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Matrix-free methods

In this article, we will delve into the fascinating world of Matrix-free methods, exploring its many facets and shedding light on issues that have sparked the interest and curiosity of many. From its impact on society to its implications in the scientific field, including its influence on popular culture, this exhaustive analysis will seek to unravel the mysteries surrounding Matrix-free methods and offer a panoramic view that allows our readers to better understand its importance and relevance. in a world in constant change and evolution. Join us on this journey of discovery and reflection about Matrix-free methods, whose impact is felt in all areas of modern life.

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In computational mathematics, a matrix-free method is an algorithm for solving a linear system of equations or an eigenvalue problem that does not store the coefficient matrix explicitly, but accesses the matrix by evaluating matrix-vector products.[1] Such methods can be preferable when the matrix is sufficiently large that storing and manipulating it would be prohibitively expensive with respect to memory or computation time, even with the use of methods for sparse matrices. Many iterative methods allow for a matrix-free implementation, including:

Distributed solutions have also been explored using coarse-grain parallel software systems to achieve homogeneous solutions of linear systems.[6]

It is generally used in solving non-linear equations like Euler's equations in computational fluid dynamics. Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver.[7] Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. To avoid this expense, matrix-free methods are employed. In order to remove the need to calculate the Jacobian, the Jacobian vector product is formed instead, which is in fact a vector itself. Manipulating and calculating this vector is easier than working with a large matrix or linear system.

References

  1. ^ Langville, Amy N.; Meyer, Carl D. (2006), Google's PageRank and beyond: the science of search engine rankings, Princeton University Press, p. 40, ISBN 978-0-691-12202-1
  2. ^ Coppersmith, Don (1993), "Solving linear equations over GF(2): Block Lanczos algorithm", Linear Algebra and Its Applications, 192: 33–60, doi:10.1016/0024-3795(93)90235-G
  3. ^ Knyazev, Andrew V. (2001). "Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method". SIAM Journal on Scientific Computing. 23 (2): 517–541. Bibcode:2001SJSC...23..517K. CiteSeerX 10.1.1.34.2862. doi:10.1137/S1064827500366124.
  4. ^ Wiedemann, D. (1986), "Solving sparse linear equations over finite fields" (PDF), IEEE Transactions on Information Theory, 32: 54–62, doi:10.1109/TIT.1986.1057137
  5. ^ Lamacchia, B. A.; Odlyzko, A. M. (1991), "Solving Large Sparse Linear Systems Over Finite Fields", Advances in Cryptology – CRYPT0' 90, Lecture Notes in Computer Science, vol. 537, p. 109, doi:10.1007/3-540-38424-3_8, ISBN 978-3-540-54508-8
  6. ^ Kaltofen, E.; Lobo, A. (1996), "Distributed Matrix-Free Solution of Large Sparse Linear Systems over Finite Fields", Algorithmica, vol. 24, no. 3–4, pp. 311–348, CiteSeerX 10.1.1.17.7470, doi:10.1007/PL00008266, S2CID 13305650
  7. ^ Prabhune, Bhagyashree C.; Krishnan, Suresh (4 March 2020). "A fast matrix-free elasto-plastic solver for predicting residual stresses in additive manufacturing". Computer Aided Design. 123 102829. doi:10.1016/j.cad.2020.102829.


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