Matrix-free methods

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. Such methods can be preferable when the matrix is so big that storing and manipulating it would cost a lot of memory and computer time, even with the use of methods for sparse matrices. Many iterative methods allow for a matrix-free implementation, including the power method and the conjugate gradient method.

It is generally used in solving non-linear equations like Euler's equations in CFD. Solving these equations needs to find jacobian which is costly in terms of CPU time and storage. To avoid this matrix free methods are employed. So one need not has to calculate jacobian, instead jacobian vector product is formed which is a vector itself. Manipulating this vector is easy than whole big matrix.

References

• 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.