Chapter 1 Homework All Calculations Were Performed Standard Double

Floating Point Arithmetic 1

1.4 Floating Point Arithmetic

1. Determine the value of each of the following expressions using 4-digit rounding

and 4-digit chopping arithmetic. For each quantity, compute the absolute and

the relative error.

(a) π+e−cos 22◦

(b) e/7 + √2 ln π

(c) πln 2 + √10 cos 22◦

(d) ln 2 −√10 + tan 22◦/73

√9

(a) To ten digits π+e−cos 22◦= 4.932690627. In chopping arithmetic, we first

calculate

In rounding arithmetic, we calculate

2Section 1.4

(b) To ten digits e/7 + √2 ln π= 2.007218505. In chopping arithmetic, we first

calculate

Finally,

In rounding arithmetic, we calculate

Finally,

Floating Point Arithmetic 3

In rounding arithmetic, we calculate

(d) To ten digits (ln 2 −√10 + tan 22◦)/(7 3

√9) = −0.1418283677. For the nu-

merator, we first calculate

4Section 1.4

For the denominator,

Finally

Working in rounding arithmetic, we first calculate

Then

Finally

In the following table, δdenotes the absolute error and ǫthe relative error.

2. Identify the potential roundoff error problems in the following algorithm for

calculating the roots of the quadratic equation ax2+bx +c= 0.

GIVEN: real coefficients a,b,c

STEP 1: calculate disc =√b2−4ac

STEP 2: calculate root1 = (−b+disc)/(2a)

STEP 3: calculate root2 = c/(a·root1)

OUTPUT: root1 and root2

Note that this algorithm uses the fact that the product of the roots of ax2+

bx +c= 0 is equal to c/a.

3. Identify the potential roundoff error problems in the following algorithm for

calculating the roots of the quadratic equation ax2+bx +c= 0.

GIVEN: real coefficients a,b,c

STEP 1: calculate disc =√b2−4ac

STEP 2: calculate root1 = −2c/(b+disc)

STEP 3: calculate root2 = −(b/a)−root1

OUTPUT: root1 and root2

Note that this algorithm uses the fact that the sum of the roots of ax2+bx+c= 0

is equal to −b/a.

4. Construct an algorithm which computes the roots of the quadratic equation

ax2+bx +c= 0 and which avoids as many roundoff error problems as possible.

Test your algorithm by computing the roots of the quadratic equations 0.2x2−

47.91x+ 6 = 0 and 0.025x2+ 7x−0.1 = 0. Use 4 decimal digit rounding

arithmetic in your calculations.

6Section 1.4

Both values are correct to the digits shown.

5. Show that the relative error incurred by using x2+x3

6to approximate ex−

cos x−xis roughly 10−24 for |x| ≤ 5×10−8.

Let f(x) = ex−cos x−x. Then, by Taylor’s theorem

6. In the ﬂoating point number system F(10,10,−98,100), subtract each of the

following pairs of numbers. How many significant decimal digits are lost in

performing the subtraction, and how does this compare with the number of

significant decimal digits to which the numbers agree?

(a) 355/113 and π

(b) 685/252 and e

(c) cos(0.1◦) and cos(0.11◦)

(d) 103/280 and 1/e

(a) In F(10,10,−98,100),

(b) In F(10,10,−98,100),

(c) In F(10,10,−98,100),

8Section 1.4

(d) In F(10,10,−98,100),

7. (a) To how many significant decimal digits do the numbers √10002 and √10001

agree?

(b) In the ﬂoating point number system F(10,10,−98,100), subtract √10001

from √10002. How many significant decimal digits are lost in performing

the subtraction?

(c) Explain how you would rearrange your computations to obtain a more

accurate answer.

(a) Because

(b) In F(10,10,−98,100),

Floating Point Arithmetic 9

8. (a) For what values of xdoes (1 −cos x)/x2= 1/2 to full machine precision.

Consider both IEEE standard single precision and double precision. (Hint:

Use Taylor series.)

(b) Repeat part (a) to determine the values of xfor which e−x= 1 to full

machine precision.

(c) Repeat part (a) to determine the positive values of xfor which

ln(1 + x)−cos x−x=−1 to full machine precision.

(a) Using the standard Taylor series expansion for cos x, we find

(b) By Taylor’s theorem,

10 Section 1.4

(c) Using the standard Taylor series expansion for ln(1 + x)and cos x, we find

9. (a) Plot the function f(x) = 1 −cos xover the interval −5×10−8≤x≤

5×10−8. Generate points at 1001 uniformly spaced abscissas and perform

all calculations in IEEE standard double precision.

(b) Reformulate fto avoid cancellation error and then repeat part (a).

10. Repeat Exercise 9 for the function f(x) = tan−1x−sin x.

(a) Here is a graph of f(x) = tan−1x−sin xover the interval −5×10−8≤x≤

(b) Expanding both tan−1xand sin xin Taylor series, we find

12 Section 1.4

11. Repeat Exercise 9 for the function f(x) = ln(1 + x)−cos x−x+ 1 over the

interval −5×10−6≤x≤5×10−6.

(a) Here is a graph of f(x) = ln(1+x)−cos x−x+1 over the interval −5×10−6≤

Floating Point Arithmetic 13

(b) Expanding both ln(1 + x)and cos xin Taylor series, we find

12. Near certain values of xeach of the following functions cannot be accurately

computed using the formula as given due to cancellation error. Identify the

values of xwhich are involved (e.g., near x= 0 or large positive x) and propose

a reformulation of the function (e.g., using Taylor series, rationalization, trig

identities, etc.) to remedy the problem.

(a) f(x) = 1 + cos x(b) f(x) = e−x+ sin x−1

(c) f(x) = ln x−ln(1/x)(d) f(x) = √x2+ 1 −√x2+ 4

(e) f(x) = 1 −2 sin2x(f) f(x) = ln(x+√x2+ 1)

(g) f(x) = x−sin x(h) f(x) = ln x−1

(a) When x≈(2n+ 1)π, for any integer n,cos x≈ −1, and there will be

14 Section 1.4

expansions for e−xand sin xto reformulate the function as

(c) When xis near 1, ln x≈ln(1/x), and there will be cancellation error in the

(d) For large positive and negative x,√x2+ 1 ≈√x2+ 4, so there will be cancel-

(e) When xis near π

(f) For large negative values of x,√x2+ 1 ≈ |x|=−x, so there will be cancella-

(g) When xis near 0, x≈sin x, so there will be cancellation error in the calculation

Floating Point Arithmetic 15

13. (a) Verify that

f(x) = 1 −sin xand g(x) = cos2x

1 + sin x

are identical functions.

(b) Which function should be used for computations when xis near π/2? Why?

(c) Which function should be used for computations when xis near 3π/2?

Why?

(a) Note that

14. It was noted that evaluation of the expression

333.5b6+a211a2b2−b6−121b4−2+ 5.5b8+a

2b

when a= 77617.0 and b= 33096.0 requires at least 37 decimal digits of precision.

(a) HP workstations have a double precision extended format which corre-

sponds to the ﬂoating point number system F(2,113,−16381,16384). Does

this system provide enough precision to evaluate the above expression?

(b) What is the smallest value for kfor which the ﬂoating point number system

F(2, k, m, M) provides 37 decimal digits of precision?

16 Section 1.4

(a) In the ﬂoating point number system F(2,113,−16381,16384), machine pre-

cision with rounding is

(b) In order for the ﬂoating point number system F(2, k, m, M)to provide at least

15. Consider the following linear system of equations





3.02 −1.05 2.53

4.33 0.56 −1.78

−0.83 −0.54 1.47







x1

x2

x3



=

−1.61

7.23

−3.38



.

(a) Determine the solution of this system using exact arithmetic during Gaus-

sian elimination.

(b) Determine the solution of this system using 3 decimal digit rounding arith-

metic during Gaussian elimination.

(c) Explain the difference between the answers found in part (a) and those

found in part (b).

(a) To carry out the calculations in exact arithmetic, we first convert all coeffi-

cients and right-hand side components to fractions. Gaussian elimination then

produces

Floating Point Arithmetic 17

(b) Starting from the original matrix, the pivot for the first pass of Gaussian elim-

ination is placed in the first row, first column. The multiplier needed to elimi-

The multiplier needed to eliminate the −0.83 entry in the first column of the

Thus, the final reduced matrix is

16. One strategy for alleviating the accumulation of roundoff error during Gaussian

elimination is known as partial pivoting (a more detailed description of this

process will be provided in Chapter 3). The basic idea is as follows. During the

i-th pass of Gaussian elimination, find the row, starting at row iand running

through the last row of the matrix, which has the largest entry in column i.

Interchange this row with the current row iand proceed with the elimination

phase.

18 Section 1.4

(a) Repeat Exercise 15(b) using this partial pivoting strategy. What is the

relative error in each component of the computed solution?

(b) Repeat the “Linear System of Equations” problem considered in the text

using the partial pivoting strategy. What is the relative error in each

component of the computed solution?

(a) For the first pass using partial pivoting on the matrix in Exercise 15, we note

that the largest entry in the first column is in the second row. We therefore

The multiplier needed to eliminate the 3.02 entry in the first column of the

The matrix for the next pass of Gaussian elimination is then

Thus, the final reduced matrix is

Floating Point Arithmetic 19

(b) For the system of equations in Example 1.11, the largest element in the first

column is originally in the first row. Thus, there is no need to interchange

We now observe that the largest element in the bottom two rows of the second

column is in the third row. We therefore interchange the second and third rows

The multiplier needed to eliminate the 0.0001 entry in the second column of