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The Krylov subspace method is an iterative method of solving a set of linear equations that is efficient for computers, making it an attractive method of solving large sets of equations in cases where it is applicable. It is iterative in that a series of approximations to the solution are produced, and using criteria to judge the closeness of the approximation to the solution, a trade-off can be made between accuracy and compute time. The method involves defining a Krylov subspace for the matrix that represents the equations, by choosing a vector, multiplying it by the matrix, and multiplying the result by the matrix repeatedly until a specified criteria is met.