ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains the procedures for multiplication of a square triangular matrix in various transpositions with a general matrix.
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Data Types | |
interface | setMatMulTri |
Return the matrix solution to the system of linear equations with a triangular coefficient matrix \(\ms{C}\). More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_matrixMulTri" |
This module contains the procedures for multiplication of a square triangular matrix in various transpositions with a general matrix.
Such matrix multiplications frequently occur in solving general or triangular systems of equations with single or multiple right-hand sides (as a matrix).
Each system of linear equation is comprised of
The procedures under the generic interface setMatMulTri return the solution of the following corresponding linear triangular system of equations,
System of Equations | Solution (corresponding matrix multiplication) | BLAS routines |
---|---|---|
\(\ms{C}^{-1} \ms{X}^{\hphantom{-T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{C}^\hphantom{T} \ms{I} \) | \(\ms{?trmv}~/~\ms{?trmm}\) |
\(\ms{C}^{-T} \ms{X}^{\hphantom{-T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{C}^{T} \ms{I} \) | \(\ms{?trmv}~/~\ms{?trmm}\) |
\(\ms{C}^{-H} \ms{X}^{\hphantom{-T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{C}^{H} \ms{I} \) | \(\ms{?trmv}~/~\ms{?trmm}\) |
\(\ms{X}^{\hphantom{-T}} \ms{C}^{-1} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{I}^{\hphantom{T}} \ms{C} \) | \(\ms{?trmv}~/~\ms{?trmm}\) |
\(\ms{X}^{\hphantom{-T}} \ms{C}^{-T} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{I}^{\hphantom{T}} \ms{C}^{T} \) | \(\ms{?trmv}~/~\ms{?trmm}\) |
\(\ms{X}^{\hphantom{-T}} \ms{C}^{-H} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{I}^{\hphantom{T}} \ms{C}^{H} \) | \(\ms{?trmv}~/~\ms{?trmm}\) |
System of Equations | Solution (corresponding matrix multiplication) | BLAS routines |
---|---|---|
\(\ms{C}^{\hphantom{T}} \ms{X}^{\hphantom{T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{C}^{-1} \ms{I} \) | \(\ms{?trsv}~/~\ms{?trsm}\) |
\(\ms{C}^T \ms{X}^{\hphantom{T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{C}^{-T} \ms{I} \) | \(\ms{?trsv}~/~\ms{?trsm}\) |
\(\ms{C}^H \ms{X}^{\hphantom{T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{C}^{-H} \ms{I} \) | \(\ms{?trsv}~/~\ms{?trsm}\) |
\(\ms{X}^{\hphantom{T}} \ms{C}^{\hphantom{T}} = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{I}^{\hphantom{-T}} \ms{C}^{-1} \) | \(\ms{?trsv}~/~\ms{?trsm}\) |
\(\ms{X}^{\hphantom{T}} \ms{C}^T = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{I}^{\hphantom{-T}} \ms{C}^{-T} \) | \(\ms{?trsv}~/~\ms{?trsm}\) |
\(\ms{X}^{\hphantom{T}} \ms{C}^H = \alpha\ms{I}\) | \( \ms{I} \leftarrow \alpha \ms{I}^{\hphantom{-T}} \ms{C}^{-H} \) | \(\ms{?trsv}~/~\ms{?trsm}\) |
where
Computational efficiency and practical considerations require the (matrix multiplication) solution to be written to \(\ms{I}\) instead of being returned explicitly to a separate argument as \(\ms{X}\).
STRMV
, DTRMV
, CTRMV
, and ZTRMV
, STRSV
, DTRSV
, CTRSV
, and ZTRSV
, STRMM
, DTRMM
, CTRMM
, and ZTRMM
, STRSM
, DTRSM
, CTRSM
, and ZTRSM
.?TPMV
, ?TPSV
, ?TBMV
and ?TBSV
for Linear Full and Band packing format of the triangular input matrix must be implemented.
Final Remarks ⛓
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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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character(*, SK), parameter pm_matrixMulTri::MODULE_NAME = "@pm_matrixMulTri" |
Definition at line 119 of file pm_matrixMulTri.F90.