[Docs] batched serial tbsv (#2796)

* fix syr routine name

Signed-off-by: yasahi-hpc <y.asahi6412@gmail.com>

* Add batched serial tbsv docs

Signed-off-by: yasahi-hpc <y.asahi6412@gmail.com>

---------

Signed-off-by: yasahi-hpc <y.asahi6412@gmail.com>

Author: yasahi-hpc

docs/source/API/batched/dense-index.rst

@@ -5,6 +5,7 @@ API: Batched Dense (DLA)
    :maxdepth: 2
    :hidden:
 
+   dense/batched_tbsv
    dense/batched_ger
    dense/batched_syr
    dense/batched_getrf
@@ -196,7 +197,7 @@ BLAS 2
      - `TeamTrsv`
      - `TeamVectorTrsv`
    * - TBSV
-     - `SerialTbsv`
+     - :doc:`SerialTbsv <dense/batched_tbsv>`
      - --
      - --
    * - TPSV

docs/source/API/batched/dense/batched_syr.rst

@@ -20,7 +20,7 @@ Perform a symmetric rank-1 update of matrix :math:`A` by vector :math:`x` with s
    A &= A + \alpha (x * x^H) \: \text{(if ArgTrans == KokkosBatched::Trans::ConjTranspose)}
    \end{align}
 
-1. If ``ArgTrans == KokkosBatched::Trans::Transpose``, this operation is equivalent to the BLAS routine ``SSYR`` (``CSYR``) or ``DGER`` (``ZSYR``) for single or double precision for real (complex) matrix.
+1. If ``ArgTrans == KokkosBatched::Trans::Transpose``, this operation is equivalent to the BLAS routine ``SSYR`` (``CSYR``) or ``DSYR`` (``ZSYR``) for single or double precision for real (complex) matrix.
 
 2. If ``ArgTrans == KokkosBatched::Trans::ConjTranspose``, this operation is equivalent to the BLAS routine ``CHER`` or ``ZHER`` for single or double precision for complex matrix.
 

docs/source/API/batched/dense/batched_tbsv.rst

@@ -0,0 +1,66 @@
+KokkosBatched::Tbsv
+###################
+
+Defined in header: :code:`KokkosBatched_Tbsv.hpp`
+
+.. code:: c++
+
+    template <typename ArgUplo, typename ArgTrans, typename ArgDiag, typename ArgAlgo>
+    struct SerialTbsv {
+      template <typename AViewType, typename XViewType>
+      KOKKOS_INLINE_FUNCTION static int invoke(const AViewType &A, const XViewType &X, const int k);
+    };
+
+
+Solves a system of the linear equations :math:`A \cdot X = B` or :math:`A^T \cdot X = B` or :math:`A^H \cdot X = B` where :math:`A` is an n-by-n unit or non-unit, upper or lower triangular band matrix with :math:`(k + 1)` diagonals.
+
+1. For a real band matrix :math:`A`, this solves a system of the linear equations :math:`A \cdot X = B` or :math:`A^T \cdot X = B`.
+   This operation is equivalent to the BLAS routine ``STBSV`` or ``DTBSV`` for single or double precision.
+
+2. For a complex band matrix :math:`A`, this solves a system of the linear equations :math:`A \cdot X = B` or :math:`A^T \cdot X = B` or :math:`A^H \cdot X = B`.
+   This operation is equivalent to the BLAS routine ``CTBSV`` or ``ZTBSV`` for single or double precision.
+
+.. note::
+
+  No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine.
+
+Parameters
+==========
+
+:A: Input view containing the upper or lower triangular band matrix. See `LAPACK reference <https://www.netlib.org/lapack/lug/node124.html>`_ for the band storage format.
+:X: Input/output view containing the right-hand side on input and the solution on output.
+:k: The number of superdiagonals or subdiagonals within the band of :math:`A`. :math:`k >= 0`
+
+
+Type Requirements
+-----------------
+
+- ``ArgUplo`` must be one of the following:
+   - ``KokkosBatched::Uplo::Upper`` for upper triangular solve
+   - ``KokkosBatched::Uplo::Lower`` for lower triangular solve
+
+- ``ArgTrans`` must be one of the following:
+   - ``KokkosBatched::Trans::NoTranspose`` to solve a system :math:`A \cdot X = B`
+   - ``KokkosBatched::Trans::Transpose`` to solve a system :math:`A^T \cdot X = B`
+   - ``KokkosBatched::Trans::ConjTranspose`` to solve a system :math:`A^H \cdot X = B`
+
+- ``ArgDiag`` must be one of the following:
+   - ``KokkosBatched::Diag::Unit`` for the unit triangular matrix :math:`A`
+   - ``KokkosBatched::Diag::NonUnit`` for the non-unit triangular matrix :math:`A`
+
+- ``ArgAlgo`` must be ``KokkosBatched::Algo::tbsv::Unblocked`` for the unblocked algorithm
+- ``AViewType`` must be a Kokkos `View <https://kokkos.org/kokkos-core-wiki/API/core/view/view.html>`_ of rank 2 containing the band matrix A
+- ``XViewType`` must be a Kokkos `View <https://kokkos.org/kokkos-core-wiki/API/core/view/view.html>`_ of rank 1 containing the right-hand side that satisfies
+  - ``std::is_same_v<typename XViewType::value_type, typename XViewType::non_const_value_type> == true``
+
+Example
+=======
+
+.. literalinclude:: ../../../../../example/batched_solve/serial_tbsv.cpp
+  :language: c++
+
+output:
+
+.. code::
+
+   tbsv works correctly!