=======================
Triangular Solver (trs)                                               [TOC:2]
=======================

*trs* (defined in namespace `flens::lapack`) solves a system of linear equations

    *--[LATEX]----------------------------------------*
    |                                                 |
    |  A X = B    \quad\text{or}\quad    A^T X = B    |
    |                                                 |
    *-------------------------------------------------*

Two variants are provided:
 - For a general $n \times n$ matrix $A$ containing the $LU$ factorization
   computed by __trf__.
 - For a triangular matrix $A$.

Further subvariants are provided for cases where the right hand side $B$ is
a matrix or vector.


Variant: System of Linear Equations
===================================

Multiple Right-Hand Sides
-------------------------
*--[CODEREF]----------------------------------------------------------------*
|                                                                           |
|  template <typename MA, typename VPIV, typename MB>                       |
|      typename RestrictTo<IsGeMatrix<MA>::value                            |
|                       && IsIntegerDenseVector<VPIV>::value                |
|                       && IsGeMatrix<MB>::value,                           |
|               void>::Type                                                 |
|      trs(Transpose trans, const MA &A, const VPIV &piv, MB &&B);          |
|                                                                           |
*---------------------------------------------------------------------------*
                         [c:@N@flens@N@lapack@FT@>3#T#T#Ttrs#$@N@cxxblas@E@T]
                         [ranspose#&1t0.0#&1t0.1#&t0.2#templatetypenameMA,ty]
                         [penameVPIV,typenameMBtypenameRestrictToIsGeMatrixM]
                         [AvalueandIsIntegerDenseVectorVPIVvalueandIsGeMatri]
                         [xMBvalue,voidType                                 ]

    trans    `(input) bool`                                                    +
             Solve $AX=B$, $A^T X=B$ or $A^H X = B$
    A        `(input) real or complex valued GeMatrix`                         +
             The factors $L$ and $U$ from the factorization $A = PLU$ as
             computed __trf__.
    piv      `(input) integer valued DenseVector`                              +
             The pivot indices as returned from __trf__.
    B        `(input/output) real or complex valued GeMatrix`                  +
             On entry, the $n \times n_{rhs}$ right hand side matrix $B$.      +
             On exit, the solution matrix $X$.


Single Right-Hand Side
----------------------
*--[CODEREF]----------------------------------------------------------------*
|                                                                           |
|  template <typename MA, typename VPIV, typename VB>                       |
|      typename RestrictTo<IsGeMatrix<MA>::value                            |
|                       && IsIntegerDenseVector<VPIV>::value                |
|                       && IsDenseVector<VB>::value,                        |
|               void>::Type                                                 |
|      trs(Transpose trans, const MA &A, const VPIV &piv, VB &&b);          |
|                                                                           |
*---------------------------------------------------------------------------*
                         [c:@N@flens@N@lapack@FT@>3#T#T#Ttrs#$@N@cxxblas@E@T]
                         [ranspose#&1t0.0#&1t0.1#&t0.2#templatetypenameMA,ty]
                         [penameVPIV,typenameVBtypenameRestrictToIsGeMatrixM]
                         [AvalueandIsIntegerDenseVectorVPIVvalueandIsDenseVe]
                         [ctorVBvalue,voidType                              ]

    trans    `(input) bool`                                                    +
             Solve $AX=B$, $A^T X=B$ or $A^H X = B$
    A        `(input) real or complex valued GeMatrix`                         +
             The factors $L$ and $U$ from the factorization $A = PLU$ as
             computed __trf__.
    piv      `(input) integer valued DenseVector`                              +
             The pivot indices as returned from __trf__.
    b        `(input/output) real or complex valued DenseVector`               +
             On entry, the right hand side vector $b$.                         +
             On exit, the solution vector $x$.



Variant: Triangular System
==========================

Multiple Right-Hand Sides
-------------------------
*--[CODEREF]----------------------------------------------------------------*
|                                                                           |
|  template <typename MA, typename MB>                                      |
|      typename RestrictTo<IsTrMatrix<MA>::value                            |
|                       && IsGeMatrix<MB>::value,                           |
|               typename RemoveRef<MA>::Type::IndexType>::Type              |
|      trs(Transpose trans, const MA &A, MB &&B);                           |
|                                                                           |
*---------------------------------------------------------------------------*
                         [c:@N@flens@N@lapack@FT@>2#T#Ttrs#$@N@cxxblas@E@Tra]
                         [nspose#&1t0.0#&t0.1#templatetypenameMA,typenameMBt]
                         [ypenameRestrictToIsTrMatrixMAvalueandIsGeMatrixMBv]
                         [alue,typenameRemoveRefMATypeIndexTypeType         ]

    trans    `(input) bool`                                                    +
             Solve $AX=B$, $A^T X=B$ or $A^H X = B$
    A        `(input) real or complex valued TrMatrix`                         +
             The $n \times n$ triangular matrix.
    B        `(input/output) real or complex valued GeMatrix`                  +
             On entry, the $n \times n_{rhs}$ right hand side matrix $B$.      +
             On exit, the solution matrix $X$.


Single Right-Hand Side
----------------------
*--[CODEREF]----------------------------------------------------------------*
|                                                                           |
|  template <typename MA, typename VB>                                      |
|      typename RestrictTo<IsTrMatrix<MA>::value                            |
|                       && IsDenseVector<VB>::value,                        |
|               typename RemoveRef<MA>::Type::IndexType>::Type              |
|      trs(Transpose trans, const MA &A, VB &&b);                           |
|                                                                           |
*---------------------------------------------------------------------------*
                         [c:@N@flens@N@lapack@FT@>2#T#Ttrs#$@N@cxxblas@E@Tra]
                         [nspose#&1t0.0#&t0.1#templatetypenameMA,typenameVBt]
                         [ypenameRestrictToIsTrMatrixMAvalueandIsDenseVector]
                         [VBvalue,typenameRemoveRefMATypeIndexTypeType      ]

    trans    `(input) bool`                                                    +
             Solve $AX=B$, $A^T X=B$ or $A^H X = B$
    A        `(input) real or complex valued TrMatrix`                         +
             The $n \times n$ triangular matrix.
    b        `(input/output) real or complex valued DenseVector`               +
             On entry, the $n \times n_{rhs}$ right hand side vector $b$.      +
             On exit, the solution vector $x$.

Notes
=====
- *Examples*: __lapack-getrs__ and __lapack-trtrs__.
- *trs* is a port of __dgetrs__, __zgetrs__, __dtrtrs__ and __ztrtrs__.  Also
  this documentation is taken from there.


:links: lapack-getrs -> doc:flens/examples/lapack-getrs
        lapack-trtrs -> doc:flens/examples/lapack-trtrs
        (d.+)        -> file:cxxlapack/netlib/lapack/$1.f
        (z.+)        -> file:cxxlapack/netlib/lapack/$1.f
        (.+)         -> doc:flens/lapack/impl/$1