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 computing time, even with the use of methods for sparse matrices. Many iterative methods allow for a matrix-free implementation, including:
* the power method,
* the Lanczos algorithm,
* Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG),
* Wiedemann's coordinate recurrence algorithm, and
* the conjugate gradient method.

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

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. 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

Numerical linear algebra

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