Applied Numerical Analysis fifth edition potx

This chapter contains a short review of those topics from elementary single-variable calculus that will be needed in later chapters, together with an introduction to the terminology used

Numerical Analysis

Richard L Burden Youngstown State University

J Douglas Faires

Youngstown State University

sáo ' PWS PUBLISHING COMPANY

‘BOSTON.

Contents

Ĩ Mathematical Preliminaries I

1.1 Review of Calculus 2 / 1.2 Round-Off Errors and Computer Arithmetic 12 1.3 Algorithms and Convergence 24

2.6 Zeros of Polynomials and Miiller’s Method 2 2.7 Survey of Methods and Software 93

3.1 Interpolation and the Lagrange Polynomial 98 3.2 Divided Differences 112

3.3 Hermite Interpolation 123 3.4 Cubic Spline Interpolation 130 3.5 Parametric Curves 148 3.6 Survey of Methods and Software 154

Contents

4.4 4.5 4.6

47 4.8 4.9

Composite Numerical Integration 784 Adaptive Quadrature Methods 192 Romberg Integration 199

Gaussian Quadrature 205 Multiple Integrals 277 Improper Integrals 224 4.10 Survey of Methods and Software 230

Initial-Value Problems for Ordinary Differential

5.1 5.2 5.3 3.4 5.5 5.6

57 3.8 5.9

Elementary Theory of Initial-Value Problems 234 Euler’s Method 239 -

Higher-Order Taylor Methods 248 Runge-Kutta Methods 254 Error Control and the Runge-Kutta-Fehlberg Method 263 Multistep Methods 270

Variable Step-Size Multistep Methods -282 Extrapolation Methods 288

Higher-Order Equations and Systems of Differential Equations 294

7 Hterative Techniques in Matrix Algebra 389

6 Direct Methods for Solving Linear Systems 323

Linear Systems of Equations 324 Pivoting Strategies 338

Linear Algebra and Matrix Inversion 345 The Determinant of a Matrix 358

Matrix Factorization 362 Special Types of Matrices 371 Survey of Methods and Software 386

7.1 Norms of Vectors and Matrices 390 7.2 Eigenvalues and Eigenvectors 407 7.3 Iterative Techniques for Solving Linear Systems 406

Contents x

7.4 Error Estimates and Iterative Refinement 424

7.5 Survey of Methods and Software 434

8.1 Discrete Least-Squares Approximation 436

8.2 Orthogonal Polynomials and Least-Squares Approximation 449

8.3 Chebyshev Polynomials and Economization of Power Series 459 -

8.4 Rational Function Approximation 469

8.5 Trigonometric Polynomial Approximation 480

8.6 Fast Fourier Transforms 485

8.7 Survey of Methods and Software 496

9.1 Linear Algebra and Eigenvalues 498

9.2 The Power Method 506

10.4 Steepest Descent Techniques 568

10.5 Survey of Methods and Software 575

41.1 The Linear Shooting Method 578

11.2 The Shooting Method for Nonlinear Problems 585

11.3 Finite-Difference Methods for Linear Problems 592

11.4 Finite-Difference Methods for Nonlinear Problems 598

11.5 Rayleigh-Ritz Method 605

11.6 Survey of Methods and Software 620

12.4 An Introduction to the Finite-Element Method 657

12.5 Survey of Methods and Software 672

Preface

TYHGIB G1 HRHHGIGIHILNEHUIH-ĐE

About the Text

ES ANTON re fae SE eS A RIN TEP SE TET

We have developed the material in this text for a sequence of courses in the theory and application of numerical approximation techniques The text is designed primarily for junior-level mathematics, science, and engineering majors who have completed at least the first year of the standard college calculus sequence and have some knowledge of a high- level programming language Familiarity with the fundamentals of matrix algebra and differential equations is also useful, but adequate introductory material on these topics is presented in the text

Previous editions of the book have been used in a wider variety of situations than we originally intended In some cases, the mathematical analysis underlying the development

of approximation techniques is emphasized, rather than the methods themselves; in others, the emphasis is reversed The book has also been used as the core reference for courses at the beginning graduate level in engineering and computer science programs, as

the basis for the actuarial examination in numerical analysis, where self-study is common,

and in first-year courses in introductory- analysis offered at international universities We have tried to adapt the book to fit these diverse requirements without compromising our otiginal purpose: to give.an introduction to modern approximation techniques; to explain how, why, and when they can be expecied to work; and to provide a firm basis for future study in numerical analysis

The book contains sufficient material for a full year of study, but we expect many readers to use the text for only a single-term course In such a course, students learn

to identify the type of problems that require numerical techniques for their solution, see examples of error propagation that can occur when numerical methods are applied, and accurately approximate the solutions of some problems that cannot be solved exactly The remainder of the text serves as a reference for methods that are not discussed in the course Either the full-year or single-course treatment is consistent with the purpose of the text

Virtually every concept in the text is illustrated by example, and this edition contains more than 2000 class-tested exercises These exercises range from elementary applications

of methods and algorithms to generalizations and extensions of theory In addition, the exercise sets include a large number of applied problems from diverse areas of engineering

Preface

as well as from the physical, computer, biological, and social sciences The applications chosen concisely demonstrate how numerical methods can be (and are) applied in “real- life” situations

e At the end of Chapters 2 through 12 we,added a section entitled “Survey of Methods and Software.” These sections summarize the methods developed in the chapter and recommend strategies for choosing tecliniques to use in various situations These sec- tions also reference appropriate programs in the International Mathematical and Statis- tics (IMSL) and Numerical Algorithms Group (NAG) libraries and refer to other professional sources when they are pertinent to the material in the chapter

® New algorithms are included for the Method of False Position, Bézier curve generation,

Gaussian quadrature for double and triple integrals, Padé approximation, and Chebyshev rational function approximation The editorial comments in all the algorithms have been rewritten, when appropriate, to more closely correspond to the discussion in the text

© The Method of False Position (Regula Falsi) has been included in Chapter 2 since this type of bracketing technique is commonly used in professional software packages

e The presentation of Lagrange interpolation in Chapter 3 has been streamlined for better continuity We also reduced the discussion of Taylor polynomials in this chapter to make

it clearer that Taylor polynomials are not used for interpolation

© Chapter 3 now concludes with a section on parametric curves In this section we de- scribe how interactive computer graphic systems permit curves to be drawn and quickly modified Many of our students are familiar with computer graphic software that permits freehand curves to be quickly drawn and modified This section describes how cubic Hermite spline functions and Bézier polynomials make this possible Although the section is intended to be informational rather than computational, we have included an algorithm for generating Bézier curves

* Chapter 5 contains a new series of examples that better illustrate the methods used for solving initial-value problems We use a single initial-value problem to illustrate all the standard techniques of approximation, which permits the methods to be compared more effectively

e The review material on linear algebra presented in Chapter 6 and at the beginning of

Chapter 7 has been condensed and reorganized to better reflect how this material is

taught at most institutions

© We reworked and reordered the exercise sets to provide better continuity both within the sections and from section to section New exercises have been added to make the book.

sosmenu enna em ROR eT UT ST ER TN STE

Algorithms

As in the previous editions, we give a detailed, structured algorithm without program

listing for each method in the text The algorithms are in a form that students with even

limited programming experience can code A Student Study Guide is available with this edition; it includes solutions to representative exercises and a disk containing programs written from the algorithms The programs are written in both FORTRAN and Pascal and the disks are formatted for a DOS platform The publisher can also provide instructors with a complete solutions manual that provides answers and solutions to all the exercises

in the book as well as a copy of the disk that is included in the study guide All the results

in the Solutions Manual were regenerated for this edition using both the FORTRAN and Pascal programs on the disk

The algorithms in the text lead to programs that give correct results for the examples and exercises in the text, but no attempt was made to write general-purpose professional software In particular, the algorithms are not always written in the form that leads to the most efficient program in terms of either time or storage requirements When a conflict occurred between writing an extremely efficient algorithm and writing a slightly different one illustrating the important features of the method, the latter path was invariably taken

The flow chart on page xiv indicates chapter prerequisites The only deviation from this chart is described in the footnote at the bottom of the first page of Section 3.4 Most

of the possible sequences that can be generated from this chart have been taught by the authors at Youngstown State University

We would like to personally thank the reviewers for this edition:

George Andrews, Oberlin College Richard O Hill, Jr., Michigan John E Buchanan, Miami University State University

Richard Franke, The Naval Postgraduate School Leonard J Lipkin, The University Richard E Goodrick, Evergreen State College of North Florida

Nathaniel Grossman, The University of California at Jim Ridenhour, Austin Peay State

Max Gunzburger, Virginia Polytechnic and State Steven E Rigdon, Southern Tili-

David R Hill, Temple University

In particular, we thank Phillip Schmidt of the University of Akron Phil is both a good friend and, when appropriate, a most critical reviewer

We were again fortunate to have an excellent team of student assistants, headed by Sharyn Campbell Included in this group were Genevieve Bundy, Beth Eggens, Melanie George, and Kim Graft Thanks for keeping us honest Finally, we would like to thank Chuck Nelson of the English Department at Youngstown State University for his assistance

- on the variety of equipment in the Professional Design and Production Center

Richard L Burden

J Douglas Faires

Suppose two experiments are conducted to test this law, using the same gas in each case In the first experiment,

CHAPTER 1 © Mathematical Preliminaries

The experiment is then repeated, using the same values of R and

N, but increasing the pressure by a factor of two while reducing the volume by the same factor Since the product PV remains the same, the predicted temperature would still be 17°C, but now we find that the actual temperature of the gas is 19°C

Clearly, the ideal gas law is suspect when an error of this magni- tude is obtained Before concluding that the law is invalid in this situ- ation, however, we should examine the data to determine whether the error can be attributed to the experimental results If so, it would be

of interest to determine how much more accurate our experimental results would need to be to ensure that an error of this magnitude could not occur /

Analysis of the error involved in calculations is an important topic in numerical analysis and will be introduced in Section 1.2 This particular application is considered in Exercise 24 of that section This chapter contains a short review of those topics from elementary single-variable calculus that will be needed in later chapters, together with an introduction to the terminology used in discussing conver- gence, error analysis, and the machine representation of numbers

8 > 0 such that | f(x) — L| < e whenever x e X and 0 < |x — xo | < ồ (See Figure 1.1.)

Let f be a function defined on a set X of real numbers and x9 € X; f is said to be continuous at x, if lim,_,, f(x) = f(%o) The function f is said to be continuous on X

if it is continuous at each number in X Bs B

C(X) denotes the set of all functions continuous on X When X is an interval of the real line, the parentheses in this notation will be omitted For example, the set of all functions continuous on the closed interval [a, b] is denoted C[a, b]

The limit of a sequence of real or complex numbers can be defined in a similar manner

Let {x,}%_, be an infinite sequence of real or complex numbers The sequence is said to converge to a number x (called the limit) if, for any ¢ > 0, there exists a positive integer

N(e) such that n > M(¢) implies |x, — x| < € The notation lim„_s„ X„ = %, OFX, > X as

n—> %, means that the sequence {x,, };-1 converges to x eB © The following theorem relates the concepts of convergence and continuity

TẾ f is a function defined on a set X of real numbers and x9 € X, then the following are equivalent:

If f is a function defined in an open interval containing xo, f is said to be differentiable

at xg if

li im —T————— ƒ@) — f&o)

x—>xq x — Xo

exists When this limit exists it is denoted by f’ (xo) and is called the derivative of f at

Xo A function that has a derivative at each number in a set X is said to be differentiable

CHAPTER 1 © Mathematical Preliminaries

on X The derivative of f at x9 is the slope of the tangent line to the graph of f at

If the function f is differentiable at x, then f is continuous at xo BoE Bo

The set of all functions that have n continuous derivatives on X is denoted C"(X), and

the set of functions that have derivatives of all orders on X is denoted C°(X) Polynomial,

rational, trigonometric, exponential, and logarithmic functions are in C”(X), where X

consists of all numbers at which the functions are defined When X is an interval of the real line, we will again omit the parentheses in this notation

The next theorems are of fundamental importance in deriving methods for error estimation The proofs of these theorems and the other unreferenced results in this section can be found in any standard calculus text

(olle’s Theorem) Suppose fe C[a, b] and f is differentiable on (a, 5) If f(a) = f() = 0, then a number

c in (a, b) exists with f’(c) = 0 (See Figure 1.3.)

f@)

VN

1.1 Review of Calculus

Theorem 1.8 (Mean Value Theorem)

lffe Cla, bland f is differentiable on (a, b), then a number c in (a, b) exists with

Theorem 1.9 (Extreme Value Theorem)

if f € C[a, bj, then c;, ¢ € [a, b] exist with f(c)) = f(x) =f (cs) for each x € [a, 8) If,

in addition, f is differentiable on (a, b), then the numbers c, and c, occur either at endpoints of [a, b] or where f’ is zero ef 6 5

‘The other basic concept of calculus that will be used extensively is the Riemann integral

Definition 1.10 The Riemann integral of the function f on the interval [a, b] is the following limit,

| A function f that is continuous on an interval [a, 5] is Riemann integrable on the

| interval This permits us to choose, for computational convenience, the points x; to be

| equally spaced in [a, b] and for eachi = 1,2, ,n, to choose z; = x; In this case,

(Weighted Mean Value Theorem for Integrals)

If fe C[a, bl, g is integrable on [a, b], and g(x) does not change sign on [a, 5], then there exists a number c in (a, b) with

(Generalized Rolle’s Theorem)

Let fe C[a, b] be n times differentiable on (a, b) If f vanishes at then + 1 distinct

numbers xọ, z„ in [4, b], then a number c in (a, D) exists with fo) = 0

The next theorem presented is the Intermediate Value Theorem Although its state- ment is intuitively clear, the proof is beyond the scope of the usual caiculus course The proof can be found in most analysis tests (see, for example, Fulks [59], p 67)

(intermediate Value Theorem) Tfƒe C{[a, b] and K is any number between f(a) and f(b), then there exists c in (a, b) for

Show that x° — 2x3 + 3x? — 1 = Ohas a solution in the interval (0, 1]

Consider f(x) = x> - 2x3 + 3x? — 1 The function f is 2 polynomial and is

continuous on [0, 1] Since

fO) = -1<0<1 =f, the Intermediate Value Theorem implies that there is a number x in (0, 1) with x° — 2x?

As seen in Example 1, the Intermediate Value Theorem is important as an aid to determine when solutions to certain problems exist It does not, however, give a means for finding these solutions This topic is considered in Chapter 2

The final theorem in this review from calculus describes the development of the Taylor polynomials The importance of the Taylor polynomials to the study of numerical analysis cannot be overemphasized, and the following result will be used repeatedly

POO yeh tee + ~ x0)"

Pula) = Fo) + FOr — ¥0) +

by taking the limit of P,,(x) as 1 — © is called the Taylor series for f about x9 In the case

Xo = 0, the Taylor polynomial is called a Maclaurin polynomial and the Taylor series is

The term truncation error generally refers to the error involved in using a truncated

or finite summation to approximate the sum of an infinite series This terminology will be reintroduced in subsequent chapters

Determine (a) the second and (b) the third Taylor polynomials for f(x) = cos x about

Xo = 0, and use these polynomials to approximate cos(0.01) (c) Use the third Taylor polynomial and its remainder term to approximate J! cos x dx

Since f c C”(Ñ), Taylor’s Theorem can be applied for any n > 0

a Forn = 2 andx 9 = 0, we have

COS + = Ì — 4x? +1 ix} sin £Œ), where &(x) is a number between 0 and x (See Figure 1.8.) With x = 0.01, the Taylor polynomial and remainder term is

cos 0.01 = 1 — ‡(0.01” + ¿(0.01 sin E@)

= 0.99995 + 0.16 X 107° sin E@),

where 0 < &(x) < 0.01 (The bar over the six in 0.16 is used to indicate that this digit

repeats indefinitely.) Since |sin €(x)| < 1, we have

|cos 0.01 — 0.99995] = 0.16 x 107%,

so the approximation 0.99995 matches at least the first five digits of cos 0.01 Using

standard tables we find that the value of cos 0.01 to 11 digits is 0.99995000042, which

gives agreement through the first nine digits

b Since f’”(0) = 0, the third Taylor polynomial and remainder term about x» = Ois

cosx = 1 — 4x24 dx4 cos EG),

where 0 < £(x) < 0.01 The approximating polynomial remains the same; and the approx- imation is still 0.99995, but we now have much better accuracy assurance since

[dex cos E()| = 4,(0.01)*) ~ 4.2 1071,

The first two parts of the example nicely illustrate the two objectives of numerical analysis The first is to find approximation, which both Taylor polynomials provide The second objective is to determine the accuracy of the approximation In this case the third Taylor polynomial was much more informative than the second, even though both poly- nomials gave the same approximation

A bound for the error in this approximation can be determined from the integral of the —

Taylor remainder term:

Ị 10 CHAPTER 1 * Mathematical Preliminaries

6 Suppose f € C fa, b] and f‘(x) exists on (a, b) Show that if f’(x) 0 for all x in (a, b), then

if there can exist at most one number p in [a, b] with f(p) = 0

a, Use P,(0.5) to approximate f (0.5) Find an upper bound for the error | (0.5) — P2(0.5)|

using the error formula and compare it to the actual error

b Find a bound for the error | f(x) — Pa(x)| in using P(x) to approximate f(x) on the interval [0, 1)

c Approximate [5 f(x) dx using [9 Pa(x) dx /

d Find an upper bound for the error in (c) using $3} |Ro(x)| dx and compare the bound to the actual error

Repeat Exercise 7 using x9 = 1/6

Find the third Taylor polynomial P(x) for the function f(@) = ( — 1) Inx expanded about Xo=l

a Use P;(0.5) to approximate f (0.5) Find an upper bound for the error | f(0.5) — P,(0.5)| using the error formula and compare it to the actual error

b Find a bound for the error | f(x) — Ps(x)| in using P(x) to approximate f(x) on the

interval (0.5, 1.5]

ce Approximate [23 f(x) de using Jq3 Ps) dx

d Find an upper bound for the error-in (c) using [33 [R(x | dx and compare the bound to the actual error

Let f(x) = 2x cos(2x) — @ ~ 23? and xe = Ô

a Find the third Taylor polynomial P;(x) and use it ïo approximate F(A)

b Use the error formula in Taylor’s Theorem to find an upper bound for the error | #4)

— P;(0.4)| Compute the actual error

c Find the fourth Taylor polynomial P,(x) and use it to approximate f (0.4)

d Use the error formula in Taylor's Theorem to find an upper bound for the error | f (0.4)

— P,{0.4)| Compute the actual error

Find the fourth Taylor polynomial P,(x) for the function f@ = xe ** expanded about

Xo = 0

a Find an upper bound for | f(x) — P.(x)| for0 =x = 0.4

bp Approximate [94 f(x) dx using [5% P4(x) dex

c Find an upper bound for the error in (b) using f 94 Pix) dx

d Approximate f’(0.2) using P¿(0.2) and find the error

Use the error term of a Taylor polynomial to estimate the error involved in using sin x + x to approximate sin 1°

Use a Taylor polynomial about a / 4 to approximate cos 42° to an accuracy of 107%

Let f(x) = (1 — x)! and xq = 0 Find the nth Taylor polynomial P,(x) = f(x) expanded about x9 Find the value of n necessary for P,(x) to approximate f(x) to within 10-6 on

[0, 0.51

Let f(x) = e* and xq = 0 Find the mth Taylor polynomial P,,(x) for f(x) expanded about

Xp Find the value of n necessary for P,(x) to approximate f (x) to within 107° on [6, 0.5] Find for an arbitrary positive integer n, the nth Maclaurin polynomial P,(x) for f(x) =

arctan x

The polynomial Pạ@œ) = 1 — $x? is used to approximate f(x) = cos x in [-3 31 Find a bound for the maximum error

The nth Taylor polynomial for a function f at xo is sometimes referred to as the polynomial

of degree at most ø that ““best'” approximates f near Xo- -

a Explain why this description is accurate

b Find the quadratic polynomial that best approximates-a function f neat x) = 1, if the function has as its tangent line the line with equation y = 4 — 1 when x = 1 and has

fr) = 6.

A Maclaurin polynomial for e* is used to give the approximation 2.5 to e The error bound

in this approximation is established to be E = ¢ 1 Find a bound for the error in E

The error function defined by

erf(x) = zh ev! at

gives the probability that any one of a series of trials will lie within x units of the mean, assuming that the trials have a standard normal distribution This integral cannot be evalu- ated in terms of elementary functions, so an approximating technique must be used

a Integrate the Maclaurin series for e~‘” to show that

2) HN 1#y 2tr1 )

SH) =7 2 Ok + DEN

b The error function can also be expressed in the form

= 2 x2rl

ents) = Fae eT 3-5-::@kE+ 1)"

Verify that the two series agree for k = 0, 1, 2, 3, and 4 [Hint: Use the Maclaurin series fore]

c Use the series in part (a) to approximate erf(1) to within 1077

Use the same number of terms used in part (c) to approximate erf(1) with the series in part (b)

e, Explain why difficulties occur using the series in part (b) to approximate erf(x)

Suppose f € C[a, b], that x, and x, are in [a, b], and that c, and c, are positive constants Show that a number ¢ exists between x, and x, with

Cif 1) + cof 2)

€ị Tứ

#@)=

Use the Mean Value Theorem to show the following:

|cos a ~ cos b| =|a — d| b |sina + sinb| =|a + b|

A function f: [a, b] > R is said to satisfy a Lipschitz condition with Lipschitz constant L

on [a, b} if, for every x, y la, b], | ƒŒœ) — ƒ(y)| = Hà - y\

a Show that if f satisfies a Lipschitz condition with Lipschitz constant L on an interval [ø, b], then ƒ e C[a, b]

b Show that if f has a derivative that is bounded on [a, b] by L, then f satisfies a Lipschitz condition with Lipschitz constant L on [a, b]

c Give an example of a function that is continuous on a closed interval but does not satisfy

a Lipschitz condition on the interval

Arithmetic

The arithmetic performed by a calculator or computer is different from the arithmetic that

we use in our algebra and calculus courses From our past experiences we expect that we will always have as true statements such things as 2 + 2 = 4, 42 = 16, and (V3)? = 3

1.2 Round-Off Errors and Computer Arithmetic 13

Jn standard computational arithmetic we will have the first two, but not the third To

understand why this is true we must explore the world of finite-digit arithmetic

Tn our traditional mathematical world we permit numbers with an infinite number

of nonperiodic digits The arithmetic we use in this world defines V3 as that unique posi- tive number that when multiplied by itself produces the integer 3 In the computational world, however, each representable number has only a fixed, finite number of digits Since

\/3 does not have a finite-digit representation, it is given an approximate representation within the machine, one whose square will not be precisely 3, although it will likely be sut- ficiently close to 3 to be acceptable in most situations In most cases, then, this machine representation and arithmetic is satisfactory and passes without notice or concern, but we must be aware that this is not always true and be alert to the problems that it can produce

Round-off error occurs when a calculator or computer is used to perform real-number calculations This error arises because the arithmetic performed in a machine involves numbers with only a finite number of digits, with the result that calculations are performed with approximate representations of the actual numbers In a typical computer, only a relatively small subset of the real number system is used for the representation of all the real numbers This subset contains only rational numbers, both positive and negative, and stores a fractional part, called the mantissa, together with an exponential part, called the characteristic For example, a single-precision floating-point number used in the IBM

3000 and 4300 series consists of a 1-binary-digit (bit) sign indicator, a 7-bit exponent with

Since 24 binary digits correspond to between 6 and 7 decimal digits, we can assume that this number has at least 6 decimal digits of precision for the floating-point number system The exponent of 7 binary digits gives a range of 0 to 127 However, using only positive integers for the characteristic does not permit an adequate representation of num- bers with small magnitude To ensure that numbers with smali magnitude are equally representable, 64 is subtracted from the characteristic, so the range of the exponential part

The leftmost bit is a zero, which indicates that the number is positive The next seven bits,

1000010, are equivalent to the decimal number

1-25 + 0-25+0-22+0-22+0:22+1-21+0-22= 66 and are used to describe the characteristic, 16°" The final 24 bits indicate that the

mantissa is

1-G2y 41-02 +1- dart 1-deay +i: (2# + 1-(12)%

As a consequence, this machine number precisely represents the decimal number

if 14

CHAPTER 1 " Mathematical Preliminaries

and the next largest machine number is

are used by this system to represent all real numbers With this representation, the number

of binary machine numbers used to represent [16", 16"*"] is constant independent of n

within the limit of the machine; that is, for —-64 = n = 63 This requirement also implies

that the smallest normalized, positive machine number that can be represented is

Numbers occurring in calculations that have a magnitude of less than 16~© result in

what is called underflow and are often set to zero, while numbers greater than 16° result

in overflow and cause the computations to halt

The arithmetic used on microcomputers differs somewhat from that used on main- frame computers In 1985, the [EER (Institute for Electrical and Electronic Engineers) published a report called Binary Floating Point Arithmetic Standard 754-1985 In this report, formats were specified for single, double, and extended precisions, and these stan- dards are generally followed by microcomputer manufacturers who use floating-point hardware For example, the numerical coprocessor for IBM-compatible microcomputers implements a 64-bit representation for a real number, called a long real The first bit is a sign indicator denoted s This is followed by an 11-bit exponent c and a 52-bit mantissa f The base for the exponent is 2 and, to obtain numbers with both large and small magni- tude, the actual exponent is c — 1023 In addition, a normalization is imposed that requires that the units digit be 1, and this digit is not stored as part of the 52-bit mantissa Using this system gives a floating-point number of the form

(- 1} * ge- 1023 * q +),

which provides between 15 and 16 decimal digits of precision and a range of approximately

10% to 10298, This is the form that compilers use with the coprocessor and refer to as

double precision

The use of binary digits tends to conceal the computational difficulties that occur when a finite collection of machine numbers is used to represent all the real numbers To

1.2 Round-Off Errors and Computer Arithmetic

explain the problems that can arise, we will now assume, for simplicity, that machine numbers are represented in the normalized decimal floating-point form

+ O.d, dh dy X 10", 14; <9, 0<a,<=9,

for cach ¡ = 2, ,, where, from what we have just discussed, the IBM mainframe

machines have approximately k = 6 and —78 =n = 76 Numbers of this form will be called decimal machine numbers

Any positive real number y can be normalized to

y = Ody dy dy bess Agag X 10”,

If y is within the numerical range of the machine, the floating-point form of y, denoted by fi), is obtained by terminating the ‘mantissa of y at k decimal digits There are two ways

of performing this termination One method is to simply chop off the digits d,4; dii2 -

to obtain

fl) = O.dy dy dy X 10”,

This method is quite accurately called chopping the number The other method is to add

5 x 10"~#*+Ð to y and then chop to obtain a number of the form

fly) = 0.88 & X 10”,

The latter method is often referred to as rounding the number In this method, dps, =

5, we add one to d; to obtain f(y); that is, we round up If d,,; <5, we merely chop off

all but the first k digits; so we round down

The number zr has an infinite decimal expansion of the form a = 3.14159265 Written in normalized decimal form, we have

If p* is an approximation to p, the absolute error is |p — p*|, and the relative error is

Consider the absolute and relative errors in representing p by p* in the following example

16

EXAMPLE 2

CHAPTER 1 °" Mathematical Preliminartes

a Ifp = 0.3000 x 10! and p* = 0.3100 x 10!, the absolute error is 0.1 and the

relative error is 0.3333 X 1071

b lfp = 0.3000 x 1073 and p* = 0.3100 X 107%, the absolute error is 0.1 X

10~* and the relative error is 0.3333 X 107)

c Ifp = 0.3000 X 10* and p* = 0.3100 X 10%, the absolute error is 0.1 x 10° and the relative error is 0.3333 xX 107!

This example shows that the same relative error, 0.3333 X 107}, occurs for widely varying absolute errors As a measure of accuracy, the absolute error may be misleading and the relative error more meaningful J4

Returning to the machine representation of numbers, we see that the floating-point representation fi( y) for the number y has the relative error

y ~ fly)

ỳ

Tf k decimal digits and chopping are used for the machine representation of

y = O.d; dy dy dy X 10",

are distributed along the real line Because of the exponential form of the characteristic,

the same number of decimal machine numbers are used to represent each of the intervals {0.1, 1], [1, 10], and [10, 100] In fact, within the limits of the machine, the number of decimal machine numbers in [10”, 10"*] is constant for all integers n

In addition to inaccurate representation of numbers, the arithmetic performed in a computer is not exact The arithmetic generally involves manipulating binary digits by various shifting or logical operations Since the actual mechanics of these operations are not pertinent to this presentation, we shall devise our own approximation to computer arithmetic Although our arithmetic will not give the exact picture, it suffices to explain the problems that occur (For an explanation of the manipulations actually involved, the

< x 107 = 10°F,

i 0.1

wb

1.2 Round-Off Errors and Computer Arithmetic 17

reader is urged to consult more technically oriented computer science texts, such as Mano, [97], Computer System Architecture.)

Assume that the floating-point representations fi(x) and f(y) are given for the real numbers x and y and that the symbols ©, ©, ®, © represent machine addition, subtrac- tion, multiplication, and division operations, respectively We will assume a finite-digit arithmetic given by

x@y = fA@ +A), x @y = MAC x OY xO©y=/Œ@ ~ #@), 2x Oy = AGH) + AO)

This arithmetic corresponds to performing exact arithmetic on the floating-point represen- tations of x and y and then converting the exact result to its finite-digit floating-point representation

Suppose that x = i, y= 3 and that five-digit chopping is used for arithmetic calculations involving x and y Table 1.1 lists the values of these computer-type operations on fl(x) =

0.33333 X 10° and f(y) = 0.71428 x 10° =z

Operation Resut Actual value Absolute error Relative error

xQy 0.10476 Xx 10! 22/21 0.196 X 1077 0.182 x 1077

yOx 0.38095 x 10° §/21 0.238 x 107 0.625 X 1075 x@y 0.23809 x 10” 5/21 0.524 x 10”? 0.220 < 10” y@x 0.21428 X 10° 15/7 0.571 x 10”! 0.267 x 107*

Operation Result Actual value Absolute error Relative error

yOu 0.30000 x 1074 0.34714 < 10”! 0.471 X 107° 0.136 (yOwuOw 0.27000 < 10' 0.31243 X 10° 0.424 0.136 (qœw@@&»y 0.29629 X 10° 0.34285 X 10’ 0.465 0.136 u@yv 0.98765 X 10° 0.98766 X 10° 0.161 x 10! 0.163 x 107

18

EXAMPLE 4

CHAPTER 1 © Mathematical Preliminartes

One of the most common error-producing calculations involves the cancellation of significant digits due to the subtraction of nearly equal numbers Suppose two nearly equal numbers x and y, with x > y, have the k-digit representations

fle) = 0.d; dy dy O41 Open - % X 10”,

and fi y) = 0.4; dy dp pc: B+2 - 8 X 10”

The floating-point form of x — y is

AAG) — AQ) = 0.6541 Ope + + OR X 10",

where 0.0541 Tyan ++ Oe = 0-41 Men + + Me — 0.Bp+i Bp+a.-cỔy

The floating-point number used to represent x — y has only k — p digits of significance However, in most calculation devices, x — y will be assigned k digits, with the last p being either zero or randomly assigned Any further calculations involving x — y retain the problem of having only k — p digits of significance, since a chain of calculations cannot

be expected to be more accurate than its weakest portion

If a finite-digit representation or calculation introduces an error, further enlargement

of the error occurs when dividing by a number with small magnitude (or equivalently, when multiplying by a number with large magnitude) Suppose, for example, that the number z has the finite-digit approximation z + 6, where the error 4 is introduced by representation or by previous calculation Dividing by ¢ # 0 results in the approximation

z/e~ ƒ( + 8/0)

Suppose € = 107”, where n > 0 Then

z/£= zx 10"

and HM + 8/9) = ( + ð) x 10"

Thus, the absolute error in this approximation,

|8| multiplied by the factor 10”

The loss of accuracy due to round-off error can often be avoided by a careful se- quencing of operations or reformulation of the problem, as illustrated in the final two examples

, is the original absolute error

The quadratic formula states that the roots of ax? + bx + c = 0, whena ¥ 0, are

In this equation, ð2 is much larger than 4ac, so the numerator in the calculation for

x, involves the subtraction of nearly equal numbers Suppose we perform the calculations for x, using four-digit rounding arithmetic First we have

On the other hand, the calculation for x, involves the addition of the nearly equal numbers

—band —Vb* — 4ac and presents no problem:

with the small relative error 6.2 x 107*

The “rationalization” technique can also be applied to give the alternate form for x2:

20

EXAMPLE 5

Table 1.3

CHAPTER 1 * Mathematical Preliminaries

Evaluate f(x) = x3 — 6x? + 3x — 0.149 atx = 4.71 using three-digit arithmetic

Table 1.3 gives the intermediate results in the calculations Note that the three-digit chopping values simply retain the leading three digits, with no rounding involved, and differ significantly from the three-digit rounding values

Polynomials should always be expressed in nested form before performing an evalu- ation, since this form minimizes the number of required arithmetic calculations The decreased error in Example 5 is due to the fact that the number of computations has been

it

an

as

1.2 Round-Off Errors and Computer Arithmetic 21

reduced from four multiplications and three additions to two multiplications and three additions One way to reduce round-off error is to reduce the number of error-producing computations

Repeat Exercise 5 using four-digit rounding arithmetic

Repeat Exercise 5 using three-digit chopping arithmetic

Repeat Exercise 5 using four-digit chopping arithmetic The first three nonzero terms of the Maclaurin series for the arctangent function are x— px + Bx, Compute the absolute error and relative error in the following approximations

of 7 using the polynomial in place of the arctangent:

a 4 [ arctan 2 + arctan ¡1 bo a 16 arctan 4 =4 arctan 7g

“

The number e is sometimes defined by e = > sọ where n! = n(n — 1)-+-2-1,ifn 0,

n=O

22 CHAPTER 1 © Mathematical Preliminaries

a gx? — 2x + b= 0 1,2 4 123 LL

b Bx Pam B= 0

c, 1.002x7 - 11.01x + 0.01265 = 0

d 1.002x? + 11.01x + 0.01265 = 0

Repeat Exercise 11 using four-digit chopping arithmetic

Using the IBM mainframe format, find the decimal equivalent of the following floating-point

a 0 1000011 10101001001 100000000000

b 1 1000011 101010010011000000000000 c0 0111111 010001 111000000000000000

a Show that both formulas are algebraically correct

b Using the data (xo, yo) = (1.31, 3.24) and Œ¡, yị) = (1.93, 4.76) and three-digit rounding arithmetic, compute the x-intercept both ways Which method is better and why?

An approximate value of e~> correct to three digits is 6.74 x 107% Which formula, (a) or

(b), gives the most accuracy, and why?

The two-by-two linear system

ax + by = e,

ex + dy =f, where a, b, c, d, e, f are given, can be solved for x and y as follows:

set m = c/a, provided a # 0;

d, =d— mb;

Repeat Exercise 17 using four-digit chopping arithmetic

a Show that the polynomial nesting technique described in Example 5 can also be applied

to the evaluation of „

ƒ@) = 10162 ~ 4.622 — 3.116” + 12.⁄2¿7 — 1.99

b Use three-digit rounding arithmetic, the assumption that e!3 = 4.62, and the fact

that e"* = (2”)” to evaluate f (1.53) as given in part (a)

c Redo the calculation in part (b) by first nesting the calculations

Compare the approximations in parts (b) and (c) to the true three-digit result f(1.53) =

~7.61

A rectangular parallelepiped has sides 3 cm (centimeters), 4 cm, and 5 cm, measured only

to the nearest centimeter What are the best upper and lower bounds for the volume of this parallelepiped? What are the best upper and lower bounds for the surface area?

Suppose that fi(y) is a k-digit rounding approximation to y Show that

y

of <05 x 107, [Hint: If dy., <5, then fly) = Od; dy dy X 10% đ,„ị = 5, then f(y) = Ody dy de x10” + 10*1

mì _ —_—_m kj— kiếm — ĐI

The binomial coefficient

describes the number of ways of choosing a subset of k objects from a set of m elements

a Suppose decimal machine numbers are of the form

+0.d;dpdydyX 10", 1ed,=9, 08489, ad ~-15Sn=15

What is the largest value of m for which the binomial coefficient (#) can be computed

by the definition without causing overflow?

b Show that (#) can also be computed by

i) (GSE)

c, What is the largest value of m for which the binomial coefficient (2) can be computed

by the formula in part (b) without causing overflow?

d Using four-digit chopping arithmetic, compute the number of possible 5-card hands in a

52-card deck Compute the actual and relative error

Let f < C[a, b] be a function whose derivative f’ exists on (a, b) Suppose f is to be evaluated at x9 in (a, b), but instead of computing the actual value f(x), the approximate

value, (x9) is the actual value of f at xo + €; that is, # (to) = f@ + 9).

24 CHAPTER 1 * Mathematical Preliminaries

a Use the Mean Value Theorem to estimate the absolute error | f(x) ~ f(xq)| and the

Fo) — Fo)

fo)

b Ife = 5 X 107° and xq = 1, find bounds for the absolute and relative errors for

i f(x) = e* ii f@) = sinx

c Repeat part (b) with e = (5 X 10° )xp and xo = 10

24, The opening example to this chapter described a physical experiment involving the temper- ature of a gas under pressure In this application, we were given P = 1.00 atm, V = 0 100mŠ,

N = 0.00420 mole, and R = 0.08206 Solving for 7 in the ideal gas law gives

ERG ROS ST RE ek ee

1 3 Algorithms and Convergence:

The examples in Section 1.2 demonstrate ways that machine calculations involving ap- proximations can result in the growth of round-off errors Throughout the text we will be examining approximation procedures, called algorithms, involving sequences of calcula- tions An algorithm is a procedure that describes, in an unambiguous manner, a finite sequence of steps to be performed in a specified order The object of the algorithm is to implement a numerical procedure to solve a problem or approximate a solution to the problem

A pseudocode is used for describing the algorithms This pseudocode specifies the form of the input to be supplied and the form of the desired output Not all numerical procedures give satisfactory output for arbitrarily chosen input As a consequence, a stop- ping technique independent of the numerical technique is incorporated into each algorithm

so that infinite loops are unlikely to occur

Two punctuation symbols are used in the algorithms: the period (.) indicates the

termination of a step, while the semicolon (;) separates tasks within a step Indentation is

used to indicate that groups of statements are to be treated as a single entity

Looping techniques in the algorithms are either counter controlled, for example,

of the more advanced topics in the latter half of the book are difficult to programm in certain languages, especially if large systems or complex arithmetic are involved

The algorithms are liberally laced with comments These are written in italics and contained within parentheses to distinguish them from the algorithmic statements

Step 1 Set SUM = 0

Siep2 Fori= 1,2, ,Ndo

set SUM = SUM + x;

Step 3 OUTPUT (SUM)

26

Definition 1.16

CHAPTER 1 * Mathematical Preliminaries

An algorithm to solve this problem is

INPUT value x, tolerance TOL, maximum number of iterations M

OUTPUT degree N of the polynomial or a message of failure

Step 2 While N = M do Steps 3-5

Step 3 Set SIGN = — SIGN,

SUM = SUM + SIGN- TERM;

POWER = POWER -y;

Whether the output is a value for N or the failure message depends on the precision

of the computational device being used

We are interested in choosing methods that will produce dependably accurate results for a wide range of problems One criterion we will impose on an algorithm whenever’ possible is that small changes in the initial data produce correspondingly small changes in the final results An algorithm that satisfies this property is called stable; it is unstable when this criterion is not fulfilled Some algorithms will be stable for certain choices of initial data but not for all choices We will characterize the stability properties of algo- rithms whenever possible

To consider further the subject of round-off error growth and its connection to algo- rithm stability, suppose an error with magnitude Ep is introduced at some stage in the calculations and that the magnitude of the error after n subsequent operations is denoted

by E,, The two cases that arise most often in practice are defined as follows

Suppose that E,, represents the magnitude of an error after n subsequent operations If

E, ~= CnEp, where C is a constant independent of n, the growth of error is said to be linear H E„ ~= C"Eạ, for some C > 1, the growth of error is called exponential

Figure 1.9

EXAMPLE 2 ở

tà "

1.3 Algorithms and Convergence

Linear growth of error is usually unavoidable and when C and Ep are small the results are generally acceptable Exponential growth of error should be avoided, since the term C” becomes large for even relatively smail values of n This leads to unacceptable inaccu- tacies, regardless of the size of Ey As @ consequence, an algorithm that exhibits linear growth of error is stable, while an algorithm exhibiting exponential error growth

is unstable (See Figure 1.9.)

0.10000 x.10', 0433333 x10, 011111 10,

0.37036 x 10”1, 0.12345 x 1071,

The round-off error introduced by replacing 3 by 0.33333 produces an error of only (0.33333)" < 107” in the nth term of the sequence This method of generating the se- quence is stable

Another way to generate the sequence is to define pp = 1 p = 3, and compute, for

Table 1.4

CHAPTER 1 © Mathematical Preliminaries

for any pair of constants C, and C) To verify this,

racies found in the entries of Table 1.4 a &

n Computed p,, Correct value p,

To reduce the effects of round-off error, we can use high-order digit arithmetic such

as the double- or multiple-precision option available on most digital computers A disad- vantage in using double-precision arithmetic is that it takes more computer time, and the growth of round-off error is not eliminated but only postponed until subsequent computa- tions are performed

One approach to estimating round-off error is to use interval arithmetic (that is, to retain the largest and smallest possible values at each step) so that, in the end, we obtain

an interval that contains the true value Unfortunately, we may have to find a very small interval for reasonable implementation It is also possible to study error from a statistical standpoint This study, however, involves considerable analysis and is beyond the scope of this text Henrici [72], pages 305-309, presents a discussion of a statistical approach to estimate accumulated round-off error

Since iterative techniques involving sequences are often used, the section concludes with a brief discussion of some terminology used to describe the rate at which convergence occurs when employing a numerical technique In general, we would like the technique to converge as rapidly as possible The following definition is used to compare the conver- gence rates of various methods

Suppose {,,} is a sequence known to converge to zero and {a,,},-) converges to a number

a If a positive constant K exists with

ja, — al = KB, for large n,

then we say that {e,}7_, converges to a a with rate of convergence O(8,,) (This is read

“big oh of 8, ”) This is indicated by writing a, = @ + O(B,) of &, > & with rate of

Suppose that the sequences {a,} and {@,} are described by a, = (1 + 1) /n? and &, =

(n + 3)/ for each integer n = 1 Although lim, @ = 0 and lim,_, &, = 0, the

sequence {@,} converges to this limit much faster than the sequence {a,,}

In fact, using five-digit rounding arithmetic gives the entries in Table 1.5

so a, =O+ O|- n while @, = 0 + Ô| =2] - 2

The rate of convergence of {a,,} to zero is similar to the convergence of {1 / n} to zero, while {@,} converges to zero at a rate similar to the more rapidly convergent sequence -

(L5) |FŒ) - L|Ị<KG(Œ) — for sufficiendy smailh,

then we write F(h) = L + O(G(h))

From Example 2(b) of Section 1.1 we know that by using a third Taylor polynomial,

cosh = 1 — }lŸ + Gat cos Eh)

30 CHAPTER 1 * Mathematical Preliminaries

for some number é(h) between zero and h

Consequently,

cosh + 3h? = 1 + x7h* cos E(h)

This implies that

cosh +ih = 1+ Oh, since |(eos h+4$h)- 1| = lủ cos 0|“ ain

The implication is that cos h + 5 i? converges to its limit, 1, at least as fast as h*

10 1

1 a Use three-digit chopping arithmetic to compute the sum =e = first by 2 Tá at aad

+ im and then by 75g te +37 Hi +: ¬ Which method is more accurate and why?

& 6 Son e~2 Gopi ao ar

3 The Maclaurin series for the arctangent function converges for all values of x and is given

by

Š x?m 1

arctan x = (—1⁄!———

n=l Qn ~ 1)

Recall that tan r/4 = 1

a Determine the number of terms of the series that need to be summed to ensure that

\4 arctan 1 — a|< 107%,

b The single precision version of the scientific programming language FORTRAN re- quires the value of a to be within 10-7 How many terms of this series must be summed

to obtain this degree of accuracy?

4, Exercise 3 details a rather inefficient means of obtaining an approximation to 7 The method can be improved substantially by observing that a /4 = arctan ị + arctan 3 and evaluating the series for arctan at 4 and at % Determine the number of terms that must be summed to ensure an approximation to a to within 1077