound for Horner s rule and can also be easily incorporated into the code for simulatneous evaluation with the values above (Higham 2002 Accuracy and Stability of Numerical Algorithms, Second Edition). 1. Add the computation of the running error bound to the code above (single and double precision should be supported) and compare this bound s prediction with those of the a priori bound above when Horner s rule and the two error bounds are evaluated in single precision and the exact value of p 9 (x) is evaluated in double precision. 2. Plot, quantify and discuss your observations. 3. Propose two other polynomials and repeat the comparisons of a priori versus running error bounds. Make sure you discuss why and how you generated the polynomials. 7