(1986), 'Solving sparse linear equations over finite fields' (PDF), IEEE Transactions on Information Theory, 32: 54–62, doi: 10.1109/TIT.1986.1057137 'Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method'. (2006), Google's PageRank and beyond: the science of search engine rankings, Princeton University Press, p. 40, ISBN 978-2-1 Manipulating and calculating this vector is easier than working with a large matrix or linear system. In order to remove the need to calculate the jacobian, the jacobian vector product is formed instead, which is in fact a vector itself. To avoid this expense, matrix free methods are employed. Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver. It is generally used in solving non-linear equations like Euler's equations in Computational Fluid Dynamics.

Wiedemann's coordinate recurrence algorithm, andĭistributed solutions have also been explored using coarse-grain parallel software systems to achieve homogeneous solutions of linear systems.

Locally Optimal Block Preconditioned Conjugate Gradient Method ( LOBPCG),.

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

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.

The matrix free