# Strassen's algorithm is not optimal trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations

@article{Pan1978StrassensAI, title={Strassen's algorithm is not optimal trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations}, author={Victor Y. Pan}, journal={19th Annual Symposium on Foundations of Computer Science (sfcs 1978)}, year={1978}, pages={166-176} }

A new technique of trilinear operations of aggregating, uniting and canceling is introduced and applied to constructing fast linear non-commutative algorithms for matrix multiplication. The result is an asymptotic improvement of Strassen's famous algorithms for matrix operations.

## 119 Citations

The Technique of Trilinear Aggregating and the Recent Progress in the Asymptotic Acceleration of Matrix Operations

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1984

Field extension and trilinear aggregating, uniting and canceling for the acceleration of matrix multiplications

- Computer Science20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
- 1979

The acceleration of matrix multiplication MM, is based on the combination of the method of algebraic field extension and trilinear aggregating, uniting and canceling and a fast algorithm of O(N2.7378) complexity for N × N matrix multiplication is derived.

Matrix Multiplication, a Little Faster

- Physics
- 2020

Strassen’s algorithm (1969) was the first sub-cubic matrix multiplication algorithm. Winograd (1971) improved the leading coefficient of its complexity from 6 to 7. There have been many subsequent ...

Recursion removal in fast matrix multiplication

- Computer Science, Engineering
- 2003

A general method of breaking recursions in fast matrix multiplication algorithms is introduced, which is generalized from recursions removal of a specificfast matrix multiplication algorithm of Winograd.

Matrix Multiplication , a Li le Faster Regular Submission

- Computer Science
- 2017

A generalization of Probert’s lower bound that holds under change of basis is proved, showing that for matrix multiplication algorithms with a 2×2 base case, the leading coecient of the Strassen-Winograd algorithm cannot be further reduced, hence optimal.

Multiplying matrices faster than coppersmith-winograd

- Computer ScienceSTOC '12
- 2012

An automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction is developed and a new improved bound on the matrix multiplication exponent ω<2.3727 is obtained.

Matrix Multiplication, a Little Faster

- Computer ScienceSPAA
- 2017

A generalization of Probert's lower bound that holds under change of basis is proved, showing that for matrix multiplication algorithms with a 2x2 base case, the leading coefficient of the algorithm cannot be further reduced, hence optimal.

Matrix Multiplication, a Little Faster

- Computer ScienceJ. ACM
- 2020

It is proved that for matrix multiplication algorithms with a 2 × 2 base case, the leading coefficient of Strassen-Winograd’s O(nlog27) algorithm cannot be further reduced, and is therefore optimal, and applied to other fast matrix multiplicationgorithms, improving their arithmetic and communication costs by significant constant factors.

Complexity Of Computations

- Mathematics, Computer ScienceACM Annual Conference
- 1978

Construction of algorithms is a time honored mathematical activity, with the whole field of Numerical Analysis devoted to finding a variety of algorithms for numerical integration of differential equations.

## References

SHOWING 1-10 OF 29 REFERENCES

Sur le calcul des produits de matrices

- Mathematics
- 1971

SummaryThe purpose of this paper is to prove that Strassen's algorithm is a representation of a product of (2,2) matrices by a Hadamard product in a space of 7 dimensions. For matrices (n, n) it is…

A Fast Non-Commutative Algorithm for Matrix Multiplication

- Computer Science, MathematicsMFCS
- 1977

It is shown that some well-known algorithms are special cases of the non-commutative algorithm for the multiplication of two square matrices of order n that is presented.

Some Techniques for Proving Certain Simple Programs Optimal

- Mathematics, Computer ScienceSWAT
- 1969

Techniques for establishing a lower bound on the number of arithmetic operations necessary for sets of simple expressions are developed and a modification of Strassen's algorithm for multiplying n × p matrices by p × q matrices is developed, establishing that matrix multiplication with elements from a commutative ring requires fewer multiplications than with elements with non-commutative elements.

A New Algorithm for Inner Product

- MathematicsIEEE Transactions on Computers
- 1968

A new way of computing the inner product of two vectors is described that can be performed using roughly n3/2 multiplications instead of the n3multiplications which the regular method necessitates.

Duality applied to the complexity of matrix multiplications and other bilinear forms

- Mathematics, Computer ScienceSTOC
- 1973

An algorithm for computing a single bilinear form over a noncommutative ring with a minimum number of multiplications is derived by considering a dual problem.

Duality Applied to the Complexity of Matrix Multiplication and Other Bilinear Forms

- Mathematics, Computer ScienceSIAM J. Comput.
- 1973

The paper considers the complexity of bilinear forms in a noncommutative ring. The dual of a computation is defined and applied to matrix multiplication and other bilinear forms. It is shown that t...

On the number of multiplications necessary to compute certain functions

- Mathematics, Computer Science
- 1970

A new algorithm for matrix multiplication, which requires about 1/2(n cubed) multiplications, is obtained following the results of Pan Motzkin about polynomial evaluation and the product of a matrix by vector.

On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

- Mathematics, Computer Science
- 1969

The algorithm minimizes the number of multiplications for matrix multiplication without commutativity for the special cases p=1 or 2, n=1,2, $\cdots$ and p = 3, n = 3.

Vermeidung von Divisionen.

- Mathematics
- 1973

The extent to whieh the use of divisions may speed up the evaluation of polynomials is estimated from above. In particular it is shown that for multiplying general matrices the use of divisions does…

On Obtaining Upper Bounds on the Complexity of Matrix Multiplication

- Computer ScienceComplexity of Computer Computations
- 1972

The search for better algorithms for AB is reduced to the decomposition of X, thus circumventing the manipulation of products which appear in the final algorithm for AB.