*************************** Size of Krylov Subspace *************************** :: Size of Krylov subspace = ----------------------- Description / Usage ----------------------- This optional card allows the user to specify the dimension (size) of the Krylov subspace for the **gmres** option of the *Solution Algorithm* card, where **m,** specifies the number of orthogonalization directions and can be any positive integer less than or equal to the order of the matrix. If the *Size of Krylov subspace* card is omitted, then the default dimension is **m** = 30. ------------ Examples ------------ The following is a sample input card: :: Size of Krylov subspace = 128 ------------------------- Technical Discussion ------------------------- If the size of the subspace is at least as large as the maximum number of iterations permitted by the solver then the **gmres** iteration will not include any restarts. Depending on the problem, restarts may be beneficial, and then again they may not. Particularly poorly conditioned linear systems may never converge below a certain tolerance if **gmres** is allowed to restart (i.e. they “level off”). However, some linear systems will admit a converged solution more rapidly with restarts than without. Consequently, the user may wish to experiment with different values of this parameter. See the Orthogonalization card for related information. **gmres’** internal iterations create a Krylov subspace up to dimension m (less in some circumstances, such as convergence). The time and space required by the internal iterations increases nonlinearly with **m** (but see the *Orthogonalization* card) - a doubling of **m** will result in more than a doubling of space and time requirements. So simply choosing a very large dimension is generally not recommended.