Gisbert Stoyan - Elementary Numerical Mathematics for Programmers and Engineers

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Springer International Publishing Switzerland 2016

Gisbert Stoyan and Agnes Baran Elementary Numerical Mathematics for Programmers and Engineers Compact Textbooks in Mathematics 10.1007/978-3-319-44660-8_1

1. Floating Point Arithmetic

Gisbert Stoyan 1 and Agnes Baran 2

(1)

Faculty of Informatics, ELTE University, Budapest, Hungary

(2)

Faculty of Informatics, University of Debrecen, Debrecen, Hungary

You know from secondary school: If the base of a power is a >0, then the following relations are valid:

If q 1

holds, moreover, if q =1, then

1.1 Integers

Integers are represented on a computer in the form of signed binary numbers. Often 2-, 4- and 8-byte integers are available where a byte possesses eight binary digits. In many computers 4 bytes are the smallest availableaddressableunit of the memory. It may turn out that we can work with one- and 16-byte integers, too.

The arithmetical operations performed with integers may be and often are faster than the operations performed with floating point numbers (which will be discussed in the following section), but this depends on their word length, the processor and the translator. Such operations can be considered as error-free, hence, using integers a given algorithm can be faster on the computer. However, all steps of a calculation with integers have to be carefully planned, because actually in this case we work in residue classes. As an example consider the addition of 1-byte integers:

The processing unit cuts offwithout any warning!the 1 which cannot be placed in the frame (it would produce an overflow ), and in this way the sum is replaced by its residual with respect to 28.

Thus, we move to the residue class of 28 and, e.g., in the case of two-byte numbers to the residue class of 216, etc.

1.2 Floating Point Numbers

This type of numbers is fundamental in the use of computers for numerical computations. The form of nonzero floating point numbers is

(1.1)

where a >1 is the base of the representation of numbers, + or is the sign, t >1 is the number of digits, k is the exponent. The digit m 1 is normalized, that is, it satisfies the inequality

This ensures the uniqueness of the representation and the full exploitation of the available digits. For the remaining digits

holds. The number zero is not normalized: in this case , and its sign is usually +.

It often happens that the base a of the representation of floating point numbers is not equal to 2. Instead of 2 it can be, e.g., 10 or 16. Some computers have a decimal CPU. However, usually the corresponding arithmetic is implemented by the programming language, and when, e.g., the main purpose is data processing, it can be more economical to use the decimal number system. We may view the computations as being done in the decimal number system and can count on about t =8 using single precision and about t =16 using double precision.

We can imagine the storage of floating point numbers in the following form (reality may be different from this, but here we do not describe these details of the technical implementation which do not affect our numerical calculations):

where the vector ( m 1,, m t )=: m is the mantissa (or significand ). The exponent k is also called the characteristic of the number. Depending on the machine and on the precision (single, double, quadruple precision) four, eight or 16 bytes are available to store m . Simultaneously, the range of k increases but, depending on the given precision,

where k ()<0, k (+)>0 and| k ()|| k (+)|(see Exercises ). Then the largest representable number is

while the smallest number is M . The floating point numbers form a discrete subset of the rational numbers from [ M , M ], and this subset is symmetric about zero.

We denote the positive floating point number nearest to zero by ,

Thus, besides zero there is no other (normalized) floating point number in the interval . (Here we disregard the fact that on a computer denormalized nonzero numbers can also be used, but to ensure the uniqueness of the representation only in the case k = k (). Practically, this means that then the formula for changes to .)

Fig. 1.1

Floating point numbers on the real axis

The interval appears as a huge gap if we represent the floating point numbers on the real axis (see Fig..) since the positive floating point number nearest to is

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