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Ifpack2 Templated Preconditioning Package
Version 1.0
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Ifpack2 Templated Preconditioning Package
Version 1.0
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Diagonally scaled Chebyshev iteration for Tpetra sparse matrices. More...
#include <Ifpack2_Chebyshev_decl.hpp>
Public Types | |
Typedefs | |
| typedef MatrixType | matrix_type |
| The template parameter of this class. More... | |
| typedef MatrixType::scalar_type | scalar_type |
| The type of the entries of the input MatrixType. More... | |
| typedef MatrixType::local_ordinal_type | local_ordinal_type |
| The type of local indices in the input MatrixType. More... | |
| typedef MatrixType::global_ordinal_type | global_ordinal_type |
| The type of global indices in the input MatrixType. More... | |
| typedef MatrixType::node_type::device_type | device_type |
| The Kokkos::Device specialization used by the input MatrixType. More... | |
| typedef MatrixType::node_type | node_type |
| The Node type used by the input MatrixType. More... | |
| typedef Teuchos::ScalarTraits < scalar_type >::magnitudeType | magnitude_type |
| The type of the magnitude (absolute value) of a matrix entry. More... | |
| typedef Tpetra::RowMatrix < scalar_type, local_ordinal_type, global_ordinal_type, node_type > | row_matrix_type |
| The Tpetra::RowMatrix specialization matching MatrixType. More... | |
| typedef Tpetra::Map < local_ordinal_type, global_ordinal_type, node_type > | map_type |
| The Tpetra::Map specialization matching MatrixType. More... | |
| typedef Tpetra::Vector < scalar_type, local_ordinal_type, global_ordinal_type, node_type > | vector_type |
| The Tpetra::Vector specialization matching MatrixType. More... | |
 Public Types inherited from Ifpack2::Preconditioner< MatrixType::scalar_type, MatrixType::local_ordinal_type, MatrixType::global_ordinal_type, MatrixType::node_type > | |
| typedef Teuchos::ScalarTraits < MatrixType::scalar_type > ::magnitudeType | magnitude_type |
| The type of the magnitude (absolute value) of a matrix entry. More... | |
Public Member Functions | |
| Chebyshev (const Teuchos::RCP< const row_matrix_type > &A) | |
| Constructor. More... | |
| virtual | ~Chebyshev () |
| Destructor. More... | |
Preconditioner computation methods | |
| void | setParameters (const Teuchos::ParameterList ¶ms) |
| Set (or reset) parameters. More... | |
| void | initialize () |
| Initialize the preconditioner. More... | |
| bool | isInitialized () const |
| void | compute () |
| (Re)compute the left scaling, and (if applicable) estimate max and min eigenvalues of D_inv * A. More... | |
| bool | isComputed () const |
Implementation of Ifpack2::Details::CanChangeMatrix | |
| virtual void | setMatrix (const Teuchos::RCP< const row_matrix_type > &A) |
| Change the matrix to be preconditioned. More... | |
Implementation of Tpetra::Operator | |
| void | apply (const Tpetra::MultiVector< scalar_type, local_ordinal_type, global_ordinal_type, node_type > &X, Tpetra::MultiVector< scalar_type, local_ordinal_type, global_ordinal_type, node_type > &Y, Teuchos::ETransp mode=Teuchos::NO_TRANS, scalar_type alpha=Teuchos::ScalarTraits< scalar_type >::one(), scalar_type beta=Teuchos::ScalarTraits< scalar_type >::zero()) const |
| Apply the preconditioner to X, returning the result in Y. More... | |
| Teuchos::RCP< const map_type > | getDomainMap () const |
| The Tpetra::Map representing the domain of this operator. More... | |
| Teuchos::RCP< const map_type > | getRangeMap () const |
| The Tpetra::Map representing the range of this operator. More... | |
| bool | hasTransposeApply () const |
| Whether it's possible to apply the transpose of this operator. More... | |
| void | applyMat (const Tpetra::MultiVector< scalar_type, local_ordinal_type, global_ordinal_type, node_type > &X, Tpetra::MultiVector< scalar_type, local_ordinal_type, global_ordinal_type, node_type > &Y, Teuchos::ETransp mode=Teuchos::NO_TRANS) const |
| Compute Y = Op(A)*X, where Op(A) is either A, \(A^T\), or \(A^H\). More... | |
Attribute accessor methods | |
| Teuchos::RCP< const Teuchos::Comm< int > > | getComm () const |
| The communicator over which the matrix is distributed. More... | |
| Teuchos::RCP< const row_matrix_type > | getMatrix () const |
| The matrix for which this is a preconditioner. More... | |
| Teuchos::RCP< const Tpetra::CrsMatrix< scalar_type, local_ordinal_type, global_ordinal_type, node_type > > | getCrsMatrix () const |
| Attempt to return the matrix A as a Tpetra::CrsMatrix. More... | |
| double | getComputeFlops () const |
| The total number of floating-point operations taken by all calls to compute(). More... | |
| double | getApplyFlops () const |
| The total number of floating-point operations taken by all calls to apply(). More... | |
| int | getNumInitialize () const |
| The total number of successful calls to initialize(). More... | |
| int | getNumCompute () const |
| The total number of successful calls to compute(). More... | |
| int | getNumApply () const |
| The total number of successful calls to apply(). More... | |
| double | getInitializeTime () const |
| The total time spent in all calls to initialize(). More... | |
| double | getComputeTime () const |
| The total time spent in all calls to compute(). More... | |
| double | getApplyTime () const |
| The total time spent in all calls to apply(). More... | |
| size_t | getNodeSmootherComplexity () const |
| Get a rough estimate of cost per iteration. More... | |
| MatrixType::scalar_type | getLambdaMaxForApply () const |
| The estimate of the maximum eigenvalue used in the apply(). More... | |
Implementation of Teuchos::Describable | |
| std::string | description () const |
| A simple one-line description of this object. More... | |
| void | describe (Teuchos::FancyOStream &out, const Teuchos::EVerbosityLevel verbLevel=Teuchos::Describable::verbLevel_default) const |
| Print the object with some verbosity level to a Teuchos::FancyOStream. More... | |
| Public Member Functions inherited from Ifpack2::Preconditioner< MatrixType::scalar_type, MatrixType::local_ordinal_type, MatrixType::global_ordinal_type, MatrixType::node_type > | |
| virtual | ~Preconditioner () |
| Destructor. More... | |
| Public Member Functions inherited from Ifpack2::Details::CanChangeMatrix< Tpetra::RowMatrix< MatrixType::scalar_type, MatrixType::local_ordinal_type, MatrixType::global_ordinal_type, MatrixType::node_type > > | |
| virtual void | setMatrix (const Teuchos::RCP< const Tpetra::RowMatrix< MatrixType::scalar_type, MatrixType::local_ordinal_type, MatrixType::global_ordinal_type, MatrixType::node_type > > &A)=0 |
| Set the new matrix. More... | |
| virtual | ~CanChangeMatrix () |
| Destructor. More... | |
Diagonally scaled Chebyshev iteration for Tpetra sparse matrices.
| MatrixType | A specialization of Tpetra::RowMatrix. |
This class implements a Chebyshev polynomial preconditioner or smoother for a Tpetra sparse matrix. Given a matrix A, it applies Chebyshev iteration to the left-scaled matrix \(D^{-1} A\), where D = diag(A) is the matrix of the diagonal entries of A. This class' constructor accepts a Tpetra::RowMatrix or any subclass thereof (including Tpetra::CrsMatrix). Its template parameter may be a specialization of either class.
Chebyshev is derived from Preconditioner, which itself is derived from Tpetra::Operator. Therefore, a Chebyshev instance may be used as an operator in any code that invokes the operator as apply().
Given a matrix A, a right-hand side X, and an initial guess Y, this class computes an approximate solution to \(AY=X\) via Chebyshev iteration using the left-scaled matrix \(D^{-1} A\), where D is the matrix of the diagonal elements of A. (You may control left scaling yourself if you wish, by providing an optional vector of the entries of \(D^{-1}\).) While Chebyshev iteration works for any matrix, we have chosen only to allow real-valued, symmetric positive definite matrices.
Chebyshev iteration was originally intended as an iterative solver for linear systems. See the following publication (the spelling of "Chebyshev" in Latin characters differs in some publications):
T. Manteuffel, "The Tchebychev iteration for nonsymmetric linear systems," Numer. Math., 28 (1977), pp. 307-327.
It also works as a smoother for algebraic multigrid, which is the target use case of this implementation.
We require that the input matrix A be real-valued and symmetric positive definite. Thus, all of its eigenvalues must lie in a positive interval on the real line. Furthermore, if D is the matrix of the diagonal elements of A, then the same is true of \(D^{-1} A\).
Suppose \([\lambda_{min}, \lambda_{max}]\) is the interval of the eigenvalues of \(D^{-1} A\). Users may either give us an estimate of the maximum eigenvalue \(\lambda_{max}\), or let us compute it (which we do with a few power iterations). They may optionally also give us the (estimated) ratio \(\eta = \lambda_{max} / \lambda_{min}\), or (an estimate of) the minimum eigenvalue \(\lambda_{min}\). The \(\eta\) parameter corresponds to the "smoother: Chebyshev alpha" parameter of ML. (We use "eta" instead of "alpha" to avoid confusion with the "alpha" argument of the apply() method of Tpetra::Operator.)
When using Chebyshev iteration by itself to solve linear systems, it is important to have good estimates of both the minimum and maximum eigenvalues. However, when using a small number of Chebyshev iterations as a smoother in multigrid, the maximum eigenvalue estimate is more important. (The point of a smoother is to smooth out the high-frequency components of the error, that is, those that correspond to the largest eigenvalues. The coarser grids below the current grid will take care of the lower-frequency components of the error.) This is why we use a ratio \(\eta = \lambda_{max} / \lambda_{min}\), rather than requiring a guess for \(\lambda_{min}\). In fact, we only use \(\lambda_{min}\) for error checking, not when determining the Chebyshev coefficients. Often, if users give us \(\lambda_{max}\), our default value of \(\eta\) suffices.
Underestimating \(\lambda_{min}\) may make Chebyshev fail to converge, or fail to reduce the highest-frequency components of the error, if used as a smoother. Thus, we always multiply the given \(\lambda_{min}\) by a small factor (1.1). This heuristic accounts for the fact that typical methods for estimating extremal eigenvalues (like Lanczos or CG) underestimate them.
If you do not give us an estimate for the maximum eigenvalue, we estimate it using a few iterations of the power method in the compute() method. We do not attempt to refine the eigenvalue bounds over Chebyshev iterations, as the typical smoother case does not use very many iterations. For an example of a Chebyshev implementation that updates eigenvalue bound estimates, see Steve Ashby's CHEBYCODE:
S. ASHBY, "CHEBYCODE: A Fortran implementation of Manteuffel's adaptive Chebyshev algorithm," Tech. Rep. UIUCDCS-R-85-1203, University of Illinois, 1985.
Call the setParameters() method to give this instance your eigenvalue bound estimates (if you have them), as well as to set other options controlling the behavior of Chebyshev iteration. The documentation of setParameters() lists all the parameters that this class accepts. Where possible, we list comparable parameters in the Ifpack package and the ML multigrid package.
Chebyshev should spend most of its time in Tpetra's native sparse matrix-vector multiply kernel. This should give good performance, since we have spent a lot of effort tuning that kernel. Depending on the Node type of your Tpetra matrix, the kernel may also exploit threads for additional parallelism within each MPI process ("hybrid parallelism" a.k.a. "MPI + X"). If your application depends on hybrid parallelism for performance, you should favor Chebyshev smoothers whenever possible over "serial within a process" smoothers like Gauss-Seidel or SOR (Symmetric Over-Relaxation).
The original implementation of this class was an adaptation of ML's ML_Cheby routine. The original author was Ulrich Hetmaniuk, a Sandia employee in what was then (2006) Org 1416. Ifpack2 has seen significant development since then.
| typedef MatrixType Ifpack2::Chebyshev< MatrixType >::matrix_type |
The template parameter of this class.
| typedef MatrixType::scalar_type Ifpack2::Chebyshev< MatrixType >::scalar_type |
The type of the entries of the input MatrixType.
| typedef MatrixType::local_ordinal_type Ifpack2::Chebyshev< MatrixType >::local_ordinal_type |
The type of local indices in the input MatrixType.
| typedef MatrixType::global_ordinal_type Ifpack2::Chebyshev< MatrixType >::global_ordinal_type |
The type of global indices in the input MatrixType.
| typedef MatrixType::node_type::device_type Ifpack2::Chebyshev< MatrixType >::device_type |
The Kokkos::Device specialization used by the input MatrixType.
| typedef MatrixType::node_type Ifpack2::Chebyshev< MatrixType >::node_type |
The Node type used by the input MatrixType.
| typedef Teuchos::ScalarTraits<scalar_type>::magnitudeType Ifpack2::Chebyshev< MatrixType >::magnitude_type |
The type of the magnitude (absolute value) of a matrix entry.
| typedef Tpetra::RowMatrix<scalar_type, local_ordinal_type, global_ordinal_type, node_type> Ifpack2::Chebyshev< MatrixType >::row_matrix_type |
The Tpetra::RowMatrix specialization matching MatrixType.
MatrixType must be a Tpetra::RowMatrix specialization. This typedef will always be a Tpetra::RowMatrix specialization.
| typedef Tpetra::Map<local_ordinal_type, global_ordinal_type, node_type> Ifpack2::Chebyshev< MatrixType >::map_type |
The Tpetra::Map specialization matching MatrixType.
| typedef Tpetra::Vector<scalar_type, local_ordinal_type, global_ordinal_type, node_type> Ifpack2::Chebyshev< MatrixType >::vector_type |
The Tpetra::Vector specialization matching MatrixType.
If you wish to supply setParameters() a precomputed vector of diagonal entries of the matrix, use a pointer to an object of this type.
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explicit |
Constructor.
| [in] | A | The sparse matrix to which to apply Chebyshev iteration. The matrix A must be square, and its domain Map and range Map must be the same. The latter means that the vectors x and y in the sparse matrix-vector product y = A*x must both have the same distribution over process(es). |
We do not require that the row Map and the range Map of A be the same. However, set-up will take less time if they are identical (in terms of pointer equality). This is because we have to extract the diagonal entries of A as a row Map vector: if the row and range Maps are not identical, we have to redistribute the vector from the row Map to the range Map.
The constructor will only check the requirements on the various Maps of A if the CMake configuration option Teuchos_ENABLE_DEBUG was set to ON before building Trilinos. The checks require \(O(1)\) global reductions over all processes in A's communicator, so we prefer to avoid them if we can.
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Destructor.
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Set (or reset) parameters.
This method fills in the input ParameterList with missing parameters set to their default values. You may call this method as many times as you want. On each call, the input ParameterList is treated as a complete list of the desired parameters, not as a "delta" or change list from the current set of parameters. (That is, if you remove parameters from the list that were there in the last call to setParameters() and call setParameters() again with the revised list, this method will use default values for the removed parameters, rather than letting the current settings remain.) However, since the method fills in missing parameters, you may keep calling it with the ParameterList used in the previous call in order to get the same behavior as before.
Parameters that govern spectral bounds of the matrix:
ScalarType): lambdaMax, an upper bound of the bounding ellipse of the eigenvalues of the matrix A. If you do not set this parameter, we will compute an approximation. See "Parameters that govern eigenvalue
analysis" to control this approximation process.ScalarType): eigRatio, the ratio of lambdaMax to the lower bound of the bounding ellipse of the eigenvalues of A. We use lambdaMax and eigRatio to determine the Chebyshev iteration coefficients. This parameter is optional and defaults to 30.ScalarType): lambdaMin, a lower bound of real part of bounding ellipse of eigenvalues of the matrix A. This parameter is optional and only used for a quick check if the matrix is the identity matrix (if lambdaMax == lambdaMin == 1).Parameters that govern the number of Chebyshev iterations:
int): numIters, the number of iterations. This overrides "relaxation: sweeps" and "smoother: sweeps" (see below).int): numIters, the number of iterations. We include this for compatibility with Ifpack. This overrides "smoother: sweeps" (see below).int): numIters, as above. We include this for compatibility with ML.Parameters that govern eigenvalue analysis:
int): eigMaxIters, the number of power method iterations used to compute the maximum eigenvalue. This overrides "eigen-analysis:
iterations" (see below).int): eigMaxIters, as above. We include this parameter for compatibility with ML.std::string): The algorithm to use for estimating the max eigenvalue. This parameter is optional. Currently, we only support "power-method" (or "power method"), which is what Ifpack::Chebyshev uses for eigenanalysis. We include this parameter for compatibility with ML.Parameters that govern other algorithmic details:
RCP<const V> or const V*): If nonnull, we will use a deep copy of this vector for left scaling as the inverse diagonal of the matrix A, instead of computing the inverse diagonal ourselves. We will make a copy every time you call setParameters(). If you ever call setParameters() without this parameter, we will clear our copy and compute the inverse diagonal ourselves again. If you choose to provide this parameter, you are responsible for updating this if the matrix has changed.ST): minDiagVal. If any entry of the diagonal of the matrix is less than this in magnitude, it will be replaced with this value in the inverse diagonal used for left scaling.bool): If true, then always use the zero vector(s) as the initial guess(es). If false, then apply() will use X on input as the initial guess(es).Both the Ifpack and ML packages implement a Chebyshev smoother. We accept Ifpack and ML names for parameters whenever Ifpack2 has an equivalent parameter. Default settings for parameters relating to spectral bounds come from Ifpack.
The following list maps from an ML parameter to its corresponding Ifpack2 parameter.
ML does not have a parameter corresponding to "chebyshev: max eigenvalue", because ML estimates the spectral radius automatically. Ifpack and Ifpack2 both can estimate this automatically, but also let the user provide an estimate. Similarly, ML does not have a parameter corresponding to "chebyshev: min eigenvalue".
The following list maps from an Ifpack parameter to its corresponding Ifpack2 parameter. Many of the parameters have the same names, in which case we simply write same.
The optional user-provided vector of diagonal entries of the matrix may have any distribution for which an Export to the range Map of the matrix is legal. However, if the vector is already distributed according to the range Map, that saves us the communication cost of an Export. We also avoid the Export in case the row Map and the range Map of the matrix are the same. If they are not the same, and if the vector is distributed according to the row Map, we will reuse the Export from the matrix. Otherwise, we have to make a fresh Export object, which is more expensive. To avoid this cost, you should always provide a row Map or range Map vector for this parameter.
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Initialize the preconditioner.
The compute() method will call initialize() automatically if it has not yet been called, so you do not normally need to call this. However, it is correct to call initialize() yourself, and compute() will not call it again if it already has been called.
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Whether the preconditioner has been successfully initialized (by calling initialize()).
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(Re)compute the left scaling, and (if applicable) estimate max and min eigenvalues of D_inv * A.
You must call this method before calling apply(),
Users have the option to supply the left scaling vector D_inv and estimates of the min and max eigenvalues of D_inv * A as parameters to setParameters(). If users did not supply a left scaling, then this method will compute it by default (if assumeMatrixUnchanged is false). Likewise, if users did not supply at least an estimate of the max eigenvalue, this method will estimate it by default. If estimation of the eigenvalues is required, this method may take as long as several Chebyshev iterations.
Advanced users may avoid recomputing the left scaling vector and eigenvalue estimates by setting the "chebyshev: assume matrix
does not change" parameter of setParameters() to true. The left scaling vector and eigenvalue estimates will always be computed if the user did not provide them and we have not yet computed them. Any changes to parameters that affect computation of the inverse diagonal or estimation of the eigenvalue bounds will not affect subsequent apply() operations, until the "chebyshev: assume matrix does not change" parameter is set back to false (its default value).
This method will call initialize() if it has not already been called. However, you may call initialize() before calling this method if you wish.
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Whether compute() has been called at least once.
Note that you must always call compute() if the matrix has changed, if you have called setParameters(), or if you have not yet called compute(). This method only tells you if compute() has been called at least once, not if you need to call compute(). Ifpack2 doesn't have an efficient way to tell if the matrix has changed, so we ask users to tell Ifpack2 if the matrix has changed.
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Change the matrix to be preconditioned.
| [in] | A | The new matrix. |
! isInitialized () ! isComputed ()Calling this method resets the preconditioner's state. After calling this method with a nonnull input, you must first call initialize() and compute() (in that order) before you may call apply().
You may call this method with a null input. If A is null, then you may not call initialize() or compute() until you first call this method again with a nonnull input. This method invalidates any previous factorization whether or not A is null, so calling setMatrix() with a null input is one way to clear the preconditioner's state (and free any memory that it may be using).
The new matrix A need not necessarily have the same Maps or even the same communicator as the original matrix.
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Apply the preconditioner to X, returning the result in Y.
This method actually computes Y = beta*Y + alpha*(M*X), where M*X represents the result of Chebyshev iteration on X, using the matrix Op(A). Op(A) is either A itself, its transpose \(A^T\), or its Hermitian transpose \(A^H\), depending on the mode argument. Since this class currently requires A to be real and symmetric positive definite, it should always be the case that \(A = A^T = A^H\), but we will still respect the mode argument.
| [in] | X | A (multi)vector to which to apply the preconditioner. |
| [in,out] | Y | A (multi)vector containing the result of applying the preconditioner to X. |
| [in] | mode | If Teuchos::NO_TRANS, apply the matrix A. If mode is Teuchos::NO_TRANS, apply its transpose \(A^T\). If Teuchos::CONJ_TRANS, apply its Hermitian transpose \(A^H\). |
| [in] | alpha | Scaling factor for the result of Chebyshev iteration. The default is 1. |
| [in] | beta | Scaling factor for Y. The default is 0. |
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The Tpetra::Map representing the domain of this operator.
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The Tpetra::Map representing the range of this operator.
| bool Ifpack2::Chebyshev< MatrixType >::hasTransposeApply | ( | ) | const |
Whether it's possible to apply the transpose of this operator.
| void Ifpack2::Chebyshev< MatrixType >::applyMat | ( | const Tpetra::MultiVector< scalar_type, local_ordinal_type, global_ordinal_type, node_type > & | X, |
| Tpetra::MultiVector< scalar_type, local_ordinal_type, global_ordinal_type, node_type > & | Y, | ||
| Teuchos::ETransp | mode = Teuchos::NO_TRANS |
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| ) | const |
Compute Y = Op(A)*X, where Op(A) is either A, \(A^T\), or \(A^H\).
| [in] | X | Input (multi)vector of sparse matrix-vector multiply. If mode == Teuchos::NO_TRANS, X must be in the domain Map of the matrix A. Otherwise, X must be in the range Map of A. |
| [out] | Y | Output (multi)vector of sparse matrix-vector multiply. If mode == Teuchos::NO_TRANS, Y must be in the range Map of the matrix A. Otherwise, Y must be in the domain Map of A. |
| [in] | mode | Whether to apply the matrix A, its transpose \(A^T\), or its conjugate transpose \(A^H\). This method applies A if mode is Teuchos::NO_TRANS, \(A^T\) if mode is Teuchos::TRANS, and \(A^H\) (the Hermitian transpose) if mode is Teuchos::CONJ_TRANS. |
Since this class currently requires A to be real and symmetric positive definite, setting mode should not affect the result.
| Teuchos::RCP< const Teuchos::Comm< int > > Ifpack2::Chebyshev< MatrixType >::getComm | ( | ) | const |
The communicator over which the matrix is distributed.
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virtual |
The matrix for which this is a preconditioner.
| Teuchos::RCP< const Tpetra::CrsMatrix< typename MatrixType::scalar_type, typename MatrixType::local_ordinal_type, typename MatrixType::global_ordinal_type, typename MatrixType::node_type > > Ifpack2::Chebyshev< MatrixType >::getCrsMatrix | ( | ) | const |
Attempt to return the matrix A as a Tpetra::CrsMatrix.
This class does not require that A be a Tpetra::CrsMatrix. If it is NOT, this method will return Teuchos::null.
| double Ifpack2::Chebyshev< MatrixType >::getComputeFlops | ( | ) | const |
The total number of floating-point operations taken by all calls to compute().
| double Ifpack2::Chebyshev< MatrixType >::getApplyFlops | ( | ) | const |
The total number of floating-point operations taken by all calls to apply().
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virtual |
The total number of successful calls to initialize().
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The total number of successful calls to compute().
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The total number of successful calls to apply().
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The total time spent in all calls to initialize().
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The total time spent in all calls to compute().
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virtual |
The total time spent in all calls to apply().
| size_t Ifpack2::Chebyshev< MatrixType >::getNodeSmootherComplexity | ( | ) | const |
Get a rough estimate of cost per iteration.
| MatrixType::scalar_type Ifpack2::Chebyshev< MatrixType >::getLambdaMaxForApply | ( | ) | const |
The estimate of the maximum eigenvalue used in the apply().
| std::string Ifpack2::Chebyshev< MatrixType >::description | ( | ) | const |
A simple one-line description of this object.
| void Ifpack2::Chebyshev< MatrixType >::describe | ( | Teuchos::FancyOStream & | out, |
| const Teuchos::EVerbosityLevel | verbLevel = Teuchos::Describable::verbLevel_default |
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| ) | const |
Print the object with some verbosity level to a Teuchos::FancyOStream.
1.8.5