pm_matrixClass::frobenius_type Type Reference

This is a concrete derived type whose instances are exclusively used to signify the frobenius class of a given matrix within an interface of a procedure of the ParaMonte library.
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Inheritance diagram for pm_matrixClass::frobenius_type:

Collaboration diagram for pm_matrixClass::frobenius_type:

Detailed Description

This is a concrete derived type whose instances are exclusively used to signify the frobenius class of a given matrix within an interface of a procedure of the ParaMonte library.

Objects instantiated from this derived type are exclusively used to differentiate the procedures within the various generic interfaces of the ParaMonte library.
As such, this concrete derived type does not contain any attributes.

A Frobenius matrix is a special kind of square matrix from numerical mathematics.
A matrix is a Frobenius matrix if it has the following three properties:

1. All entries on the main diagonal are ones.
   
2. The entries below the main diagonal of at most one column are arbitrary.
   
3. Every other entry is zero.
   

The following matrix is an example.

\begin{equation} A = \begin{pmatrix} 1&0&0&\cdots &0\\ 0&1&0&\cdots &0\\ 0&a_{32}&1&\cdots &0\\ \vdots &\vdots &\vdots &\ddots &\vdots \\ 0&a_{n2}&0&\cdots &1 \end{pmatrix} \end{equation}

Frobenius matrices are invertible.
The inverse of a Frobenius matrix is again a Frobenius matrix, equal to the original matrix with changed signs outside the main diagonal.
The inverse of the example above is therefore:

\begin{equation} A^{-1} = \begin{pmatrix} 1&0&0&\cdots &0\\ 0&1&0&\cdots &0\\ 0&-a_{{32}}&1&\cdots &0\\ \vdots &\vdots &\vdots &\ddots &\vdots \\ 0&-a_{{n2}}&0&\cdots &1 \end{pmatrix} \end{equation}

Note

This concrete derived type is not meant to be directly accessed by the end users.
Instead, the end users should use the specific object parameter instance of this derived type (e.g., frobenius) as directed by the documentation of the specific procedure they intend to use.

See also

genrecmat
posdefmat
symmetric
hermitian
triang_type
posdefmat_type
symmetric_type
hermitian_type
upperDiag_type
lowerDiag_type
upperZero_type
lowerZero_type
upperUnit_type
lowerUnit_type
unitTriang_type
atomicTriang_type
frobenius_type
genrecmat_type
matrix_type
gauss_type

Final Remarks ⛓

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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
   
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This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, September 1, 2017, 12:00 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas Austin

Definition at line 1889 of file pm_matrixClass.F90.

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The documentation for this type was generated from the following file:

• src/fortran/main/pm_matrixClass.F90