Creating an Epetra_CrsMatrix

This file first gives an example of how to create an Epetra_CrsMatrix object, then it details the supported matrices and gives a list of required parameters.

Given an already created Epetra_Map, Galeri can construct an Epetra_CrsMatrix object that has this Map as RowMatrixRowMap(). A simple example is as follows. Let `Map`

be an already created `Epetra_Map*`

object; then, a diagonal matrix with on the diagonal can be created using the instructions

#include "Galeri_CrsMatrices.h" using namespace Galeri; ... string MatrixType = "Diag"; List.set("a", 2.0); Epetra_CrsMatrix* Matrix = CreateCrsMatrix(MatrixType, Map, List);

More interesting matrices can be easily created. For example, a 2D biharmonic operator can be created like this:

List.set("nx", 10); List.set("ny", 10); Epetra_CrsMatrix* Matrix = Galeri.Create("Biharmonic2D", Map, List);

For matrices arising from 2D discretizations on Cartesian grids, it is possible to visualize the computational stencil at a given grid point by using function PrintStencil2D, defined in the Galeri namespace:

#include "Galeri_Utils.h" using namespace Galeri; ... // Matrix is an already created Epetra_CrsMatrix* object // and nx and ny the number of nodes along the X-axis and Y-axis, // respectively. PrintStencil2D(Matrix, nx, ny);

The output is:

2D computational stencil at GID 12 (grid is 5 x 5) 0 0 1 0 0 0 2 -8 2 0 1 -8 20 -8 1 0 2 -8 2 0 0 0 1 0 0

To present the list of supported matrices we adopt the following symbols:

- MATLAB: please refer to the MATLAB documentation for more details on the properties of this matrix;
- DENSE: the matrix is dense (but still stored as Epetra_CrsMatrix);
- MAP: the number of elements and its distribution are determined from the input map;
- MAP2D: the input map has been created by CreateMap(), using MapType =
`Cartesian2D`

. The values of`nx`

and`ny`

are still available in the input list; - MAP3D: the input map has been created by CreateMap(), using MapType =
`Cartesian3D`

. The values of`nx`

,`ny`

and`nz`

are still available in the input list; - SER: the matrix can be obtained in serial environments only;
- PAR: the matrix can be obtained in parallel (and serial) environments.
- The symbol indicates the matrix of the element in MATLAB notation (that is, starting from 1).

The list of supported matrices is now reported in alphabetical order.

`BentPipe2D`

(MAP2D, PAR): Returns a matrix corresponding to the finite-difference discretization of the problem`diff`

, and that of using`conv`

. The default values are`diff=1e-5`

,`conv=1`

.`BigCross2D`

(MAP2D, PAR): Creates a matrix corresponding to the following stencil:`Laplace2DFourthOrder`

. A non-default value must be set in the input parameter list before creating the matrix. For example, to specify the value of , one should doList.set("ee", 12.0); Matrix = Galeri.Create("BigCross2D", Map, List);

`BigStar2D`

(MAP2D, PAR): Creates a matrix corresponding to the stencil`Biharmonic2D`

.`Biharmonic2D`

(MAP2D, PAR): Creates a matrix corresponding to the discrete biharmonic operator,`Cauchy`

(MAP, MATLAB, DENSE, PAR): Creates a particular instance of a Cauchy matrix with elements . Explicit formulas are known for the inverse and determinant of a Cauchy matrix. For this particular Cauchy matrix, the determinant is nonzero and the matrix is totally positive.`Cross2D`

(MAP2D, PAR): Creates a matrix with the same stencil of`Laplace2D}`

, but with arbitrary values. The computational stencil isThe default values are

List.set("a", 4.0); List.set("b", -1.0); List.set("c", -1.0); List.set("d", -1.0); List.set("e", -1.0);

For example, to approximate the 2D Helmhotlz equation

with the standard 5-pt discretization stencil

and , one can set

List.set("a", 4 - 0.1 * h * h); List.set("b", -1.0); List.set("c", -1.0); List.set("d", -1.0); List.set("e", -1.0);

The factor can be considered by scaling the input parameters.

`Cross3D`

(MAP3D, PAR): Similar to the Cross2D case. The matrix stencil correspond to that of a 3D Laplace operator on a structured 3D grid. On a given x-y plane, the stencil is as in`Laplace2D`

. The value on the plane below is set using`f`

, the value on the plane above with`g`

.`Diag`

(MAP, PAR): Creates , where is the identity matrix of size`n`

. The default value isList.set("a", 1.0);

`Fiedler`

(MAP, MATLAB, DENSE, PAR): Creates a matrix whose element are . The matrix is symmetric, and has a dominant positive eigenvalue, and all the other eigenvalues are negative.`Hanowa`

(MAP, MATLAB, PAR): Creates a matrix whose eigenvalues lie on a vertical line in the complex plane. The matrix has the 2x2 block structure (in MATLAB's notation)List.set("a", -1.0);

`Hilbert`

(MAP, MATLAB, DENSE, PAR): This is a famous example of a badly conditioned matrix. The elements are defined in MATLAB notation as .`JordanBlock`

(MAP, MATLAB, PAR): Creates a Jordan block with eigenvalue`lambda`

. The default value is`lambda=0.1`

;`KMS`

(MAP, MATLAB, DENSE, PAR): Create the Kac-Murdock-Szego Toepliz matrix such that (for real only). Default value is , or can be using`rho`

. The inverse of this matrix is tridiagonal, and the matrix is positive definite if and only if . The default value is`rho=-0.5`

;`Laplace1D`

(MAP, PAR): Creates the classical tridiagonal matrix with stencil .`Laplace1DNeumann`

(MAP, PAR): As for`Laplace1D`

, but with Neumann boundary conditioners. The matrix is singular.`Laplace2D`

(MAP2D, PAR): Creates a matrix corresponding to the stencil of a 2D Laplacian operator on a structured Cartesian grid. The matrix stencil is:`Laplace2DFourthOrder`

(MAP2D, PAR): Creates a matrix corresponding to the stencil of a 2D Laplacian operator on a structured Cartesian grid. The matrix stencil is:`Laplace3D`

(MAP3D, PAR): Creates a matrix corresponding to the stencil of a 3D Laplacian operator on a structured Cartesian grid.`Lehmer`

(MAP, MATLAB, DENSE, PAR): Returns a symmetric positive definite matrix, such that`Minij`

(MAP, MATLAB, DENSE, PAR): Returns the symmetric positive definite matrix defined as .`Ones`

(MAP, PAR): Returns a matrix with . The default value is`a=1`

;`Parter`

(MAP, MATLAB, DENSE, PAR): Creates a matrix . This matrix is a Cauchy and a Toepliz matrix. Most of the singular values of A are very close to .`Pei`

(MAP, MATLAB, DENSE, PAR): Creates the matrix`Recirc2D`

(MAP2D, PAR): Returns a matrix corresponding to the finite-difference discretization of the problem`diff`

, and that of using`conv`

. The default values are`diff=1e-5`

,`conv=1`

.`Ris`

(MAP, MATLAB, PAR): Returns a symmetric Hankel matrix with elements , where is problem size. The eigenvalues of A cluster around and .`Star2D`

(MAP2D, PAR): Creates a matrix with the 9-point stencil:List.set("a", 8.0); List.set("b", -1.0); List.set("c", -1.0); List.set("d", -1.0); List.set("e", -1.0); List.set("z1", -1.0); List.set("z2", -1.0); List.set("z3", -1.0); List.set("z4", -1.0);

`Stretched2D`

(MAP2D, PAR): Creates a matrix corresponding to the following stencil:`epsilon=0.1`

;`Tridiag`

(MAP, PAR): Creates a tridiagonal matrix with stencilList.set("a", 2.0); List.set("b", -1.0); List.set("c", -1.0);

`UniFlow2D`

(MAP2D, PAR): Returns a matrix corresponding to the finite-difference discretization of the problemList.set("alpha", .0); List.set("diff", 1e-5); List.set("conv", 1.0);

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