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chore: update tex eqns
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‎lib/node_modules/@stdlib/lapack/base/dgttrf/README.md

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@@ -40,31 +40,37 @@ where `L` is a product of permutation and unit lower bidiagonal matrices and `U`
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For a tridiagonal matrix `A`, its elements are stored in three arrays:
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43-
<!-- <equation class="equation" label="eq:matrix_a" align="center" raw="A = \begin{bmatrix}
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d_1 & du_1 & 0 & \cdots & 0 \\
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dl_1 & d_2 & du_2 & \cdots & 0 \\
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0 & dl_2 & d_3 & \ddots & \vdots \\
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\vdots & \ddots & \ddots & \ddots & du_{n-1}\\
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0 & \cdots & 0 & dl_{n-1} & d_n
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\end{bmatrix}" alt="Representation of matrix A."> -->
43+
<!-- <equation class="equation" label="eq:matrix_a" align="center" raw="A = \left[
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\begin{array}{rrrrr}
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d_1 & du_1 & 0 & \cdots & 0 \\
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dl_1 & d_2 & du_2 & \ddots & \vdots \\
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0 & dl_2 & d_3 & \ddots & 0 \\
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\vdots & \ddots & \ddots & \ddots & du_{n-1} \\
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0 & \cdots & 0 & dl_{n-1} & d_n
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\end{array}
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\right]" alt="Representation of matrix A."> -->
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```math
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A = \begin{bmatrix}
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d_1 & du_1 & 0 & \cdots & 0 \\
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dl_1 & d_2 & du_2 & \cdots & 0 \\
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0 & dl_2 & d_3 & \ddots & \vdots \\
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\vdots & \ddots & \ddots & \ddots & du_{n-1}\\
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0 & \cdots & 0 & dl_{n-1} & d_n
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\end{bmatrix}
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A = \left[
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\begin{array}{rrrrr}
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d_1 & du_1 & 0 & \cdots & 0 \\
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dl_1 & d_2 & du_2 & \ddots & \vdots \\
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0 & dl_2 & d_3 & \ddots & 0 \\
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\vdots & \ddots & \ddots & \ddots & du_{n-1} \\
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0 & \cdots & 0 & dl_{n-1} & d_n
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\end{array}
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\right]
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```
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<!-- <div class="equation" align="center" data-raw-text="A = \begin{bmatrix}
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d_1 & du_1 & 0 & \cdots & 0 \\
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dl_1 & d_2 & du_2 & \cdots & 0 \\
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0 & dl_2 & d_3 & \ddots & \vdots \\
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\vdots & \ddots & \ddots & \ddots & du_{n-1}\\
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0 & \cdots & 0 & dl_{n-1} & d_n
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\end{bmatrix}" data-equation="eq:matrix_a"></div> -->
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<!-- <div class="equation" align="center" data-raw-text="A = \left[
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\begin{array}{rrrrr}
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d_1 & du_1 & 0 & \cdots & 0 \\
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dl_1 & d_2 & du_2 & \ddots & \vdots \\
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0 & dl_2 & d_3 & \ddots & 0 \\
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\vdots & \ddots & \ddots & \ddots & du_{n-1} \\
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0 & \cdots & 0 & dl_{n-1} & d_n
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\end{array}
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\right]" data-equation="eq:matrix_a"></div> -->
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<!-- </equation> -->
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@@ -82,59 +88,75 @@ After factorization, the elements of `L` and `U` overwrite the input arrays, whe
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The resulting `L` and `U` matrices have the following structure:
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<!-- <equation class="equation" label="eq:matrix_l" align="center" raw="L = \begin{bmatrix}
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1 & 0 & 0 & \cdots & 0 \\
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l_1 & 1 & 0 & \cdots & 0 \\
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<!-- <equation class="equation" label="eq:matrix_l" align="center" raw="L = \left[
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\begin{array}{rrrrr}
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1 & 0 & \cdots & \cdots & 0 \\
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l_1 & 1 & \ddots & \ddots & \vdots \\
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0 & l_2 & 1 & \ddots & \vdots \\
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\vdots & \ddots & \ddots & \ddots & 0 \\
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\vdots & \ddots & \ddots & \ddots & 0 \\
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0 & \cdots & 0 & l_{n-1} & 1
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\end{bmatrix}" alt="Representation of matrix L as derived from DL."> -->
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\end{array}
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\right]" alt="Representation of matrix L as derived from DL."> -->
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```math
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L = \begin{bmatrix}
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1 & 0 & 0 & \cdots & 0 \\
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l_1 & 1 & 0 & \cdots & 0 \\
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L = \left[
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\begin{array}{rrrrr}
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1 & 0 & \cdots & \cdots & 0 \\
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l_1 & 1 & \ddots & \ddots & \vdots \\
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0 & l_2 & 1 & \ddots & \vdots \\
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\vdots & \ddots & \ddots & \ddots & 0 \\
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\vdots & \ddots & \ddots & \ddots & 0 \\
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0 & \cdots & 0 & l_{n-1} & 1
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\end{bmatrix}
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\end{array}
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\right]
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```
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<!-- <div class="equation" align="center" data-raw-text="L = \begin{bmatrix}
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1 & 0 & 0 & \cdots & 0 \\
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l_1 & 1 & 0 & \cdots & 0 \\
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<!-- <div class="equation" align="center" data-raw-text="L = \left[
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\begin{array}{rrrrr}
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1 & 0 & \cdots & \cdots & 0 \\
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l_1 & 1 & \ddots & \ddots & \vdots \\
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0 & l_2 & 1 & \ddots & \vdots \\
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\vdots & \ddots & \ddots & \ddots & 0 \\
118+
\vdots & \ddots & \ddots & \ddots & 0 \\
108119
0 & \cdots & 0 & l_{n-1} & 1
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\end{bmatrix}" data-equation="eq:matrix_l"></div> -->
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\end{array}
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\right]" data-equation="eq:matrix_l"></div> -->
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<!-- </equation> -->
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<!-- <equation class="equation" label="eq:matrix_u" align="center" raw="U = \begin{bmatrix}
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u_{1,1} & u_{1,2} & u_{1,3} & \cdots & 0 \\
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0 & u_{2,2} & u_{2,3} & u_{2,4} & 0 \\
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0 & 0 & u_{3,3} & \ddots & \ddots \\
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\vdots & \vdots & \ddots & \ddots & u_{n-1,n}\\
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0 & 0 & \cdots & 0 & u_{n,n}
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\end{bmatrix}" alt="Representation of matrix U as derived from D, DU, DU2."> -->
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<!-- <equation class="equation" label="eq:matrix_u" align="center" raw="U = \left[
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\begin{array}{rrrrrr}
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u_{1,1} & u_{1,2} & u_{1,3} & 0 & \cdots & 0 \\
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0 & u_{2,2} & u_{2,3} & u_{2,4} & \ddots & \vdots \\
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\vdots & \ddots & u_{3,3} & u_{3,4} & \ddots & 0 \\
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\vdots & \ddots & \ddots & \ddots & \ddots & u_{n-2,n} \\
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\vdots & \ddots & \ddots & \ddots & \ddots & u_{n-1,n} \\
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0 & \cdots & \cdots & \cdots & 0 & u_{n,n}
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\end{array}
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\right]" alt="Representation of matrix U as derived from D, DU, DU2."> -->
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121136
```math
122-
U = \begin{bmatrix}
123-
u_{1,1} & u_{1,2} & u_{1,3} & \cdots & 0 \\
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0 & u_{2,2} & u_{2,3} & u_{2,4} & 0 \\
125-
0 & 0 & u_{3,3} & \ddots & \ddots \\
126-
\vdots & \vdots & \ddots & \ddots & u_{n-1,n}\\
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0 & 0 & \cdots & 0 & u_{n,n}
128-
\end{bmatrix}
137+
U = \left[
138+
\begin{array}{rrrrrr}
139+
u_{1,1} & u_{1,2} & u_{1,3} & 0 & \cdots & 0 \\
140+
0 & u_{2,2} & u_{2,3} & u_{2,4} & \ddots & \vdots \\
141+
\vdots & \ddots & u_{3,3} & u_{3,4} & \ddots & 0 \\
142+
\vdots & \ddots & \ddots & \ddots & \ddots & u_{n-2,n} \\
143+
\vdots & \ddots & \ddots & \ddots & \ddots & u_{n-1,n} \\
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0 & \cdots & \cdots & \cdots & 0 & u_{n,n}
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\end{array}
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\right]
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```
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<!-- <div class="equation" align="center" data-raw-text="U = \begin{bmatrix}
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u_{1,1} & u_{1,2} & u_{1,3} & \cdots & 0 \\
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0 & u_{2,2} & u_{2,3} & u_{2,4} & 0 \\
134-
0 & 0 & u_{3,3} & \ddots & \ddots \\
135-
\vdots & \vdots & \ddots & \ddots & u_{n-1,n}\\
136-
0 & 0 & \cdots & 0 & u_{n,n}
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\end{bmatrix}" data-equation="eq:matrix_u"></div> -->
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<!-- <div class="equation" align="center" data-raw-text="U = \left[
151+
\begin{array}{rrrrrr}
152+
u_{1,1} & u_{1,2} & u_{1,3} & 0 & \cdots & 0 \\
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0 & u_{2,2} & u_{2,3} & u_{2,4} & \ddots & \vdots \\
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\vdots & \ddots & u_{3,3} & u_{3,4} & \ddots & 0 \\
155+
\vdots & \ddots & \ddots & \ddots & \ddots & u_{n-2,n} \\
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\vdots & \ddots & \ddots & \ddots & \ddots & u_{n-1,n} \\
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0 & \cdots & \cdots & \cdots & 0 & u_{n,n}
158+
\end{array}
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\right]" data-equation="eq:matrix_u"></div> -->
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<!-- </equation> -->
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