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/Eigen3Eigen::IDRS
template<typename _MatrixType, typename _Preconditioner>
class Eigen::IDRS< _MatrixType, _Preconditioner >
The Induced Dimension Reduction method (IDR(s)) is a short-recurrences Krylov method for sparse square problems.
This class allows to solve for A.x = b sparse linear problems. The vectors x and b can be either dense or sparse. he Induced Dimension Reduction method, IDR(), is a robust and efficient short-recurrence Krylov subspace method for solving large nonsymmetric systems of linear equations.
For indefinite systems IDR(S) outperforms both BiCGStab and BiCGStab(L). Additionally, IDR(S) can handle matrices with complex eigenvalues more efficiently than BiCGStab.
Many problems that do not converge for BiCGSTAB converge for IDR(s) (for larger values of s). And if both methods converge the convergence for IDR(s) is typically much faster for difficult systems (for example indefinite problems).
IDR(s) is a limited memory finite termination method. In exact arithmetic it converges in at most N+N/s iterations, with N the system size. It uses a fixed number of 4+3s vector. In comparison, BiCGSTAB terminates in 2N iterations and uses 7 vectors. GMRES terminates in at most N iterations, and uses I+3 vectors, with I the number of iterations. Restarting GMRES limits the memory consumption, but destroys the finite termination property.
- Template Parameters
-
_MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
This class follows the sparse solver concept .
The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations and NumTraits<Scalar>::epsilon() for the tolerance.
The tolerance corresponds to the relative residual error: |Ax-b|/|b|
Performance: when using sparse matrices, best performance is achied for a row-major sparse matrix format. Moreover, in this case multi-threading can be exploited if the user code is compiled with OpenMP enabled. See Eigen and multi-threading for details.
By default the iterations start with x=0 as an initial guess of the solution. One can control the start using the solveWithGuess() method.
IDR(s) can also be used in a matrix-free context, see the following example .
- See also
- class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
| template<typename Rhs , typename Dest > | |
| void | _solve_vector_with_guess_impl (const Rhs &b, Dest &x) const |
| IDRS () | |
| template<typename MatrixDerived > | |
| IDRS (const EigenBase< MatrixDerived > &A) | |
| void | setAngle (RealScalar angle) |
| void | setResidualUpdate (bool update) |
| void | setS (Index S) |
| void | setSmoothing (bool smoothing) |
IDRS() [1/2]
| inline |
Default constructor.
IDRS() [2/2]
| inlineexplicit |
Initialize the solver with matrix A for further Ax=b solving.
This constructor is a shortcut for the default constructor followed by a call to compute().
- Warning
- this class stores a reference to the matrix A as well as some precomputed values that depend on it. Therefore, if A is changed this class becomes invalid. Call compute() to update it with the new matrix A, or modify a copy of A.
_solve_vector_with_guess_impl()
| inline |
Loops over the number of columns of b and does the following:
- sets the tolerence and maxIterations
- Calls the function that has the core solver routine
setAngle()
| inline |
The angle must be a real scalar. In IDR(s), a value for the iteration parameter omega must be chosen in every s+1th step. The most natural choice is to select a value to minimize the norm of the next residual. This corresponds to the parameter omega = 0. In practice, this may lead to values of omega that are so small that the other iteration parameters cannot be computed with sufficient accuracy. In such cases it is better to increase the value of omega sufficiently such that a compromise is reached between accurate computations and reduction of the residual norm. The parameter angle =0.7 (”maintaining the convergence strategy”) results in such a compromise.
setResidualUpdate()
| inline |
The parameter replace is a logical that determines whether a residual replacement strategy is employed to increase the accuracy of the solution.
setS()
| inline |
Sets the parameter S, indicating the dimension of the shadow space. Default is 4
setSmoothing()
| inline |
Switches off and on smoothing. Residual smoothing results in monotonically decreasing residual norms at the expense of two extra vectors of storage and a few extra vector operations. Although monotonic decrease of the residual norms is a desirable property, the rate of convergence of the unsmoothed process and the smoothed process is basically the same. Default is off
The documentation for this class was generated from the following file:
© Eigen.
Licensed under the MPL2 License.
https://eigen.tuxfamily.org/dox/unsupported/classEigen_1_1IDRS.html