docs: update copy

Signed-off-by: Athan <[email protected]>

Author: kgryte

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

@@ -34,14 +34,14 @@ A general band matrix of size `M`-by-`N` with `KL` subdiagonals and `KU` superdi
 
 **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)`
+-   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)
+#### Example
 
-Original band matrix `A`:
+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."> -->
 
@@ -59,7 +59,7 @@ A = \left[
 
 <!-- </equation> -->
 
-Band storage in array `AB` (4×5, since KL+KU+1 = 2+1+1 = 4):
+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."> -->
 
@@ -76,34 +76,34 @@ AB = \left[
 
 <!-- </equation> -->
 
-Elements marked `*` need not be set and are not referenced by LAPACK routines.
+`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, 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.
+**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
+-   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.
+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)`
+-   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)`
+-   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)
+#### Example
 
-For `UPLO = 'U'` (upper triangle stored):
+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)."> -->
 
@@ -121,7 +121,7 @@ A = \left[
 
 <!-- </equation> -->
 
-Band storage in array `AB` (3×5, since KD+1 = 2+1 = 3):
+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)."> -->
 
@@ -137,7 +137,7 @@ AB = \left[
 
 <!-- </equation> -->
 
-For `UPLO = 'L'` (lower triangle stored):
+The matrix is 3×5 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)."> -->
 
@@ -155,7 +155,7 @@ A = \left[
 
 <!-- </equation> -->
 
-Band storage in array `AB` (3×5, since KD+1 = 2+1 = 3):
+the band storage matrix `AB` (3×5, since KD+1 = 2+1 = 3) 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)."> -->