Philipp O. J. Scherer - Computational Physics

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Part I
Numerical Methods

Springer International Publishing AG 2017

Philipp O.J. Scherer Computational Physics Graduate Texts in Physics

1. Error Analysis

Philipp O. J. Scherer 1

(1)

Physikdepartment T38, Technische Universitt Mnchen, Garching, Germany

Philipp O. J. Scherer

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Several sources of errors are important for numerical data processing:

Experimental uncertainty: Input data from an experiment have a limited precision. Instead of the vector of exact values the calculation uses , with an uncertainty . This can lead to large uncertainties of the calculated results if an unstable algorithm is used or if the unavoidable error inherent to the problem is large.

Rounding errors: The arithmetic unit of a computer uses only a subset of the real numbers, the so called machine numbers . The input data as well as the results of elementary operations have to be represented by machine numbers whereby rounding errors can be generated. This kind of numerical error can be avoided in principle by using arbitrary precision arithmetics or symbolic algebra programs. But this is unpractical in many cases due to the increase in computing time and memory requirements.

Truncation errors: Results from more complex operations like square roots or trigonometric functions can have even larger errors since series expansions have to be truncated and iterations can accumulate the errors of the individual steps.

1.1 Machine Numbers and Rounding Errors

Floating point numbers are internally stored as the product of sign, mantissa and a power of 2. According to the IEEE754 standard [1] single, double and quadruple precision numbers are stored as 32, 64 or 128 bits (Table ).

Table 1.1

Binary floating-point formats

Format	Sign	Exponent	Hidden bit	Fraction	Precision
Float	s
Double	s
Quadruple	s

The sign bit s is 0 for positive and 1 for negative numbers. The exponent b is biased by adding E which is half of its maximum possible value (Table The value of a number is given by

(1.1)

The mantissa a is normalized such that its first bit is 1 and its value is between 1 and 2

(1.2)

Table 1.2

Exponent bias E

Decimal value	Binary value	Hexadecimal value	Data type
		$ 3F	Single
		$ 3FF	Double
		$3FFF	Quadruple

Since the first bit of a normalized floating point number always is 1, it is not necessary to store it explicitly (hidden bit or J-bit). However, since not all numbers can be normalized, only the range of exponents from is used for normalized numbers. An exponent of signals that the number is not normalized (zero is an important example, there exist even two zero numbers with different sign) whereas the exponent is reserved for infinite or undefined results (Table ).

Table 1.3

Special double precision numbers

Hexadecimal value	Symbolic value
$ 000 0000000000000	0
$ 080 00000000000000
$ 7FF 0000000000000	inf
$ FFF 0000000000000	-inf
$ 7FF 0000000000001 7FF FFFFFFFFFFFFF	NAN
$ 001 0000000000000	Min_Normal
$ 7FE FFFFFFFFFFFFF	Max_Normal
$ 000 0000000000001	Min_Subnormal
$ 000 FFFFFFFFFFFFF

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