docs: add band matrix storage format documentation in the README of lapack/base

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

@@ -22,6 +22,213 @@ 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 arranged in linear memory in order to save memory and improve computational efficiency. The following are the different band storage formats used throughout LAPACK.
+
+### 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
+
+Let `M = N = 5`, `KL = 2`, and `KU = 1`. Given an 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> -->
+
+the band storage matrix `AB` is then
+
+<!-- <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> -->
+
+`AB` is a 4×5 matrix as `KL+KU+1 = 2+1+1 = 4`. 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, which are 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 an `UPLO` parameter) 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
+
+Let `N = 5` and `KD = 2`. Given the following matrix `A`,
+
+<!-- <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> -->
+
+the band storage matrix `AB` when `UPLO = 'U'` (i.e., upper triangle) is
+
+<!-- <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> -->
+
+`AB` is a 3×5 matrix as `KD+1 = 2+1 = 3`. Similarly, given the following matrix `A`,
+
+<!-- <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> -->
+
+the band storage matrix `AB` when `UPLO = 'L'` (i.e., lower triangle) is
+
+<!-- <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]
+```
+
+`AB` is a 3×5 matrix as `KD+1 = 2+1 = 3`.
+
+<!-- </equation> -->
+
+### Example
+
+Consider a 4×4 general band matrix with `KL = 2` subdiagonals and `KU = 1` superdiagonal:
+
+<!-- <equation class="equation" label="eq:band_matrix_numeric_a" align="center" raw="A = \left[\begin{array}{rrrr} 1.0 & 2.0 & 0.0 & 0.0 \\ 3.0 & 4.0 & 5.0 & 0.0 \\ 6.0 & 7.0 & 8.0 & 9.0 \\ 0.0 & 10.0 & 11.0 & 12.0 \end{array}\right]" alt="Representation of band matrix A with numeric values."> -->
+
+```math
+A = \left[
+\begin{array}{rrrr}
+    1.0 & 2.0 & 0.0 & 0.0 \\
+    3.0 & 4.0 & 5.0 & 0.0 \\
+    6.0 & 7.0 & 8.0 & 9.0 \\
+    0.0 & 10.0 & 11.0 & 12.0
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+#### Band Storage Representation
+
+The band storage matrix `AB` has dimensions `(KL+KU+1) × N = (2+1+1) × 4 = 4 × 4`:
+
+<!-- <equation class="equation" label="eq:band_storage_numeric_a" align="center" raw="AB = \left[\begin{array}{rrrr} * & 2.0 & 5.0 & 9.0 \\ 1.0 & 4.0 & 8.0 & 12.0 \\ 3.0 & 7.0 & 11.0 & * \\ 6.0 & 10.0 & * & * \end{array}\right]" alt="Band storage representation of matrix A with numeric values."> -->
+
+```math
+AB = \left[
+\begin{array}{rrrr}
+    *   & 2.0 & 5.0 & 9.0 \\
+    1.0 & 4.0 & 8.0 & 12.0 \\
+    3.0 & 7.0 & 11.0 & * \\
+    6.0 & 10.0 & *   & *
+\end{array}
+\right]
+```
+
+<!-- </equation> -->
+
+Here's how to represent this band matrix in JavaScript using `Float64Array`:
+
+##### Row-Major Layout
+
+```javascript
+var AB = new Float64Array( [ 0.0, 2.0, 5.0, 9.0, 1.0, 4.0, 8.0, 12.0, 3.0, 7.0, 11.0, 0.0, 6.0, 10.0, 0.0, 0.0 ] );
+```
+
+##### Column-Major Layout
+
+```javascript
+var AB = new Float64Array( [ 0.0, 1.0, 3.0, 6.0, 2.0, 4.0, 7.0, 10.0, 5.0, 8.0, 11.0, 0.0, 9.0, 12.0, 0.0, 0.0 ] );
+```
+
+</section>
+
+<!-- /.intro -->
+
 <section class="usage">
 
 ## Usage