Numerical Methods

By Germund Dahlquist and Åke Björck

"Substantial, detailed and rigorous . . . readers for whom the book is intended are admirably served." — MathSciNet (Mathematical Reviews on the Web), American Mathematical Society.
Practical text strikes fine balance between students' requirements for theoretical treatment and needs of practitioners, with best methods for large- and small-scale computing. Prerequisites are minimal (calculus, linear algebra, and preferably some acquaintance with computer programming). Text includes many worked examples, problems, and an extensive bibliography.

• Language: English
• Publisher: Dover Publications
• Release date: Apr 26, 2012
• ISBN: 9780486139463

Book preview

7.

1.1.INTRODUCTION

Mathematics is used in one form or another within most of the areas of science and industry. There has always been a close interaction between mathematics on the one hand and science and technology on the other. During the present century, advanced mathematical models and methods have been used more and more even within other areas—for example, in medicine, economics, and social science.

Very often, applications lead to mathematical problems which in their complete form cannot be conveniently solved with exact formulas. One often restricts oneself then to special cases or simplified models which can be exactly analyzed. In most cases, one thereby reduces the problem to a linear problem—for example, a linear differential equation. Such an approach can be very effective, and leads quite often to concepts and points of view which can at least qualitatively be used even in the unreduced problem.

But occasionally such an approach does not suffice. One can instead treat a less simplified problem with the use of a large amount of numerical calculation. The amount of work depends on the demand for accuracy. With computers, which have been developed during the past twenty-five years, the possibilities of using numerical methods have increased enormously. The points of view which one has taken toward them have also changed.

To develop a numerical method means, in most cases, that one applies a small number of general and relatively simple ideas. One combines these ideas in an inventive way with one another and with such knowledge of the given problem as one can obtain in other ways—for example, with the methods of mathematical analysis. Some knowledge of the background of the problem is also of value; among other things, one should take into account the order of magnitude of certain numerical data of the problem.

In this book we shall illustrate the use of the general ideas behind numerical methods on some problems which often occur as subproblems or computational details of larger problems, though as a rule they occur in a less pure form and on a larger scale than they do here. When we present and analyze numerical methods, we use to some degree the same approach which was described first above: we study in detail special cases and simplified situations, with the aim of uncovering more generally applicable concepts and points of view which can be a guide in more difficult problems.

In this chapter we shall throw some light upon some important ideas and problems in abbreviated form. A more systematic treatment comes in the chapters following.

1.2.SOME COMMON IDEAS AND CONCEPTS IN NUMERICAL METHODS

One of the most frequently recurring ideas in many contexts is iteration (from the Latin iteratio, repetition) or successive approximation. Taken generally, iteration means the repetition of a pattern of action or process. Iteration in this sense occurs, for example, in the repeated application of a numerical process—perhaps very complicated and itself containing many instances of the use of iteration in the somewhat narrower sense to be described below—in order to successively improve previous results. To illustrate a more specific use of the idea of iteration, we consider the problem of solving an equation of the form

Here F is a differentiable function whose value we can compute for any given value of the real variable x (within a certain interval). Using the method of iteration, one starts with an initial approximation x0, and computes the sequence

Each computation of the type

is called an iteration. If the sequence {xn} converges to a limiting value α, then we have lim F(xn) = F(α), so x = α satisfies the equation x = F(x). As n grows, we would like the numbers xn to be better and better estimates of the desired root. One stops the iterations when sufficient accuracy has been attained.

A geometric interpretation is shown in Fig. 1.2.1. A root of Eq. (1.2.1) is given by the abscissa (and ordinate) of an intersection point of the curve y = F(x) and the line y = x. Using iteration and starting from (x0, F(x0)) we obtain x1 = F(x0) and the point x1 on the x-axis is obtained by first drawing a horizontal line from the point (x0, F(x0)) = (x0, x1,) until it intersects the line y = x in the point (x1, x1). From there we draw a vertical line to (x1, F(x1)) = (x1, x2) and so on. In Fig. 1.2.1 it is obvious that {xn} converges monotonely to α. Figure 1.2.2 shows a case where F is a decreasing function. There we also have convergence but not monotone convergence: the successive iterates xn are alternately to the right and to the left the of root α.

But there are also divergent cases, exemplified by Figs. 1.2.3 and 1.2.4. One can see geometrically that the quantity which determines the rate of convergence (or divergence) is the slope of the curve y = F(x) in the neighborhood of the root. Indeed, from the mean value theorem we have

where ξn lies between xn−1 and xn. Thus convergence is faster the smaller |F′(x)| is in a neighborhood of the root. Convergence is assured if |F′(x)| < 1 for all x in a neighborhood of the root containing x0 and x1. But if |F′(α)| > 1, xn converges to α only in very exceptional cases, no matter how close to α one chooses to x0 (x0 ≠ α).

Fig. 1.2.1

Fig. 1.2.2

Fig. 1.2.3

Fig. 1.2.4

Example 1.2.1.A Fast Method for Calculating Square Roots

The equation x² = c can be written in the form x = F(x), where

(Fig. 1.2.5). The limiting value is α = c¹/² and F′(α) = 0. (Show this!) Thus we set

Fig.1.2.5

For c = 2, x0 = 1.5, we get x(1.5 + 2/1.5) = 1.4167, x= 1.414214 ....

One can get a good value for x0 with a slide rule, but, as can be seen, a rougher value for x0 suffices. One can in fact show that if xn has t correct digits, then xn+1 will have at least 2t − 1 correct digits. The above iterative method for calculating square roots is used quite generally on both desktop calculators and computers.

Iteration is one of the most important aids for the practical as well as theoretical treatment of both linear and nonlinear problems. One very common application of iteration is to the solution of systems of equations. In this case {xn} is a sequence of vectors, and F is a vector-valued function. When iteration is applied to differential equations, {xn} means a sequence of functions, and F(x) means an expression in which integration or other operations on functions may be involved. A number of other variations on the very general idea of iteration will be given in later chapters. For each application it is of course necessary to find a suitable way to put the equations in a form similar to Eq. (1.2.1) and to choose a suitable initial approximation. One has a certain amount of choice in these matters which should be used in order to reduce the number of iterations one will have to make.

Example 1.2.2

The equation x² = 2 can also be written, among other ways, in the form x = 2/xgave rapid convergence when iteration was applied. On the other hand, the formula xn+1 = 2/xn gives a sequence which goes back and forth between x0 (for even n) and 2/x0 (for n odd)—the sequence does not converge.

Another often recurring idea is that one locally (that is, in a small neighborhood) approximates a complicated function with a linear function. We shall illustrate the use of this idea in the solution of the equation f(x) = 0. Geometrically, this means that we are seeking the intersection point between the x-axis and the curve y = f(x) (Fig. 1.2.6). Assume that we have an approximating value x0 to the root. We then approximate the curve with its tangent at the point (x0, f(x0)). Let x1 be the abscissa of the point of intersection between the x-axis and the tangent. Normally x1 will be a much better approximation to the root than x0. In most cases, x1 will have nearly twice as many correct digits as x0, but if x0 is a very poor initial approximation, then it is possible that x1 will be worse than x0.

A combination of the ideas of iteration and local linear approximation gives rise to a much used and, ordinarily, rapidly convergent process which is called Newton-Raphson’s method (Fig. 1.2.7). In this iterative method xn+1 is defined as the abscissa of the point of intersection between the x-axis and the tangent to the curve y = f(x) in the point (xn, f(xn)).

The approximation of the curve y = f(x) with its tangent at the point (x0, f(x0)) is equivalent to replacing the function with the first-degree terms in its Taylor series about x = x0. The corresponding approximation for functions of many variables also has important uses.

Fig. 1.2.6

Fig. 1.2.7

Another way (instead of drawing the tangent) to approximate a curve locally is to choose two neighboring points on the curve and to approximate the curve with the secant which joins the two points (Fig. 1.2.8). In a later chapter, we shall discuss more closely the secant method for the solution of equations, which is based on the above approximation.

The same secant approximation is useful in many other contexts. It is, for instance, generally used when one reads between the lines in a table of numerical values. In this case the secant approximation is called linear interpolation.

When the secant approximation is used in the approximate calculation of a definite integral,

numerical integration (Fig. 1.2.9), it is called the trapezoidal rule. With this method, the area between the curve y = y(x) and the x-axis is approximated with sum T(h) of the areas of a series of parallel trapezoids. Using the notation of Fig. 1.2.9 we have

(in the figure, n = 4). We shall show in a later chapter that the error (T(h) — I) in the above approximation is very nearly proportional to h² when h is small. One can then, in principle, attain arbitrarily high accuracy by choosing h sufficiently small, except that the computational work involved (the number of points where y(x) must be computed) is inversely proportional to h. Thus the computational work grows rapidly as one demands higher accuracy (smaller h).

Fig. 1.2.8

Fig. 1.2.9

Numerical integration is a fairly common problem because in fact it is quite seldom that the primitive function can be analytically calculated in a finite expression containing only elementary functions. It is not possible, for example, for such simple functions as exp(x²) or (sin x)/x. In order to obtain higher accuracy with significantly less work than the trapezoidal rule requires, one can use one of the following two important ideas:

(a)Local approximation of the integrand with a polynomial of higher degree (or with a function of some other class, for which one knows the primitive function).

(b)Computation with the trapezoidal rule for several values of h and then extrapolation to h = 0, so-called Richardson extrapolation or the deferred approach to the limit, with the use of general results concerning the dependence of the error upon h.

The technical details for the various ways of approximating a function with a polynomial, among others Taylor expansions, interpolation, and the method of least squares, are treated in later chapters.

has an error approximately proportional to the square of the step size. Thus, using two step sizes, h and 2h, one has:

(T(h) — T(2h)), to T(h), one should get an estimate of I which is much better than T(h). In Chap. 7 we shall see that the improvement is in most cases quite striking. That chapter also contains a further development of the extrapolation idea, Romberg’s method.

Example 1.2.3

Compute

for f(x) = x³ and f(x) = x⁴ by the trapezoidal method. Extrapolate and compare with the exact results.

We have seen above that some knowledge of the behavior of the error can, together with the idea of extrapolation, lead to a powerful method for improving results. Such a line of reasoning is useful not only for the common problem of numerical integration, but also in many other types of problems.

Approximate solution of differential equations is a very important problem which now, since the development of computers, one has the possibility of treating to a much larger extent than previously. Nearly all the areas of science and technology contain mathematical models which lead to systems of ordinary or partial differential equations. Let us consider a case of just one ordinary differential equation,

with initial condition y(0) = p. The differential equation indicates, at each point (x, y), the direction of the tangent to the solution curve which passes through the point in question. The direction of the tangent changes continuously from point to point, but the simplest approximation (which was proposed as early as the 18th century, by Euler) is that one studies the solution for only certain values of x = 0, h, 2h, 3h, ... (h is called the step or step length) and assumes that dy/dx is constant between the points. In this way, the solution curve is approximated by a polygon segment (Fig. 1.2.10) which joins the points (0, y0), (h, y1), (2h, y2), ... where

Fig. 1.2.10

Thus we have a simple recursion formula, (Euler’s method):

During the computation, each yn occurs first on the left-hand side, then recurs later on the right-hand side of an equation: hence the name recursion formula. (One could also call Eq. (1.2.4) an iteration formula, but one usually reserves the word iteration for the special case where a recursion formula is used solely as a means of calculating the limiting value.)

The only disadvantage of the above method is that the step length must be quite short if reasonable accuracy is desired. In order to improve the method one can—just as in the case of numerical integration—choose either the use of local approximation with a polynomial of higher degree or the use of extrapolation to h = 0. As we shall see in Chap. 8, a combination of these two possibilities gives good results.

In Eq. (1.2.3) the derivative y′(nh) is replaced by a difference quotient (yn+1, — yn)/h. The approximation of derivatives with difference quotients is also one of the most frequently encountered devices in the construction of numerical methods, among other things in the numerical treatment of more complicated differential equations. Observe, though, that (yn+1 — yn)/h, which is the slope of the secant line between (nh, yn) and ((n + 1)h, yn+1), in reality is a better approximation for the derivative at the midpoint of the interval [nh, (n + 1)/h] than at its left end point (see Fig. 1.2.11). The value of the derivative at the point x = nh is more accurately approximated by a centered difference quotient (see Fig. 1.2.11),

Fig. 1.2.11

The above approximation is in most situations preferable to the one mentioned previously. There are, however, situations where the first mentioned suffices, but where the centered difference quotient is entirely unusable, for reasons which have to do with how errors are propagated to later stages in the calculation. We shall not discuss this more closely here, but mention it only to intimate some of the surprising and fascinating mathematical questions which can arise in the study of numerical methods.

Higher derivatives are approximated with higher differences, that is, differences of differences, another central concept in numerical calculation. We define:

etc.

For simplicity one often omits the parentheses and writes, for example, Δ²y5 instead of (Δ²y)5 The coefficients that appear here in the expressions for the higher differences are, by the way, the binomial coefficients. In addition, if we denote the step length by Δx instead of by h, we get the following formulas, which are easily remembered:

Here we mean the value of the derivative for an x which lies right between the largest and the smallest x where the corresponding value of y is needed, so that the difference is defined. Hence, for example,

The approximation of Eq. (1.2.5) can be interpreted as an application of Eq. (1.2.6) with Δx = 2h (or else as the mean of the estimates which one gets according to Eq. (1.2.6) for y′((n + )h) and y′((n )h)).

The estimates which one gets with Eqs. (1.2.5), (1.2.6), and (1.2.7) have errors which are approximately proportional to h², assuming that the values of y are exact. With the use of the difference quotient in Eq. (1.2.3), however, the error is approximately proportional to h. This can be shown to imply that the error in the results obtained with Euler’s method is also proportional to h (not h²).

When the values of the function have errors (for example, when they are rounded numbers), the difference quotients become more and more uncertain the less h is. Thus if one wishes to compute the derivatives of a function given by a table, one should as a rule use a step length which is greater than the table step.

Example 1.2.4

For y(x) = cos (x) one has, using a. six-figure table:

Using Eq. (1.2.5) one gets y′(0.60) ≈ (0.819648 − 0.830941)/0.02 = −0.56465.Using Eq. (1.2.7) one gets y″(0.60) ≈ −83·10−6/(0.01)² = −0.83. The correct values are, with six decimals, y′(0.60) = −0.564642, y″(0.60) = −0.825336. The arrangement of the numbers in the example is called a difference scheme.

In many of the applications in which differential equations occur, x is the time variable and the differential equation expresses the rule or law of nature which directs the changes in the given system. The method of calculation of Eq. (1.2.4) means, then, that one simulates the passing of time. Simulations analogous to the above-mentioned are often used whenever the changes in the system are described with a mathematically much more complicated type of equation than an ordinary differential equation. In recent years, computers have been used to simulate automobile traffic flow and battles between tank units, among other things. Using simulation, one often bypasses the conventional mathematical formulation of the problem, as, for example, a system of differential equations. Instead, one proceeds directly from a verbal or graphical description of the system to a computer program. The technique of simulation is also of great value in studying the influence of random factors on a complicated system. In this connection one uses so-called random numbers; the values of certain variables are determined by a process comparable to dice throwing.

We have now seen a variety of ideas and concepts which can be used in the development of numerical methods. A small warning is perhaps warranted here: it is not certain that the methods will work as well in practice as one might expect. This is because approximations and the restriction of numbers to a certain number of decimals or digits introduce errors which are propagated to later stages of a calculation. The manner in which errors are propagated is decisive for the practical usefulness of a given numerical method. We shall examine such questions first in Example 1.3.3 and, above all, in Chap. 2. Later chapters will treat propagation of errors in connection with various typical problems. The risk that error propagation may upstage the desired result of a numerical process should not, however, dissuade one from the use of numerical methods. It is often wise, though, to experiment with a proposed method on a simplified problem before using it in a larger context. As a rule, a mixture of careful experiment and analysis leads to the best results.

REVIEW QUESTIONS

1.Make a list of the concepts and ideas which have been introduced. Review their use in the various types of problems mentioned.

2.Discuss the convergence condition and the rate of convergence of the method of iteration for solving x = F(x).

3.What is the trapezoidal rule? What is said about the dependence of its error on the step length h?

PROBLEMS

(In these problems, six-place tables should be used.)

to five decimal places using the method in Example 1.2.1. Begin with x0 = 3 and check the final result in a table of square roots.

(a)to six decimals, by determining the primitive function.

(b)with the trapezoidal rule, step length h .

(c)using extrapolation to h = 0 on the results which one gets with h .

(d)Compute the ratio between the error in the result in (c) to that of (b).

3.What is the relationship between xn+1 and xn in the application of Newton’s method to the equation f(x) = 0? (Set up the equation for the tangent to the curve y = f(x) in the point (xn, f(xn)).) What formula does one get when f(x) = x² — c? Have you seen this before?

4.Integrate numerically the differential equation dy/dx = y, with initial condition y(0) = 1, to x = 0.4. Use Euler’s method:

(a)with step length h = 0.2.

(b)with h = 0.1.

(c)Extrapolate to h = 0, using the fact that the error is approximately proportional to the step length (not to the square of the step length). Compare the result with the exact solution to the differential equation. What is the ratio between the errors in the results in (b) and (c), respectively.

(d)How many steps would one have needed in order to attain, without using extrapolation, the same accuracy as was obtained in (c)?

5.In Example 1.2.4 we computed y″(0.6) for y = cos (x), with step length h = 0.01.Make similar calculations using h = 0.1, h = 0.05, and h = 0.001.Which value of h gives the best result (using a six-place table)? Discuss qualitatively the influences of both the rounding errors in the table values and the error in the approximation of a derivative with a difference quotient on the result for various values of h.

6.Show that F′(α) = 0 in Example 1.2.1.

1.3.NUMERICAL PROBLEMS AND ALGORITHMS

1.3.1.Definitions

By a numerical problem we mean a clear and unambiguous description of the functional connection between input data—that is, the independent variables in the problem—and output data—that is, the desired results. Input and output data consist of a finite number of real quantities. (Since a complex number is a pair of real numbers, complex input and output data is included in this definition.) Input and output data are thus representable by finite dimensional vectors. The functional connection can be expressed in either explicit or implicit form.

By an algorithm for a given numerical problem we mean a complete description of well-defined operations through which each permissible input data vector is transformed into an output data vector. By operations we mean here arithmetic and logical operations which a computer can perform, together with references to previously defined algorithms. (The concept algorithm can be analogously defined for problems completely different from numerical problems, with other types of input data and fundamental operations—for example, inflection, merging of words, and other transformations of words in a given language.)

For a given numerical problem one can consider many differing algorithms. These can give approximate answers which have widely varying accuracy.

Example 1.3.1

To determine the largest real root of the equation

x³ + a2x² + a1x + a0 = 0,

with real coefficients a0, a1, a2, is a numerical problem. The input data vector is (a0, a1, a2). The output data is the root x; it is an implicitly defined function of the input data. An algorithm for this problem can be based on Newton-Raphson’s method, supplemented with rules for how the initial approximation should be chosen and how the iteration process is to be terminated. One could also use other iterative methods, or even algorithms based upon Cardan’s exact solution of the cubic equation. Cardan’s solution uses square roots and cube roots, so one needs to assume that algorithms for the computation of these functions have been specified previously.

One often begins the construction of an algorithm for a given problem by breaking down the problem into subproblems in such a way that the output data from one subproblem is the input data to the next subproblem. Thus the distinction between problem and algorithm is not always so easy to make. The essential point is that, in the formulation of the problem, one is only concerned with the initial state and the final state. In an algorithm, however, one should clearly define each step along the way, from start to finish.

Example 1.3.2

The problem of solving the differential equation

with boundary conditions y(0) = 0, y(5) = 1, is, according to the definition stated above, not a numerical problem. This is because the output data is the function y, which cannot, in any conspicuous way, be specified by a finite number of parameters. The above problem is a mathematical problem, which can be approximated with a numerical problem if one specifies the output data to be values approximating y(x) for x = h, 2h, 3h, ..., 5 — h, and one approximates the derivative with a difference quotient according to Eq. (1.2.7). In this way, one gets a system of nonlinear equations with 5/h − 1 unknowns. We shall not go further here into how the domain of variation of the unknowns must be restricted in order to show that the problem has a unique solution. This can be done, however, and one can also give a number of algorithms for solving the system, some good and some bad with respect to the number of calculations made and the accuracy obtained.

1.3.2.Recursive Formulas; Horner’s Rule

One of the most important and interesting parts of the preparation of a problem for a computer is to find a recursive description of the task. Sometimes an enormous amount of computation can be described by a small set of recursive formulas. Euler’s method for the step-by-step solution of ordinary differential equations (p. 9) is an example. Other examples will be given in this section and in Sec. 1.3.3. See also the problems at the end of the chapter.

A common computational task is the evaluation of a polynomial, at a given point z where, say,

p(z) = a0z³ + a1z² + a2z + a3.

This can be reformulated as

p(z) = ((a0z + a1)·z + a2)·z + a3.

For computation by hand, the following scheme, Horner’s rule, illustrates the algorithm indicated by the above reformulation:

Example 1.3.3

Compute p(8), where p(x) = 2x³ + x + 7.

Horner’s rule for evaluating a polynomial of degree n,

p(x) = a0xn + a1xn − ¹ + ... + an − 1x + an,

at a point z, is described by the recursive formula:

If the intermediate bi, are of no interest, then, in most programming languages, the algorithm can be described without subscripts for the bi, such as in the flowchart in Fig. 1.3.1 and the corresponding Algol-fragment:

(The symbol := is read is given the value of.)

Fig. 1.3.1

Sometimes, however, the bi are of intrinsic interest because of the following result, often called synthetic division:

THEOREM 1.3.1

where the bi are defined by Eq. (1.3.1).

Proof. Denote the right-hand side by g(x). Then

Hence, by Eq. (1.3.1),

and the theorem is proved.

Synthetic division is used, for instance, in the solution of algebraic equations, when already-computed roots are sucessively eliminated. Then, after each elimination, one can deal with an equation of lower degree. This process is called deflation. In Chap. 6, however, it is shown that some care is necessary in the numerical application of this idea.

The proof of the following useful relation is left as an exercise to the reader (the bi are defined by Eq. (1.3.1)): If

Further applications of synthetic division are given in the problems at the end of this chapter.

1.3.3.An Example of Numerical Instability

Thus far, we have not said much about round-off errors. The terminology necessary to a study of this type of error will be developed in the next chapter. Now we shall only give an example to show how round-off errors can completely destroy the result of a computation if one chooses a bad algorithm. In the example, we shall use a recursion formula. Recursion formulas are among the most valuable aids in numerical calculation—if they are used in the right way. As intimated in Sec. 1.3.2, one can specify very extensive calculations in relatively short computer programs with the help of such formulas.

Example 1.3.4

Compute for n = 0, 1, ..., 8

Use the recursion formula,

which follows from

We use three decimals throughout the example.

Algorithm 1. Compute

in y0, whose magnitude can be as high as 5·10−4 is multiplied by −5 in the calculation of y. That error produces an error in y, etc. Thus the error in y, the value of which can be as large as 625·5·10−4 = 0.3125. On top of this comes the round-off error committed in the various steps of the calculation, which, however, in this case can be shown to be relatively unimportant.

If one uses more decimal places of accuracy throughout the calculation, the absurd results will show up at a later stage. The above algorithm is an example of a disagreeable phenomenon, called numerical instability. We shall now see that one can avoid numerical instability by choosing a more suitable algorithm.

Algorithm 2. We use the recursion formula in the other direction,

Now the error will be divided by −5 in each step. But we need a starting value. We can see directly from the definition that yn decreases as n increases. One can also surmise that yn decreases slowly when n is large (the reader is recommended to motivate this). Thus we try setting y10 ≈ y9, and it follows that:

Algorithm 3. The same as Algorithm 2 except that one takes as starting value y10 = 0. One then gets y9 = 0.020, y8 = 0.018, and the rest of the yn have the same values as in Algorithm 2. The difference in the values for y10 in the two algorithms is 0.017.The subsequent values y9, y8, ..., y0 in the two algorithms are quite close because the error is divided by −5 in each step. A closer analysis is given in Example 2.2.12; the results obtained with Algorithm 2 have errors which are less than 10−3 for n ≤ 8.

The reader is warned, however, not to draw erroneous conclusions from the above example. The use of a recursion formula backwards is not a universal recipe! Compare Problem 10 at the end of this section!

In this book, we mean by the term numerical method a procedure which is often useful, either to approximate a mathematical problem with a numerical problem or to solve a numerical problem (or at least to reduce the numerical problem to a simpler problem). The transformation of a differential equation problem to a system of nonlinear equations (as in Example 1.3.2) is a numerical method—even without instructions as to how to solve the system of equations. When, as in Example 1.3.4, we specify a recursion formula for a sequence of integrals, we are also giving a numerical method—even without instruction as to how the recursion formula is to be used. Thus we require that a numerical method be more generally applicable than an algorithm, and set lesser emphasis on the completeness of the computational details. Newton-Raphson’s method is, for example, a numerical method for determining a root of an almost arbitrary equation, but in order to get an algorithm one must add conditions for starting and stopping the iteration process; these should be designed with regard to the type of equation and the context in which the equation occurs.

REVIEW QUESTIONS

1.Explain the concepts numerical problem, algorithm, and numerical method.

2.Give a concise explanation why Algorithm 1 of Example 1.3.4 didn’t work, and why the other two algorithms did work.

PROBLEMS

1.Use Horner’s scheme to compute p(2) where

p(x) = 2 − 3x² + 2x³ + x⁴.

2.Count the number of multiplications and additions required for the calculation of p(z) (see Sec. 1.3.2) by Horner’s rule. Compare with the work needed when the powers of x are calculated by xi = x·xi−1 and subsequently multiplied by an−i

3.(a)Prove formula (1.3.2) in Sec. 1.3.2.

(b)If

If you find this difficult, study Problem 4, below, first.)

4.From the computational scheme given below, one can read off that the polynomial x⁴ + 2x³ − 3x² + 2, after the substitution y = x − 2, is transformed to y⁴ + 10y³ + 33y² + 44y + 22. Investigate and give a theoretical explanation for how the scheme is constructed.

5.Write a program for the computation of a scalar product

6.Given the continued fraction

(a)Show that f can be computed using the algorithm

where f = d0.

(b)Write a program which reads in n, b0, ..., bn, a1 ..., an, performs the calculation, and prints the value of f.

7.Write a program which reads in a sequence of equidistant function values f0, f1 ..., to the variables y[0], y[l], ..., y[n], (n ≤ 20), and then computes the differences f0, Δf0, ..., Δnf0 and stores them where the fi were stored earlier. (Thus all the values of the function except f0 are destroyed.) Differences were defined in Sec. 1.2.The program should not use any memory space (variables) other than the y[i].

8.The coefficients of two polynomials f and g,

are given. Derive recursive formulas and write a program for the computation of the coefficients of the product of the polynomials.

9.Let x, y be nonnegative integers, with y ≠ 0. The division x/y yields the quotient q and remainder r. Show that if x and y have a common factor, then that number is a factor of r as well. Use this remark to design an algorithm for the determination of the greatest common factor of x and y (Euclid’s algorithm). Write a program which uses this algorithm and prints out the reduction of a fraction to lowest terms.

10.Derive a recursion formula for calculating the integrals

Give one algorithm that works well and another that works poorly (both based on the recursion formula).

2.1.BASIC CONCEPTS IN ERROR ESTIMATION

2.1.1.Introduction

Approximation is a central concept in almost all the uses of mathematics. One must often be satisfied with approximate values of the quantities with which one works. Another type of approximation occurs when one ignores some quantities which are small compared to other quantities. Such approximations are often necessary to insure that the mathematical and numerical treatment of a problem does not become hopelessly complicated.

We shall now introduce some notations, useful in practice, though their definitions are not exact in a mathematical sense:

a b (or b a) is read: "a is much smaller than b (or b is much greater than a). What is meant by much smaller (or much greater") depends on the context—among other things, on the desired precision. In a given instance it can be sufficient that a b; in another instance perhaps a < b/100 is necessary.

a ≈ b is read: "a is approximately equal to b and means the same as |a — bc, where c is chosen appropriate to the context. We cannot generally say, for example, that 10−6 ≈ 0.

a b (or b a) is read: "a is less than or approximately equal to b and means the same as a < b or a ≈ b."

Occasionally we shall have use for the following more precisely defined mathematical concepts:

f(x) = O(g(x)) when x → a, which means that |f(x)/g(x)| is bounded as x → a (a can be finite, +∞, or −∞).

f(x) = o(g(x)) when x → a

2.1.2.Sources of Error

Numerical results are influenced by many types of errors. Some sources of error are difficult to influence; others can be reduced or even eliminated by, for example, rewriting formulas or making other changes in the computational sequence.

A. Errors in Given Input Data. Input data can be the result of measurements which have been influenced by systematic errors or by temporary disturbances. Round-off errors occur, for example, whenever an irrational number is shortened (rounded off) to a fixed number of decimals. Round off errors can also occur when a decimal fraction is converted to the form used in the computer.

B. Round-off Errors During the Computations. If the calculating device which one is using cannot handle numbers which have more than, say, s digits, then the exact product of two s-digit numbers (which contains 2s or 2s − 1 digits) cannot be used in the subsequent calculations; the product must be rounded off. The effect of such roundings can be quite noticeable in an extensive calculation, or in an algorithm which is numerically unstable (defined in Example 1.3.3).

C. Truncation Errors. These are errors committed when a limiting process is truncated (broken off) before one has come to the limiting value. Truncation occurs, for example, when an infinite series is broken off after a finite number of terms, or when a derivative is approximated with a difference quotient (although in this case the term discretization error is better). Another example is when a nonlinear function is approximated with a linear function. Observe the distinction between truncation error and round-off error.

D. Simplifications in the Mathematical Model. In most of the applications of mathematics, one makes idealizations. In a mechanical problem, for example, one might assume that a string in a pendulum has zero mass. In many other types of problems it is advantageous to consider a given body to be homogeneously filled with matter, instead of being built up of atoms. For a calculation in economics, one might assume that the rate of interest is constant over a given period of time. The effects of such sources of error are usually more difficult to estimate than the types named in A, B, and C.

E. Human Errors and Machine Errors. In all numerical work, one must expect that clerical errors, errors in hand calculation, and misunderstandings will occur. One should even be aware that printed tables, etc., may contain errors. When one uses computers, one can expect errors in the program itself, errors in the punched cards, operator errors, and machine errors.

Errors which are purely machine errors are responsible for only a very small part of the strange results which (occasionally with great publicity) are produced by computers year after year. Most of the errors depend on the so-called human factor. As a rule, the effect of this type of error source cannot be analyzed with the help of the theoretical considerations of this chapter! We take up these sources of error in order to emphasize that both the person who carries out a calculation and the person who guides the work of others can plan so that such sources of error are not damaging. One can reduce the risk for such errors by suitable adjustments in working conditions and routines. Stress and tiredness are common causes of such errors.

One should also carefully consider what kind of checks can be made, either in the final result or in certain stages of the work, to prevent the necessity of redoing a whole project for the sake of a small error in an early stage. One can often discover whether calculated values are of the wrong order of magnitude or are not sufficiently regular (see difference checks, Chap. 7). Occasionally one can check the credibility of several results at the same time by checking that certain relations are true. In linear problems, one often has the possibility of sum checks. In physical problems, one can check, for example, to see whether energy is conserved, although because of the error sources A–D one cannot expect that it will be exactly conserved. In some situations, it can be best to treat a problem in two independent ways, although one can usually (as intimated above) check a result with less work than this.

2.1.3.Absolute and Relative Errors

Let ã be an approximate value for a quantity whose exact value is a. We define:

The absolute error in ã is ã — a.

The relative error in ã is (ã — a)/a if a ≠ 0. The relative error is often given as a percentage—for example, 3 percent relative error means that the relative error is 0.03.

In some books the error is defined with opposite sign to that which we use here. It makes almost no difference which convention one uses, as long as one is consistent. Using our definition, then, a — ã is the correction which should be added to ã to get rid of the error, ã — a. The correction and the error have, then, the same magnitude but different signs.

It is important to make a distinction between the error, which can be positive or negative, and a positive bound for the magnitude of the error, an error bound. We shall have reason to compute error bounds in many situations.

In the above definition of absolute error, a and ã need not be real numbers: they can also be vectors or matrices. (If we let ||·|| denote a vector norm (see Sec. 5.5.2), then the magnitude of the absolute and relative errors for the vector ã are defined by ||ã — a|| and ||ã — a||/||a|| respectively.)

The notation a = ã means, in this book, |ã — a. For example, a = 0.5876 ± 0.0014 means 0.5862 ≤ a ≤ 0.5890. In many applications, the same notation as above denotes the standard error (see Sec. 2.2.2) or some other measure of deviation of a statistical nature.

2.1.4.Rounding and Chopping

When one gives the number of digits in a numerical value one should not include zeros in the beginning of the number, as these zeros only help to denote where the decimal point should be. If one is counting the number of decimals, one should of course include leading zeros to the right of the decimal point.

Example

The number 0.00147 is given with three digits but has five decimals. The number 12.34 is given with four digits but has two decimals.

If the magnitude of the error in ã · 10−t, then ã is said to have t correct decimals. The digits in ã which occupy positions where the unit is greater than or equal to 10−t are called, then, significant digits (any initial zeros are not counted).

Example

0.001234 ± 0.000004 has five correct decimals and three significant digits, while 0.001234 ± 0.000006 has four correct decimals and two significant digits.

The number of correct decimals gives one an idea of the magnitude of the absolute error, while the number of significant digits gives a rough idea of the magnitude of the relative error.

There are two ways of rounding off numbers to a given number (t) of decimals. In chopping, one simply leaves off all the decimals to the right of the tth. That way of abridging a number is not recommended since the error has, systematically, the opposite sign of the number itself. Also, the magnitude of the error can be as large as 10−t. A surprising number of computers use chopping on the results of every arithmetical operation. This usually does not do so much harm, because the number of digits used in the operations is generally far greater than the number of significant digits in the data.

In rounding (sometimes called correct rounding), one chooses, among the numbers which can be expressed with t decimals, a number which is closest to the given number. Thus if the part of the number which stands to the right of the t· 10−t, in magnitude, then one should leave the t· 10−t, then one raises the tth decimal by 1. In the boundary case, when that which stands to the right of the t· 10−t, one should raise the t· 10−t (or the corresponding operation in a base other than 10), because this is easier to realize technically. Whichever convention one chooses in the boundary case, the error