This method ( and the general idea of an FFT ) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered ( Heideman & Burrus, 1984 ) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 ( and subsequently rediscovered several times in limited forms ).

The most well-known use of the Cooley – Tukey algorithm is to divide the transform into two pieces of size at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general ( as was known to both Gauss and Cooley / Tukey ).

The Rader-Brenner algorithm ( 1976 ) is a Cooley – Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability ; it was later superseded by the split-radix variant of Cooley – Tukey ( which achieves the same multiplication count but with fewer additions and without sacrificing accuracy ).

( 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 ).

John Wilder Tukey ForMemRS (; June 16, 1915 – July 26, 2000 ) was an American statistician best known for development of the FFT algorithm and box plot.

Tukey was born in New Bedford, Massachusetts in 1915, and obtained a B. A.

The term " software ", which Paul Niquette claims he coined in 1953, was first used in print by Tukey in a 1958 article in American Mathematical Monthly, and thus some attribute the term to him.

Exploratory data analysis was promoted by John Tukey to encourage statisticians visually to examine their data sets, to formulate hypotheses that could be tested on new data-sets.

Tukey held that too much emphasis in statistics was placed on statistical hypothesis testing ( confirmatory data analysis ); more emphasis needed to be placed on using data to suggest hypotheses to test.

Dr. Kurtz graduated from Knox College in 1950, and was awarded a Ph. D. degree from Princeton University in 1956, where his advisor was John Tukey, and joined the Mathematics Department of Dartmouth College that same year.

His results, however, have been disputed, especially in 1954 by a team consisting of John Tukey, Frederick Mosteller and William G. Cochran, who stated much of Kinsey's work was based on convenience samples rather than random samples, and thus would have been vulnerable to bias.

According to, this name was coined by John Tukey in the 1950s.

City Marshal Francis Tukey resisted mayor John Prescott Bigelow's appointment of McGinniskin, expressing the predominant anti-Irish sentiments in the city by arguing it was done at " the expense of an American.

By far the most commonly used FFT is the Cooley – Tukey algorithm.

Even the " exact " FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small ; most FFT algorithms, e. g. Cooley – Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms.

Cooley and John Tukey, is the most common fast Fourier transform ( FFT ) algorithm.

A radix-2 decimation-in-time ( DIT ) FFT is the simplest and most common form of the Cooley – Tukey algorithm, although highly optimized Cooley – Tukey implementations typically use other forms of the algorithm as described below.

Tukey coined many statistical terms that have become part of common usage, but the two most famous coinages attributed to him were related to computer science.

His most significant contribution to the world of mathematics and digital signal processing is the Fast Fourier transform, which he co-developed with John Tukey ( see Cooley – Tukey FFT algorithm ) while working for the research division of IBM in 1965.

Therefore, for N even the convolution is cyclic, but in this case N is composite and one would normally use a more efficient FFT algorithm such as Cooley – Tukey.

Bruun's algorithm has not seen widespread use, however, as approaches based on the ordinary Cooley – Tukey FFT algorithm have been successfully adapted to real data with at least as much efficiency.

" On January 5, 1852, shortly before the newly elected mayor Benjamin Seaver ( who had been supported by Tukey ) took office, Tukey fired McGinniskin without giving a reason.

Duncan's test has been criticised as being too liberal by many statisticians including Henry Scheffé, and John W. Tukey.

In fact, the root mean square ( rms ) errors are much better than these upper bounds, being only O ( ε √ log N ) for Cooley – Tukey and O ( ε √ N ) for the naïve DFT ( Schatzman, 1996 ).

Some FFTs other than Cooley – Tukey, such as the Rader-Brenner algorithm, are intrinsically less stable.

Thus, Bluestein's algorithm provides an O ( N log N ) way to compute prime-size DFTs, albeit several times slower than the Cooley – Tukey algorithm for composite sizes.

Furthermore, there is evidence that Bruun's algorithm may be intrinsically less accurate than Cooley – Tukey in the face of finite numerical precision ( Storn, 1993 ).

Tukey promoted the use of five number summary of numerical data — the two extremes ( maximum and minimum ), the median, and the quartiles — because these median and quartiles, being functions of the empirical distribution are defined for all distributions, unlike the mean and standard deviation ; moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries ( the mean and standard deviation ).

While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method.

The well-known radix-2 Cooley – Tukey algorithm, for N a power of 2, can compute the same result with only ( N / 2 ) log < sub > 2 </ sub > N complex multiplies ( again, ignoring simplifications of multiplications by 1 and similar ) and N log < sub > 2 </ sub > N complex additions.

Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm ; it also re-expresses a DFT as a convolution, but this time of the same size ( which can be zero-padded to a power of two and evaluated by radix-2 Cooley – Tukey FFTs, for example ), via the identity.

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 ).

In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as O (√ N ) for the Cooley – Tukey algorithm ( Welch, 1969 ).

In particular, one can pad to a power of two or some other highly composite size, for which the FFT can be efficiently performed by e. g. the Cooley – Tukey algorithm in O ( N log N ) time.

If the log is taken base 2, the unit of information is the binary digit or bit ( so named by John Tukey ); if we use a natural logarithm instead, we might call the resulting unit the " nat.

Instead of recursively factorizing directly, though, Cooley – Tukey instead first computes x < sub > 2 </ sub >( z ω < sub > N </ sub >), shifting all the roots ( by a twiddle factor ) so that it can apply the recursive factorization of to both subproblems.

For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley – Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors.

* Uncomfortable science, due to statistician John Tukey: Inference from a limited sample of data, where further samples influenced by the same causality, a finite natural phenomenon for which it is difficult to overcome the problem of using a common sample of data for both exploratory data analysis and confirmatory data analysis.

These statistical developments, all championed by Tukey, were designed to complement the analytic theory of testing statistical hypotheses, particularly the Laplacian tradition's emphasis on exponential families.

Uncomfortable science is the term coined by statistician John Tukey for cases in which there is a need to draw an inference from a limited sample of data, where further samples influenced by the same cause system will not be available.

* 1965-Cooley – Tukey algorithm rediscovered by James Cooley and John Tukey

An important practical application of smooth numbers is for fast Fourier transform ( FFT ) algorithms such as the Cooley – Tukey FFT algorithm that operate by recursively breaking down a problem of a given size n into problems the size of its factors.