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For and algorithm
: For a detailed presentation of the various points of view around the definition of " algorithm " see Algorithm characterizations.
" For some people, a program is only an algorithm if it stops eventually ; for others, a program is only an algorithm if it stops before a given number of calculation steps.
For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise.
For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.
: For an example of the simple algorithm " Add m + n " described in all three levels see Algorithm examples.
Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist.
For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a " modulus " ( division ) instruction available rather than just subtraction ( or worse: just Minsky's " decrement ").
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.
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.
For his PhD thesis, he co-developed rsync, including the rsync algorithm, a highly efficient file transfer and synchronization tool.
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.
For example, algorithms are known for factoring an n-bit integer using just over 2n qubits ( Shor's algorithm ).
For example, the security available with a 1024-bit key using asymmetric RSA is considered approximately equal in security to an 80-bit key in a symmetric algorithm ( Source: RSA Security ).
For this reason cryptologists tend to look at indicators that an algorithm or key length shows signs of potential vulnerability, to move to longer key sizes or more difficult algorithms.
For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.
For illustration, the Euclidean algorithm can be used to find the greatest common divisor of a = 1071 and b = 462.
For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.
For the case of power-of-two, Papadimitriou ( 1979 ) argued that the number of complex-number additions achieved by Cooley – Tukey algorithms is optimal under certain assumptions on the graph of the algorithm ( his assumptions imply, among other things, that no additive identities in the roots of unity are exploited ).
For example, an approximate FFT algorithm by Edelman et al.
For case # 1, the painter paints ( fills ) the pixel the painter is standing upon and stops the algorithm.
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.
For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε ; that is, it is always possible to transmit with arbitrarily small block error.

For and above
( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.
For proper accreditation of schools, teachers in any course must have a degree at least one level above that for which the student is a candidate.
For example, farm equipment shipments of International Harvester in August climbed about 5% above a year earlier, Mr. Keeler reports.
For example, if the water is 25 ft ( 8 m ) deep, and the anchor roller is 3 ft ( 1 m ) above the water, the scope is the ratio between the amount of cable let out and 28 ft ( 9 m ).
For instance, iron changes from a body-centered cubic structure ( ferrite ) to a face-centered cubic structure ( austenite ) above 906 ° C, and tin undergoes a transformation known as tin pest from a metallic phase to a semiconductor phase below 13. 2 ° C.
For different reasons, various constitutionalists have praised the Act of Settlement: Henry Hallam called the act in the United Kingdom " the seal of our constitutional laws " and David Lindsay Keir placed its importance above the Bill of Rights 1689.
' For example, Tagore's idea of these two concepts should be way above any common man's and many perceive Tagore as a ' Mahana ' Artist in the realm of literature.
For example, the Arabic letters transliterated as and have the same basic shape, but has one dot below,, and has two dots above,.
For long periods, urban areas such as Buenos Aires, Rosario, and Córdoba welcomed European immigrants, including, above all, those of Italian and Spanish descent.
For the analysis to correspond usefully to the actual execution time, the time required to perform a step must be guaranteed to be bounded above by a constant.
: For example, picture 1 above illustrates the concrete relationship " Cat sits on Mat ".
: For example, graph 1 above expresses the abstraction " agent sits on location ".
For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.
For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.
For more details on geological crystal formation, see above.
For this to occur, energy is required, as in the semiconductor the next higher states lie above the band gap.
For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).
For example, PAL-M ( Brazil ) uses a 3. 58 MHz subcarrier, and SECAM uses two different frequencies, 4. 250 MHz and 4. 40625 MHz above the video carrier.
For example, The Shining might be sorted as Shining, The ( see Alphabetical order above ), but it may still be desired to display it as The Shining.
For measurements at experimentally controlled volume, one can use the assumption, stated above, that the pressure of the body of calorimetric material is can be expressed as a function of its volume and temperature.
For measurements at experimentally controlled temperature, it is again assumed that the volume of the body of calorimetric material can be expressed as a function of its temperature and pressure, with the same provisos as mentioned just above.
For instance, a Gravitron's spin forces riders against a wall and allows riders to be elevated above the machine's floor in defiance of Earth's gravity.
For trajectories other than circular motion, for example, the more general trajectory envisioned in the image above, the instantaneous center of rotation and radius of curvature of the trajectory are related only indirectly to the coordinate system defined by u < sub > ρ </ sub > and u < sub > θ </ sub > and to the length | r ( t )|
For the simple electric dipole given above, the electric dipole moment points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges.

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