docs: add banded matrices intro section in lapack/base

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Author: aayush0325

lib/node_modules/@stdlib/lapack/base/README.md

@@ -22,6 +22,159 @@ limitations under the License.
 
 > Base (i.e., lower-level) linear algebra package (LAPACK) routines.
 
+<section class="intro">
+
+## Band Storage
+
+Many LAPACK routines work with banded matrices, which are stored compactly in two-dimensional arrays to save memory and improve computational efficiency. This section describes the different band storage formats used throughout the LAPACK library.
+
+### General Band Matrix Storage
+
+A general band matrix of size `M`-by-`N` with `KL` subdiagonals and `KU` superdiagonals is stored in a two-dimensional array `AB` with `KL+KU+1` rows and `N` columns.
+
+**Storage Mapping:**
+
+-   Columns of the original matrix are stored in corresponding columns of the array `AB`
+-   Diagonals of the matrix are stored in rows of the array `AB`
+-   Element `A( i, j )` from the original matrix is stored in `AB(KU+1+i-j, j)`
+-   Valid range for `i`: `max(1, j-KU) <= i <= min(M, j+KL)`
+
+#### Example (M=N=5, KL=2, KU=1)
+
+Original band matrix `A`:
+
+<!-- <equation class="equation" label="eq:band_matrix_a" align="center" raw="A = \left[\begin{array}{rrrrr}a_{11} & a_{12} & 0      & 0      & 0 \\a_{21} & a_{22} & a_{23} & 0      & 0 \\a_{31} & a_{32} & a_{33} & a_{34} & 0 \\0      & a_{42} & a_{43} & a_{44} & a_{45} \\0      & 0      & a_{53} & a_{54} & a_{55}\end{array}\right]" alt="Representation of band matrix A."> -->
+
+```math
+A = \left[
+\begin{array}{rrrrr}
+    a_{11} & a_{12} & 0      & 0      & 0 \\
+    a_{21} & a_{22} & a_{23} & 0      & 0 \\
+    a_{31} & a_{32} & a_{33} & a_{34} & 0 \\
+    0      & a_{42} & a_{43} & a_{44} & a_{45} \\
+    0      & 0      & a_{53} & a_{54} & a_{55}
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+Band storage in array `AB` (4×5, since KL+KU+1 = 2+1+1 = 4):
+
+<!-- <equation class="equation" label="eq:band_storage_ab" align="center" raw="AB = \left[\begin{array}{rrrrr}*      & a_{12} & a_{23} & a_{34} & a_{45} \\a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\a_{21} & a_{32} & a_{43} & a_{54} & *      \\a_{31} & a_{42} & a_{53} & *      & *\end{array}\right]" alt="Band storage representation of matrix A."> -->
+
+```math
+AB = \left[
+\begin{array}{rrrrr}
+    *      & a_{12} & a_{23} & a_{34} & a_{45} \\
+    a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\
+    a_{21} & a_{32} & a_{43} & a_{54} & *      \\
+    a_{31} & a_{42} & a_{53} & *      & *
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+Elements marked `*` need not be set and are not referenced by LAPACK routines.
+
+**Note:** When a band matrix is supplied for LU factorization, space must be allowed to store an additional `KL` superdiagonals, generated by fill-in as a result of row interchanges. This means that the matrix is stored according to the above scheme, but with `KL + KU` superdiagonals.
+
+### Triangular Band Matrices
+
+Triangular band matrices are stored in the same format as general band matrices:
+
+-   If `KL = 0`, the matrix is upper triangular
+-   If `KU = 0`, the matrix is lower triangular
+
+### Symmetric or Hermitian Band Matrices
+
+For symmetric or Hermitian band matrices with `KD` subdiagonals or superdiagonals, only the upper or lower triangle (as specified by `UPLO`) needs to be stored.
+
+**Upper Triangle Storage (UPLO = 'U'):**
+
+-   Element `A( i, j )` is stored in `AB(KD+1+i-j, j)`
+-   Valid range for `i`: `max(1, j-KD) <= i <= j`
+
+**Lower Triangle Storage (UPLO = 'L'):**
+
+-   Element `A( i, j )` is stored in `AB(1+i-j, j)`
+-   Valid range for `i`: `j <= i <= min(N, j+KD)`
+
+#### Example (N=5, KD=2)
+
+For `UPLO = 'U'` (upper triangle stored):
+
+<!-- <equation class="equation" label="eq:symmetric_upper_a" align="center" raw="A = \left[\begin{array}{rrrrr}a_{11} & a_{12} & a_{13} & 0      & 0 \\{a_{12}} & a_{22} & a_{23} & a_{24} & 0 \\{a_{13}} & {a_{23}} & a_{33} & a_{34} & a_{35} \\0      & {a_{24}} & {a_{34}} & a_{44} & a_{45} \\0      & 0      & {a_{35}} & {a_{45}} & a_{55}\end{array}\right]" alt="Representation of symmetric band matrix A (upper triangle)."> -->
+
+```math
+A = \left[
+\begin{array}{rrrrr}
+    a_{11} & a_{12} & a_{13} & 0      & 0 \\
+    {a_{12}} & a_{22} & a_{23} & a_{24} & 0 \\
+    {a_{13}} & {a_{23}} & a_{33} & a_{34} & a_{35} \\
+    0      & {a_{24}} & {a_{34}} & a_{44} & a_{45} \\
+    0      & 0      & {a_{35}} & {a_{45}} & a_{55}
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+Band storage in array `AB` (3×5, since KD+1 = 2+1 = 3):
+
+<!-- <equation class="equation" label="eq:symmetric_upper_ab" align="center" raw="AB = \left[\begin{array}{rrrrr}*      & *      & a_{13} & a_{24} & a_{35} \\*      & a_{12} & a_{23} & a_{34} & a_{45} \\a_{11} & a_{22} & a_{33} & a_{44} & a_{55}\end{array}\right]" alt="Band storage representation of symmetric matrix A (upper triangle)."> -->
+
+```math
+AB = \left[
+\begin{array}{rrrrr}
+    *      & *      & a_{13} & a_{24} & a_{35} \\
+    *      & a_{12} & a_{23} & a_{34} & a_{45} \\
+    a_{11} & a_{22} & a_{33} & a_{44} & a_{55}
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+For `UPLO = 'L'` (lower triangle stored):
+
+<!-- <equation class="equation" label="eq:symmetric_lower_a" align="center" raw="A = \left[\begin{array}{rrrrr}a_{11} & {a_{21}} & {a_{31}} & 0      & 0 \\a_{21} & a_{22} & {a_{32}} & {a_{42}} & 0 \\a_{31} & a_{32} & a_{33} & {a_{43}} & {a_{53}} \\0      & a_{42} & a_{43} & a_{44} & {a_{54}} \\0      & 0      & a_{53} & a_{54} & a_{55}\end{array}\right]" alt="Representation of symmetric band matrix A (lower triangle)."> -->
+
+```math
+A = \left[
+\begin{array}{rrrrr}
+    a_{11} & {a_{21}} & {a_{31}} & 0      & 0 \\
+    a_{21} & a_{22} & {a_{32}} & {a_{42}} & 0 \\
+    a_{31} & a_{32} & a_{33} & {a_{43}} & {a_{53}} \\
+    0      & a_{42} & a_{43} & a_{44} & {a_{54}} \\
+    0      & 0      & a_{53} & a_{54} & a_{55}
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+Band storage in array `AB` (3×5, since KD+1 = 2+1 = 3):
+
+<!-- <equation class="equation" label="eq:symmetric_lower_ab" align="center" raw="AB = \left[\begin{array}{rrrrr}a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\a_{21} & a_{32} & a_{43} & a_{54} & *      \\a_{31} & a_{42} & a_{53} & *      & *\end{array}\right]" alt="Band storage representation of symmetric matrix A (lower triangle)."> -->
+
+```math
+AB = \left[
+\begin{array}{rrrrr}
+    a_{11} & a_{22} & a_{33} & a_{44} & a_{55} \\
+    a_{21} & a_{32} & a_{43} & a_{54} & *      \\
+    a_{31} & a_{42} & a_{53} & *      & *
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage