Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm.

An example of using Euclid's algorithm will be shown below.

In all, the Blowfish encryption algorithm will run 521 times to generate all the subkeys-about 4KB of data is processed.

On any given run of the algorithm, it has a probability of at most 1 / 3 that it will give the wrong answer.

If a " weak " character is followed by another " weak " character, the algorithm will look at the first neighbouring " strong " character.

If a naive rendering algorithm is used without any filtering, high frequencies in the image function will cause ugly aliasing to be present in the final image.

However it does not mean that the algorithm will converge rapidly to this solution, just that it won't diverge arbitrarily because of inaccuracy on the source data ( backward error ), provided that the forward error introduced by the algorithm does not diverge as well because of accumulating intermediate rounding errors.

The condition number may also be infinite, in which case the algorithm will not reliably find a solution to the problem, not even a weak approximation of it ( and not even its order of magnitude ) with any reasonable and provable accuracy.

In automatic systems this can be done using a binary search algorithm or interpolation search ; manual searching may be performed using a roughly similar procedure, though this will often be done unconsciously.

A good checksum algorithm will yield a different result with high probability when the data is accidentally corrupted ; if the checksums match, the data is very likely to be free of accidental errors.

The goal of a good checksum algorithm is to spread the valid corners as far from each other as possible, so as to increase the likelihood that " typical " transmission errors will end up in an invalid corner.

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q.

Oskar Sandberg's research ( during the development of version 0. 7 ) shows that this " path folding " is critical, and that a very simple routing algorithm will suffice provided there is path folding.

For general problem classes there may be no way to show that Meta GP will reliably produce results more efficiently than a created algorithm other than exhaustion.

Rowing machines with monitors calculate performance using an algorithm unique to the individual manufacturer ; it will be affected by the type of resistance used and other factors.

The values of ipad and opad are not critical to the security of the algorithm, but were defined in such a way to have a large Hamming distance from each other and so the inner and outer keys will have fewer bits in common.

In other words, for any ( lossless ) data compression algorithm, there will be an input data set that does not get smaller when processed by the algorithm.

* Suppose that there is a compression algorithm that transforms every file into a distinct file which is no longer than the original file, and that at least one file will be compressed into something that is shorter than itself.

To choose an algorithm always means implicitly to select a subset of all files that will become usefully shorter.

But if the learning algorithm is too flexible, it will fit each training data set differently, and hence have high variance.

If the true function is simple, then an " inflexible " learning algorithm with high bias and low variance will be able to learn it from a small amount of data.

But if the true function is highly complex ( e. g., because it involves complex interactions among many different input features and behaves differently in different parts of the input space ), then the function will only be learnable from a very large amount of training data and using a " flexible " learning algorithm with low bias and high variance.

While there is no generally accepted formal definition of " algorithm ," an informal definition could be " a set of rules that precisely defines a sequence of operations.

" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

Thus an algorithm can be an algebraic equation such as y

In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension.

Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.

Gurevich: "... Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage, an algorithm is a computational process defined by a Turing machine ".

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise.

In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended " target " computer ( s ), in order for the target machines to produce output from given input ( perhaps null ).

If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types — conditional GOTO, unconditional GOTO, assignment, HALT.

From this follows a simple algorithm, which can be stated in a high-level description English prose, as:

So to be precise the following is really Nicomachus ' algorithm.

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.

However, the algorithm can be improved by adopting the method of the assumed mean.

This algorithm is often more numerically reliable than the naïve algorithm for large sets of data, although it can be worse if much of the data is very close to but not precisely equal to the mean and some are quite far away from it.

For such an online algorithm, a recurrence relation is required between quantities from which the required statistics can be calculated in a numerically stable fashion.

Bruun's algorithm ( above ) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.

The algorithm starts with an initially empty ( and therefore trivially sorted ) list.

While the patent was initially rejected by the patent office as being a purely mathematical invention, following 12 years of appeals, Pardo and Landau won a landmark court case at the CCPA ( Predecessor Court of the Federal Circuit ) overturning the Patent Office in 1983 — establishing that " something does not cease to become patentable merely because the point of novelty is in an algorithm.

As a matter of fact, the term " algorithm " was not commonly extended to approximation algorithms until later ; the Christofides algorithm was initially referred to as the Christofides heuristic.

As initially mentioned, the second main stage in the link-state algorithm is to produce routing tables, by inspecting the maps.

The algorithm was initially proposed by Marco Dorigo in 1992, and has since been diversified to solve a wider class of numerical problems.

( Although the PFA is distinct from the Cooley – Tukey algorithm, Good's 1958 work on the PFA was cited as inspiration by Cooley and Tukey in their famous 1965 paper, and there was initially some confusion about whether the two algorithms were different.

Good on what is now called the prime-factor FFT algorithm ( PFA ); although Good's algorithm was initially mistakenly thought to be equivalent to the Cooley – Tukey algorithm, it was quickly realized that PFA is a quite different algorithm ( only working for sizes that have relatively prime factors and relying on the Chinese Remainder Theorem, unlike the support for any composite size in Cooley – Tukey ).

Skipjack was invented by the National Security Agency of the U. S. Government ; this algorithm was initially classified SECRET, which prevented it from being subjected to peer review from the encryption research community.

The not frequently used ( NFU ) page replacement algorithm requires a counter, and every page has one counter of its own which is initially set to 0.

This algorithm maintains a partition of the vertices of the graph into a sequence of sets ; initially this sequence consists of a single set with all vertices.

For example, the SQL: 1999 standard includes recursive queries, and the Magic Sets algorithm ( initially developed for the faster evaluation of Datalog queries ) is implemented in IBM's DB2.

It initially observes the pattern of word hits, word-to-word matches of a given length, and marks potential matches before performing a more time-consuming optimized search using a Smith-Waterman type of algorithm.

CARINE ( Computer Aided Reasoning engINE ) is a resolution based theorem prover initially built for the study of the enhancement effects of the strategies delayed clause-construction ( DCC ) and attribute sequences ( ATS ) in a depth-first search based algorithm 2005.

While the development of the algorithm initially generated some enthusiasm, partly because of its apparent relation to biological mechanisms, the later discovery of this inadequacy caused such models to be abandoned until the introduction of non-linear models into the field.

The algorithm begins in the exponential growth phase initially with a congestion window size ( cwnd ) of 1 or 2 segments and increases it by 1 Segment Size ( SS ) for each ACK received.