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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains abstract and concrete derived types and procedures related to various common matrix transposition operations for which there is a corresponding matrix class defined in pm_matrixClass.
More...
Data Types | |
| interface | setMatTrans |
| Generate and return the transpose of the input matrix of arbitrary type and kind using a cache-oblivious approach. More... | |
| type | trans_type |
| This is a derived type for constructing concrete derived types to distinguish various procedure signatures that require different forms of transposition (Symmetric, Hermitian, ...). More... | |
| type | transHerm_type |
| This is a concrete derived type whose instances are exclusively used to request Hermitian (conjugate) transpose ( \(\cdot^H\)) of a given matrix within an interface of a procedure of the ParaMonte library. More... | |
| type | transHermSkew_type |
| This is a concrete derived type whose instances are exclusively used to request Skew-Hermitian transpose ( \(-\cdot^H\)) of a given matrix within an interface of a procedure of the ParaMonte library. More... | |
| type | transOrth_type |
| This is a concrete derived type whose instances are exclusively used to request Orthogonal Transpose ( \(\cdot^-T\)) of a given matrix within an interface of a procedure of the ParaMonte library. More... | |
| type | transSymm_type |
| This is a concrete derived type whose instances are exclusively used to request Symmetric transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library. More... | |
| type | transSymmSkew_type |
| This is a concrete derived type whose instances are exclusively used to request Skew-Symmetric transpose ( \(-\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library. More... | |
| type | transUnit_type |
| This is a concrete derived type whose instances are exclusively used to request Unitary Transpose ( \(\cdot^{-H}\)) of a given matrix within an interface of a procedure of the ParaMonte library. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_matrixTrans" |
| type(trans_type), parameter | trans = trans_type() |
This is a scalar parameter object of type trans_type that is exclusively used to request no transpose of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
| type(transSymm_type), parameter | transSymm = transSymm_type() |
This is a scalar parameter object of type transSymm_type that is exclusively used to request Symmetric transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
| type(transHerm_type), parameter | transHerm = transHerm_type() |
This is a scalar parameter object of type transHerm_type that is exclusively used to request Hermitian (conjugate) transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
| type(transOrth_type), parameter | transOrth = transOrth_type() |
This is a scalar parameter object of type transOrth_type that is exclusively used to request Orthogonal Transpose ( \(\cdot^-T\)) of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
| type(transUnit_type), parameter | transUnit = transUnit_type() |
This is a scalar parameter object of type transUnit_type that is exclusively used to request Unitary Transpose ( \(\cdot^{-H}\)) of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
| type(transSymmSkew_type), parameter | transSymmSkew = transSymmSkew_type() |
This is a scalar parameter object of type transSymmSkew_type that is exclusively used to request Skew-Symmetric transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
| type(transHermSkew_type), parameter | transHermSkew = transHermSkew_type() |
This is a scalar parameter object of type transHermSkew_type that is exclusively used to request Skew-Hermitian transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.More... | |
This module contains abstract and concrete derived types and procedures related to various common matrix transposition operations for which there is a corresponding matrix class defined in pm_matrixClass.
There are a myriad of operations that can be applied to modify matrices, for example, matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix.
The entities and derived types of this module currently focus on operations that are required for compile-time resolution of procedures within the generic interfaces of the ParaMonte library for Linear Algebra operations.
Such procedures frequently need to work on either an Identical, Inverse, Symmetric transpose, Hermitian transpose, Orthogonal Transpose, Unitary transpose, or other forms of some of their input matrix arguments.
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal.
It switches the row and column indices of a given matrix \(A\) by producing another (transposed) matrix.
The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.
Notation
The transpose of a given matrix \(A\) is frequently denoted by \(A^T\).
In the case of square matrices, \(A^T\) may also denote the \(T\)th power of the matrix \(A\).
To avoid a possible confusion, some use left upperscripts to denote transpose, that is, \({}^TA\).
An advantage of this notation is that no parentheses are needed when exponents are involved as \(({}^TA)^n = {}^T(A^n)\), notation \({}^TA^n\) is not ambiguous.
Definition
The transpose of a matrix \(A\) may be constructed by any one of the following methods,
Formally, the \(i\)-th row, \(j\)-th column element of \(A^T\) is the \(j\)-th row, \(i\)-th column element of \(A\),
\begin{equation} \left[\mathbf{A}^{\up{T}}\right]_{ij} = \left[\mathbf{A} \right]_{ji} ~. \end{equation}
If \(A\) is an \(m \times n\) matrix, then \(A^T\) is an \(n \times m\) matrix.
Matrix transposition types and correspondence to matrix classes:
There are also special matrix operations that mix inversion with Symmetric and Hermitian each having a corresponding matrix classes:**
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
| character(*,SK), parameter pm_matrixTrans::MODULE_NAME = "@pm_matrixTrans" |
Definition at line 123 of file pm_matrixTrans.F90.
| type(trans_type), parameter pm_matrixTrans::trans = trans_type() |
This is a scalar parameter object of type trans_type that is exclusively used to request no transpose of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 185 of file pm_matrixTrans.F90.
| type(transHerm_type), parameter pm_matrixTrans::transHerm = transHerm_type() |
This is a scalar parameter object of type transHerm_type that is exclusively used to request Hermitian (conjugate) transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 328 of file pm_matrixTrans.F90.
| type(transHermSkew_type), parameter pm_matrixTrans::transHermSkew = transHermSkew_type() |
This is a scalar parameter object of type transHermSkew_type that is exclusively used to request Skew-Hermitian transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 612 of file pm_matrixTrans.F90.
| type(transOrth_type), parameter pm_matrixTrans::transOrth = transOrth_type() |
This is a scalar parameter object of type transOrth_type that is exclusively used to request Orthogonal Transpose ( \(\cdot^-T\)) of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 399 of file pm_matrixTrans.F90.
| type(transSymm_type), parameter pm_matrixTrans::transSymm = transSymm_type() |
This is a scalar parameter object of type transSymm_type that is exclusively used to request Symmetric transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 256 of file pm_matrixTrans.F90.
| type(transSymmSkew_type), parameter pm_matrixTrans::transSymmSkew = transSymmSkew_type() |
This is a scalar parameter object of type transSymmSkew_type that is exclusively used to request Skew-Symmetric transpose ( \(\cdot^T\)) of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 541 of file pm_matrixTrans.F90.
| type(transUnit_type), parameter pm_matrixTrans::transUnit = transUnit_type() |
This is a scalar parameter object of type transUnit_type that is exclusively used to request Unitary Transpose ( \(\cdot^{-H}\)) of a given matrix within an interface of a procedure of the ParaMonte library.
For example usage, see the documentation of the target procedure requiring this object.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
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.
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.
Definition at line 470 of file pm_matrixTrans.F90.
1.9.3