" 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

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 ".

The programmer must translate the algorithm into a language that the simulator / computer / computor can effectively execute.

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:

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.

The algorithm can be extended to handle unequal sample weights, replacing the simple counter n with the sum of weights seen so far.

Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals.

He was highly influential in the development of computer science, giving a formalisation of the concepts of " algorithm " and " computation " with the Turing machine, which can be considered a model of a general purpose computer.

Solomonoff used this algorithm, and the code lengths it allows, to define a " universal probability " of a string on which inductive inference of the subsequent digits of the string can be based.

Exact ( not asymptotic ) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called model of computation.

Run-time efficiency is a topic of great interest in computer science: A program can take seconds, hours or even years to finish executing, depending on which algorithm it implements ( see also performance analysis, which is the analysis of an algorithm's run-time in practice ).

Since algorithms are platform-independent ( i. e. a given algorithm can be implemented in an arbitrary programming language on an arbitrary computer running an arbitrary operating system ), there are significant drawbacks to using an empirical approach to gauge the comparative performance of a given set of algorithms.

For example, if for a given problem size a parallelized implementation of an algorithm can run 12 % of the algorithm's operations arbitrarily quickly ( while the remaining 88 % of the operations are not parallelizable ), Amdahl's law states that the maximum speedup of the parallelized version is times as fast as the non-parallelized implementation.

The same algorithm can actually also be used to correct for direct line overlap of two atomic absorption lines, making HR-CS AAS the only AAS technique that can correct for this kind of spectral interference.

The algorithm is hereby placed in the public domain, and can be freely used by anyone.

Although this problem seems easier, it has been shown that if there is a practical ( randomized polynomial-time ) algorithm to solve this problem, then all problems in NP can be solved just as easily.

As a generalization of this, following easily from Euclidean algorithm in base n > 1:

The suspicion was that the algorithm had been covertly weakened by the intelligence agency so that they — but no-one else — could easily read encrypted messages.

Although a digital filter can be an IIR filter if the algorithm implementing it includes feedback, it is also possible to easily implement a filter whose impulse truly goes to zero after N time steps ; this is called a finite impulse response ( FIR ) filter.

Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1 / N factor, any FFT algorithm can easily be adapted for it.

which can easily be proven by considering the Euclidean algorithm in base n. Another useful identity relates to the Euler's totient function:

The " trick " that allows lossless compression algorithms, used on the type of data they were designed for, to consistently compress such files to a shorter form is that the files the algorithms are designed to act on all have some form of easily modeled redundancy that the algorithm is designed to remove, and thus belong to the subset of files that that algorithm can make shorter, whereas other files would not get compressed or even get bigger.

Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Jenkins-Traub method, Laguerre's method, Durand – Kerner method or by some other root-finding algorithm.

* Longer constraint lengths produce more powerful codes, but the complexity of the Viterbi algorithm increases exponentially with constraint lengths, limiting these more powerful codes to deep space missions where the extra performance is easily worth the increased decoder complexity.

The nearest neighbour algorithm is easy to implement and executes quickly, but it can sometimes miss shorter routes which are easily noticed with human insight, due to its " greedy " nature.

This kind of algorithm can be easily integrated with the Phong reflection model, the Z-buffer algorithm, and many other graphics techniques.

) The brute-force search algorithm above will call output for every candidate that is a solution to the given instance P. The algorithm is easily modified to stop after finding the first solution, or a specified number of solutions ; or after testing a specified number of candidates, or after spending a given amount of CPU time.

The 802. 11 encryption algorithm called WEP ( short for Wired Equivalent Privacy ) used a short, 24-bit IV, leading to reused IVs with the same key, which led to it being easily cracked.

Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman-Ford algorithm or the Floyd-Warshall algorithm does.

The acceleration () caused by the thruster force is just an additional acceleration to be added to the other accelerations ( force per unit mass ) affecting the spacecraft and the orbit can easily be propagated with a numerical algorithm including also this thruster force .< ref > The thrust force per unit mass being where and are given functions of time But for many purposes, typically for studies or for maneuver optimization, they are approximated by impulsive maneuvers as illustrated in figure 1 with a as given by ().

The pseudo-code above will call output for all candidates that are a solution to the given instance P. The algorithm is easily modified to stop after finding the first solution, or a specified number of solutions ; or after testing a specified number of partial candidates, or after spending a given amount of CPU time.

The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only extreme points ( vertices of the convex hull ) or all points that lie on the convex hull.

They can also be used to easily build an optimal selection algorithm, as well as near-sorting algorithms, which are algorithms that place every element near its final position, a situation in which insertion sort is fast.

The Master theorem allows us to easily calculate the running time of such a recursive algorithm in Θ-notation without doing an expansion of the recursive relation above.

To easily multiply 2 digit numbers together between 11 and 19 a simple algorithm is as follows ( where a is the ones digit of the first number and b is the ones digit of the second number ):

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.

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

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

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

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 ).

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

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

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.