改进版《数值分析》(2nd Ed)：2003年版升级

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《数值分析(第二版)》由G. Shanker Rao撰写，是2003年版本的修订版。相较于早期版本，作者在编写第二版时给予了极大的关注，旨在提升内容质量和准确性。编者努力确保书中的错误，包括排版和文字上的问题，得到了大幅度减少。这一版本进行了多项改动，以反映最新的研究进展和技术发展，使读者能够获取更为精确和实用的数值分析知识。该书作为数学分析领域的重要参考书籍，涵盖了数值方法的基础理论和应用技巧。它深入探讨了数值积分、微分方程数值解法、优化算法、线性代数在数值计算中的应用、误差分析以及数值逼近等方面。读者可以从中学习如何通过计算机模拟和数值计算解决实际问题，尤其是在工程、科学和经济领域，如气象预测、金融建模和材料科学等。第二版特别强调了算法的可读性和实用性，通过丰富的实例和清晰的解释，帮助读者理解复杂概念。同时，为了满足现代教学和研究的需求，书中还包含了最新的编程语言和技术，例如Python、MATLAB等，以及对并行计算和分布式处理的介绍。版权方面，此书受到严格保护，未经出版商书面许可，任何形式的复制、扫描、影印或电子存储都不被允许。读者如果有关于版权的问题，可以通过电子邮件联系出版社获取授权。本书的国际标准书号(ISBN)分别为10位的81-224-2295-0和13位的978-81-224-2295-5，这些信息对于购买和查找图书非常关键。此外，该书的出版背景显示，NewAge International有限公司作为全球化的出版社，其发行网络遍布印度多个城市，包括新德里、班加罗尔、钦奈等地。封面扉页上，作者向他的父亲G. Narayana Rao致敬，这体现了作者的个人情感和对知识传承的重视。《数值分析(第二版)》是一本全面而实用的教材，适合大学生、研究生以及在科研工作中需要处理数值计算问题的专业人士使用，它不仅提供了深入的理论指导，还为实践应用提供了强有力的工具。无论是理论学习还是实践经验，这本书都是一个不可或缺的参考资料。

1.1 INTRODUCTION

There are two kinds of numbers—exact and approximate numbers.

An approximate number x is a number that differs, but slightly, from an exact number X and

is used in place of the latter in calculations.

The numbers 1, 2, 3, …,

…, etc., are all exact, and

,2,,e

…, etc., written in this

manner are also exact.

1.41 is an approximate value of

and 1.414 is also an approximate value of

Similarly

3.14, 3.141, 3.14159, …, etc., are all approximate values of

π.

1.2 SIGNIFICANT DIGITS

The digits that are used to express a number are called significant digits. Figure is synonymous with

digit.

Definition 1 A significant digit of an approximate number is any non-zero digit in its decimal

representation, or any zero lying between significant digits, or used as place holder to indicate a

retained place.

The digits 1, 2, 3, 4, 5, 6, 7, 8, 9 are significant digits. ‘0’ is also a significant figure except

when it is used to fix the decimal point, or to fill the places of unknown or discarded digits.

For example, in the number 0.0005010, the first four ‘0’s’ are not significant digits, since they

serve only to fix the position of the decimal point and indicate the place values of the other digits.

The other two ‘0’s’ are significant.

Two notational conventions which make clear how many digits of a given number are signifi-

cant are given below.

1. The significant figure in a number in positional notation consists of:

(a) All non-zero digits and

(b) Zero digits which

ERRORS

2 NUMERICAL ANALYSIS

(i) lie between significant digits

(ii) lie to the right of decimal point, and at the same time to the right of a non-zero

digit

(iii) are specifically indicated to be significant

2. The significant figure in a number written in scientific notation (M × 10

) consists of all

the digits explicitly in M.

Significant figures are counted from left to right starting with the left most non zero digit.

Example 1.1

Number Significant figures No. of Significant figures

37.89 3, 7, 8, 9 4

5090 5, 0, 9 3

7.00 7, 0, 0 3

0.00082 8, 2 2

0.000620 6, 2, 0 3

5.2 × 10

5, 2 2

3.506 × 10 3, 5, 0, 6 4

8 × 10

–3

1.3 ROUNDING OFF NUMBERS

With a computer it is easy to input a vast number of data and perform an immense number of

calculations. Sometimes it may be necessary to cut the numbers with large numbers of digits. This

process of cutting the numbers is called rounding off numbers. In rounding off a number after a

computation, the number is chosen which has the required number of significant figures and which

is closest to the number to be rounded off. Usually numbers are rounded off according to the

following rule.

Rounding-off rule In order to round-off a number to n significant digits drop all the digits to the

right of the nth significant digit or replace them by ‘0’s’ if the ‘0’s’ are needed as place holders,

and if this discarded digit is

1. Less than 5, leave the remaining digits unchanged

2. Greater than 5, add 1 to the last retained digit

3. Exactly 5 and there are non-zero digits among those discarded, add unity to the last retained

digit

However, if the first discarded digit is exactly 5 and all the other discarded digits are ‘0’s’, the

last retained digit is left unchanged if even and is increased by unity if odd.

In other words, if the discarded number is less than half a unit in the nth place, the nth digit

is unaltered. But if the discarded number is greater than half a unit in the nth place, the nth digit is

increased by unity.

And if the discarded number is exactly half a unit in the nth place, the even digit rule is applied.

4 NUMERICAL ANALYSIS

−+ −

−=

involves an analytical error.

The magnitude of the error in the value of the function due to cutting (truncation) of its series

is equal to the sum of all the discarded terms. It may be large and may even exceed the sum of the

terms retained, thus making the calculated result meaningless.

(iii) Round-off errors. When depicting even rational numbers in decimal system or some other

positional system, there may be an infinity of digits to the right of the decimal point, and it may not

be possible for us to use an infinity of digits, in a computational problem. Therefore it is obvious

that we can only use a finite number of digits in our computations. This is the source of the so-

called rounding errors. Each of the FORTRAN Operations +, –, *, /, **, is subject to possible round-

off error.

To denote the cumulative effect of round-off error in the computation of a solution to a given

computational problem, we use the computational error and the computational error can be made

arbitrarily small by carrying all the calculations to a sufficiently high degree of precision.

Definition 2 By the error of an approximate number we mean the difference between the exact

number X, and the given approximate number x.

It is denoted by E (or by ∆)

E= =X x.

∆

–

Note An exact number may be regarded as an approximate number with error zero.

Definition 3 The absolute error of an approximate number x is the absolute value of the difference

between the corresponding exact number X and the number x. It is denoted by E

. Thus

E=X x

−

Definition 4 The limiting error of an approximate number denoted by ∆x is any number not less

than the absolute error of that number.

Note From the definition we have

E=X x x.

−≤∆

Therefore X lies within the range

xxXxx–

∆∆≤≤+

Thus we can write

X=x x

±∆

Definition 5 The relative error of an approximate number x is the ratio of the absolute error of

the number to the absolute value of the corresponding exact number X, where

≠

. It is denoted

by E

(or by δ)

==δ .

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