Can we create a faster FFT? A faster Fast Fourier Transform?
Our answer is YES. This question is a call for collaboration with that goal, when we can already use the first results. It is not an utopian goal.
Today, the FFT is already tremendously important, from digital communication, to DVDs, worldwide atomic weapons test ban, and much more.
Fast, inexpensive digital devices are readily available from Intel Corporation, AMD and others today.
These devices are also using quantum mechanics and multiple states of spin -- as mass-produced quantum computers working with repeatability, precision, understanding, and flawlessly. A faster FFT will make them even more useful.
Can different, cheap software make a difference, even with current hardware? The article at https://lnkd.in/g2H6_5n2
and the preprints
https://lnkd.in/dNU-M8Av
https://lnkd.in/eBmAYKD4
show that much faster digital calculation of the Fourier Transform (faster than the FFT) using digital computers is possible today, using absolutely exact rational numbers, and just different software. It is not an utopian goal.
Quantum evolution seems to be facing Noah's ark moment, the moment of true need.
We need quantum computing now, and a faster FFT can also bring it. For calculating long prime numbers, all that is needed is to use the well-known Shor's algorithm in a faster FFT setting, to find periodicity.
Other algorithms can follow, to solve faster and more efficiently other difficult problems, such as in gaming and error correction. Science can also benefit, such as in protein folding, new medicines, new sources of energy, and propulsion means.
How can you help?
Two notes that may help:
The preprint "FT=FFT" can be made available by request. The abstract is in ResearchGate.
This question follows a known process, where a solution is easier to find when an equation is seen through a connection, as a method for easy solution of difficult problems.
This is an exact, “click-mathematics”, with pieces that fit together like Lego. But seeking a different solution only for prominence or solipsistic purposes, has been an unfortunate metaphor in academic works, and that is well-known to afflict students as well as teachers. No one is looking for “octopuses on Mars”.
To bring this question to quantum computing terms.
Quantum levels are isolated. This involves two aspects: separation and isolation. Any two levels must be separated from each other, and not be in contact. In addition, they are exact, a 0-point.
Each member of the set N is also showing the same 3 quantum properties: discrete, rigorous, and isolated.
Each of the infinite members of the set N is:
#Discrete: #digital, to use a 21st century term, being separated from each other by exactly 1;
#Rigorous: showing absolute accuracy with width 0; and
#Isolated: surrounded by "nothingness", where even the word "nothingness" may be too much.
One understands that numbers are not digits, as one can use different digits to represent the same number. But numbers can be thought of as a 1:1 mapping between a symbol and a value. Digits become a “name”, a reference, and it is clear that one can use different “names” for the same number.
Irrational numbers, which are unnamed, are neither proved nor disproved. That is, they are binarily independent of the Field Axioms of the mathematical real-numbers --- or, mathematically undecidable in the language of Kurt Gödel. Not the set Q, used in the FFT.
More in our Amazon book "Quickest Calculus", to better understand calculus, as we did above with the set N.
This further brings this comment to quantum computing terms!
The integers are now elements of quantum computing.
The prime numbers are now "lumps" of the set N, and can be found by periodicity, increasing monotonically and less than quadratically -- much like quantum levels of a possible potential following Schrödinger equation in quantum physics, as bound states. This can be treated by using the new FFT.
This also make the integers show 4 quantum properties, and motivates the expression in Ramanujan's partition number formula -- that looks like a quantum physics formula.
Mathematics Number Theory and quantum physics share a deep connection -- a "wormhole".
Support comes from:
https://www.researchgate.net/publication/355406882/
https://www.researchgate.net/publication/355406639/
https://www.researchgate.net/publication/286722113/
https://www.researchgate.net/publication/236420748/
https://www.researchgate.net/publication/286640675/
https://www.researchgate.net/publication/286640638/
https://www.researchgate.net/publication/243470610/
https://www.researchgate.net/publication/286625459/
Do not try to calculate with R, C, or G. Only use computer models with B, N, Z, and Q, where B is the set of Booleans.
Computers use only B, and 0^n = 0, 1^n=1, so any multiplication can be effectively avoided. Because division can be exact with modular arithmetic, any division can be avoided as well. The result is that the FFT can be much faster, and no trigonometric result has to be calculated.
Because with R=Q multiplication is repeated addition, and exponentiation is repeated multiplication, both can be calculated by the FFT using B, further accelerating and accelerated. Nothing is lost, as R=Q.
There have been 500 years or more of illusions and mirage.
A new mathematics was started in [1] (search reference above or online).
Responding to a private message, complex numbers were already denied, when we denied mathematical real-numbers [1]. But, how about imaginary numbers, or the Gauss set, using rational numbers to resurrect complex numbers?
No, already denied by Erwin Schrödinger in 1926. And, we deny that today -- using computers, where R=Q and C does not exist.
Soon, we will reveal how the Fourier Transform and the FFT can be a mapping from Q to Q -- and much faster --without passing through C, G, or R. The beginning of that is in the RG preprint "FT=FFT".
We are not saying the last word, but just the first.
A private message asks, "How about my training on quantum circuits using non-abelian anyons?".
There is more room at the top.
Any training is useful, but that training leads to no future. So, end it and start on quantum computing as done at the Max Planck Institute of Quantum Optics in 1982, using the set Q.
Be able to easily solve the Schrödinger equation for bound states, with any potential, and collect the dust of failed attempts.
Understand continued fractions, as providing a solution to the Schrödinger equation for bound states-- with only addition and encoding in the boolean set, where 0^n=0 and 1^n=1.
For example, help improve the FFT -- this question!
A public message by GP in another question asks about reactive current in electrical engineering.
I studied electrical engineering first, in order to better understand physics.
Ask your electricity company: all electrons are real, as well as any field.
What engineers wrongly model as imaginary is a rotation by 90 degrees, not the sqrt(-1).
Forget the illusion. One can use the new set Q* or computers can use the set Q.
Computers do not find imaginary numbers in the Boolean set of {0,1}; only 0 and 1, and yet can calculate reactive current!
Yes, it is possible to create a faster FFT. In fact, there have been a number of new algorithms proposed over the years that improve on the speed of the original Cooley-Tukey FFT algorithm. Some of these algorithms, such as the Bluestein FFT and the split-radix FFT, can be significantly faster than the Cooley-Tukey FFT for certain input sizes.
Here are some examples of faster FFT algorithms:
- Bluestein FFT: This algorithm is based on the idea of padding the input data with zeros, which allows the FFT to be computed recursively in a way that is more efficient than the Cooley-Tukey algorithm.
- Split-radix FFT: This algorithm is a variation of the Cooley-Tukey algorithm that uses a different set of twiddle factors, which can lead to significant speed improvements for certain input sizes.
- Integer FFT: This algorithm is designed for integer data, and it can be significantly faster than the Cooley-Tukey algorithm for certain input sizes.
However, it is important to note that the speed of the FFT is not just a function of the algorithm itself. The hardware on which the FFT is implemented can also have a significant impact on its performance. For example, a GPU can often be used to accelerate the FFT by orders of magnitude.
As a result, it is difficult to say definitively whether a faster FFT can be created. However, it is clear that there is still room for improvement in the speed of the FFT, and new algorithms are being proposed all the time.
In addition to the algorithms mentioned above, there are a number of other approaches to creating a faster FFT. For example, some researchers are working on using quantum computers to perform the FFT, which could potentially lead to orders-of-magnitude speed improvements.
Overall, the field of FFT research is still very active, and there is a lot of potential for further speed improvements. As hardware platforms continue to improve, and as new algorithms are developed, it is likely that we will see even faster FFTs in the future.
MS: thanks, agreed. A further improvement comes from the scaling law.
To be able to use the resonant condition in physics, we need to not consider infinitesimal differences in frequencies ... and in the FFT.
This means that we do not consider irrational numbers to exist, and no Problem of Closure [1].
We consider only rational numbers and can differentiate discontinuous functions (not just distributions) [1].
Taking advantage of the well-known 1:1 relationship between rational numbers and positive natural numbers, we can consider the set N.
Similar to computers, we work only with the Boolean set B={0,1}, and build B^n, as is well-known. Thus, imaginary numbers and p-adic numbers are also not included.
Now, in the Boolean set --
1^n = 1, 0^n = 0 Eq.(1)
offering instant calculations.
Thus, quantum computing can work fast with "all states at once", with no mysterious quantum mechanics, by following the 1:1 ladder from the sets R and C, to N+ and then to B={0,1}, where there is no actual calculation for any expression. The negative values are also handled as is well-known.
Thus, a difficult problem in the set R (or C) is transformed into a trivial problem in the set B, considering quantum computing. The equivalence of the problems has been hidden by considering imaginary numbers as the basis in mathematics, along with irrational numbers.
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