For a particularly robust two-pass algorithm for computing the variance, first compute and subtract an estimate of the mean, and then use this algorithm on the residuals.

A more numerically stable two-pass algorithm first computes the sample means, and then the covariance:

In computer graphics, photon mapping is a two-pass global illumination algorithm developed by Henrik Wann Jensen that approximately solves the rendering equation.

The furnace can be situated at one end of a fire-tube which lengthens the path of the hot gases, thus augmenting the heating surface which can be further increased by making the gases reverse direction through a second parallel tube or a bundle of multiple tubes ( two-pass or return flue boiler ); alternatively the gases may be taken along the sides and then beneath the boiler through flues ( 3-pass boiler ).

For example, releases using the Xvid encoder must use the two-pass encoding method, which takes twice as long as a single pass, but achieves much higher quality ; similarly, DVD-R releases that must be re-encoded often use 6 or 8 passes to get the best quality.

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist.

While this loss of precision may be tolerable and viewed as a minor flaw of Algorithm I, it is easy to find data that reveal a major flaw in the naive algorithm: Take the sample to be ( 10 < sup > 9 </ sup > + 4, 10 < sup > 9 </ sup > + 7, 10 < sup > 9 </ sup > + 13, 10 < sup > 9 </ sup > + 16 ).

Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency.

Since the reference model given for traffic policing in the network is the GCRA, this algorithm is normally used for shaping as well, and single and dual leaky bucket implementations may be used as appropriate.

After one pair of Bézout coefficients ( x, y ) has been computed ( using extended Euclid or some other algorithm ), all pairs may be found using the formula

In information theory and computer science, a code is usually considered as an algorithm which uniquely represents symbols from some source alphabet, by encoded strings, which may be in some other target alphabet.

If one key cannot be deduced from the other, the asymmetric key algorithm has the public / private key property and one of the keys may be made public without loss of confidentiality.

# the hardware implementation, how algorithm and representation may be physically realized.

Common families include symmetric systems ( e. g. AES ) and asymmetric systems ( e. g. RSA ); they may alternatively be grouped according to the central algorithm used ( e. g. elliptic curve cryptography ).

56 bits is now considered insufficient length for symmetric algorithm keys, and may have been for some time.

The AKS primality test, published in 2002, proves that primality testing also lies in P, while factorization may or may not have a polynomial-time algorithm.

However, the condition number does not give the exact value of the maximum inaccuracy that may occur in the algorithm.

When the condition number is exactly one, then the algorithm may find an approximation of the solution with an arbitrary precision.

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.

Using to denote the size of the original grammar, the size blow-up in the worst case may range from to, depending on the transformation algorithm used.

) The inherent latency of the coding algorithm can be critical ; for example, when there is two-way transmission of data, such as with a telephone conversation, significant delays may seriously degrade the perceived quality.

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.

Depending on the algorithm used, other properties may be required as well, such as double hashing and linear probing.

In order to find the greatest common divisor, the Euclidean algorithm may be used.

With non-idempotent operations, the algorithm may have to keep track of whether the operation was already performed or not.

However, this is not the case with a special-purpose factorization algorithm, since it may not apply to the smaller factors that occur during decomposition, or may execute very slowly on these values.

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.

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.

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.