Concept

Floating-point arithmetic

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Unit in the last place

In computer science and numerical analysis, unit in the last place or unit of least precision (ulp) is the spacing between two consecutive floating-point numbers, i.e., the value the least significant digit (rightmost digit) represents if it is 1. It is used as a measure of accuracy in numeric calculations. One definition is: In radix with precision , if , then . Another definition, suggested by John Harrison, is slightly different: is the distance between the two closest straddling floating-point numbers and (i.

Interval arithmetic

[[File:Set of curves Outer approximation.png|345px|thumb|right|Tolerance function (turquoise) and interval-valued approximation (red)]] Interval arithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically correct results.

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Machine epsilon

Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of computational science. The quantity is also called macheps and it has the symbols Greek epsilon . There are two prevailing definitions. In numerical analysis, machine epsilon is dependent on the type of rounding used and is also called unit roundoff, which has the symbol bold Roman u.

Z1 (computer)

The Z1 was a motor-driven mechanical computer designed by Konrad Zuse from 1936 to 1937, which he built in his parents' home from 1936 to 1938. It was a binary electrically driven mechanical calculator with limited programmability, reading instructions from punched celluloid film. The “Z1” was the first freely programmable computer in the world that used Boolean logic and binary floating-point numbers, however, it was unreliable in operation. It was completed in 1938 and financed completely by private funds.

Motorola 6809

The Motorola 6809 ("sixty-eight-oh-nine") is an 8-bit microprocessor with some 16-bit features. It was designed by Motorola's Terry Ritter and Joel Boney and introduced in 1978. Although source compatible with the earlier Motorola 6800, the 6809 offered significant improvements over it and 8-bit contemporaries like the MOS Technology 6502, including a hardware multiplication instruction, 16-bit arithmetic, system and user stack registers allowing re-entrant code, improved interrupts, position-independent code and an orthogonal instruction set architecture with a comprehensive set of addressing modes.

Data structure alignment

Data structure alignment is the way data is arranged and accessed in computer memory. It consists of three separate but related issues: data alignment, data structure padding, and packing. The CPU in modern computer hardware performs reads and writes to memory most efficiently when the data is naturally aligned, which generally means that the data's memory address is a multiple of the data size. For instance, in a 32-bit architecture, the data may be aligned if the data is stored in four consecutive bytes and the first byte lies on a 4-byte boundary.

Half-precision floating-point format

In computing, half precision (sometimes called FP16 or float16) is a binary floating-point computer number format that occupies 16 bits (two bytes in modern computers) in computer memory. It is intended for storage of floating-point values in applications where higher precision is not essential, in particular and neural networks. Almost all modern uses follow the IEEE 754-2008 standard, where the 16-bit base-2 format is referred to as binary16, and the exponent uses 5 bits.

Kahan summation algorithm

In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floating-point numbers, compared to the obvious approach. This is done by keeping a separate running compensation (a variable to accumulate small errors), in effect extending the precision of the sum by the precision of the compensation variable.

IEEE 754

The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.