Algoritmo de Doolittle
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
l_{21} & 1 & 0 & 0 \\
l_{31} & l_{32} & 1 & 0 \\
l_{41} & l_{42} & l_{43} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} & u_{13} & u_{14} \\
0 & u_{22} & u_{23} & u_{24} \\
0 & 0 & u_{33} & u_{34} \\
0 & 0 & 0 & u_{44}
\end{bmatrix} =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{bmatrix}
\end{equation*}
$$
Multiplicando
$$
\begin{equation*}
\begin{bmatrix}
u_{11} & u_{12} & u_{13} & u_{14} \\
l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} & l_{21} u_{14} + u_{24} \\
l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} & l_{31} u_{14} + l_{32} u_{24} + u_{34} \\
l_{41} u_{11} & l_{41} u_{12} + l_{42} u_{22} & l_{41} u_{13} + l_{42} u_{23} + l_{43} u_{33 } & l_{41} u_{14} + l_{42} u_{24} + l_{43} u_{34} + u_{44}
\end{bmatrix} =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{bmatrix}
\end{equation*}
$$
Despejando y reemplazando
$$
\begin{equation}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
\frac{a_{21}}{u_{11}} & 1 & 0 & 0 \\
\frac{a_{31}}{u_{11}} & \frac{a_{32} - l_{31} u_{12}}{u_{22}} & 1 & 0 \\
\frac{a_{41}}{u_{11}} & \frac{a_{42} - l_{41} u_{12}}{u_{22}} & \frac{a_{43} - l_{41} u_{13} - l_{42} u_{23}}{u_{33}} & 1
\end{bmatrix}
\begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
0 & a_{22} - l_{21} u_{12} & a_{23} - l_{21} u_{13} & a_{24} - l_{21} u_{14} \\
0 & 0 & a_{33} - l_{31} u_{13} - l_{32} u_{23} & a_{34} - l_{31} u_{14} - l_{32} u_{24} \\
0 & 0 & 0 & a_{44} - l_{41} u_{14} - l_{42} u_{24} - l_{43} u_{34}
\end{bmatrix} =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}
\end{bmatrix}
\tag{7.1}
\end{equation}
$$
Ejemplo
Factorizar la matriz \( \mathbf{A} \)
$$
\begin{equation*}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
2 & 1 & -1 & 1 \\
3 & -1 & -1 & 2 \\
-1 & 2 & 3 & -1
\end{bmatrix}
\end{equation*}
$$
Las matrices \( \mathbf{L} \) y \( \mathbf{U} \) iniciales son
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
& 1 & 0 & 0 \\
& & 1 & 0 \\
& & & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
2 & 1 & -1 & 1 \\
3 & -1 & -1 & 2 \\
-1 & 2 & 3 & -1
\end{bmatrix}
\end{equation*}
$$
Primera fila pivote
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
\frac{2}{1} & 1 & 0 & 0 \\
\frac{3}{1} & & 1 & 0 \\
\frac{-1}{1} & & & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
2 - \frac{2}{1} (1) & 1 - \frac{2}{1} (1) & -1 - \frac{2}{1} (2) & 1 - \frac{2}{1} (3) \\
3 - \frac{3}{1} (1) & -1 - \frac{3}{1} (1) & -1 - \frac{3}{1} (2) & 2 - \frac{3}{1} (3) \\
-1 - \bigl( \frac{-1}{1} \bigr) (1) & 2 - \bigl( \frac{-1}{1} \bigr) (1) & 3 - \bigl( \frac{-1}{1} \bigr) (2) & -1 - \bigl( \frac{-1}{1} \bigr) (3)
\end{bmatrix}
\end{equation*}
$$
Simplificando
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
3 & & 1 & 0 \\
-1 & & & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
0 & -1 & -5 & -5 \\
0 & -4 & -7 & -7 \\
0 & 3 & 5 & 2
\end{bmatrix}
\end{equation*}
$$
Segunda fila pivote
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
3 & \frac{-4}{-1} & 1 & 0 \\
-1 & \frac{3}{-1} & & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
0 & -1 & -5 & -5 \\
0 & -4 - \bigl( \frac{-4}{-1} \bigr) (-1) & -7 - \bigl( \frac{-4}{-1} \bigr) (-5) & -7 - \bigl( \frac{-4}{-1} \bigr) (-5) \\
0 & 3 - \bigl( \frac{3}{-1} \bigr) (-1) & 5 - \bigl( \frac{3}{-1} \bigr) (-5) & 2 - \bigl( \frac{3}{-1} \bigr) (-5)
\end{bmatrix}
\end{equation*}
$$
Simplificando
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
3 & 4 & 1 & 0 \\
-1 & -3 & & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
0 & -1 & -5 & -5 \\
0 & 0 & 13 & 13 \\
0 & 0 & -10 & -13
\end{bmatrix}
\end{equation*}
$$
Tercera fila pivote
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
3 & 4 & 1 & 0 \\
-1 & -3 & \frac{-10}{13} & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
0 & -1 & -5 & -5 \\
0 & 0 & 13 & 13 \\
0 & 0 & -10 - \bigl( \frac{-10}{13} \bigr) (13) & -13 - \bigl( \frac{-10}{13} \bigr) (13)
\end{bmatrix}
\end{equation*}
$$
Simplificando
$$
\begin{equation*}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 \\
3 & 4 & 1 & 0 \\
-1 & -3 & -0.769231 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 2 & 3 \\
0 & -1 & -5 & -5 \\
0 & 0 & 13 & 13 \\
0 & 0 & 0 & -3
\end{bmatrix}
\end{equation*}
$$
Lo anterior puede escribirse como
$$
\begin{equation*}
\begin{array}{llll:llll}
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = \frac{a_{21}}{a_{11}} & 1 & 0 & 0 & a_{21} = a_{21} - \frac{a_{21}}{a_{11}} a_{11} & a_{22} = a_{22} - \frac{a_{21}}{a_{11}} a_{12} & a_{23} = a_{23} - \frac{a_{21}}{a_{11}} a_{13} & a_{24} = a_{24} - \frac{a_{21}}{a_{11}} a_{14} \\
l_{31} = \frac{a_{31}}{a_{11}} & l_{32} = 0 & 1 & 0 & a_{31} = a_{31} - \frac{a_{31}}{a_{11}} a_{11} & a_{32} = a_{32} - \frac{a_{31}}{a_{11}} a_{12} & a_{33} = a_{33} - \frac{a_{31}}{a_{11}} a_{13} & a_{34} = a_{34} - \frac{a_{31}}{a_{11}} a_{14} \\
l_{41} = \frac{a_{41}}{a_{11}} & l_{42} = 0 & l_{43} = 0 & 1 & a_{41} = a_{41} - \frac{a_{41}}{a_{11}} a_{11} & a_{42} = a_{42} - \frac{a_{41}}{a_{11}} a_{12} & a_{43} = a_{43} - \frac{a_{41}}{a_{11}} a_{13} & a_{44} = a_{44} - \frac{a_{41}}{a_{11}} a_{14} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{32} = \frac{a_{32}}{a_{22}} & 1 & 0 & a_{31} = a_{31} - \frac{a_{32}}{a_{22}} a_{21} & a_{32} = a_{32} - \frac{a_{32}}{a_{22}} a_{22} & a_{33} = a_{33} - \frac{a_{32}}{a_{22}} a_{23} & a_{34} = a_{34} - \frac{a_{32}}{a_{22}} a_{24} \\
l_{41} = l_{41} & l_{42} = \frac{a_{42}}{a_{22}} & l_{43} = 0 & 1 & a_{41} = a_{41} - \frac{a_{42}}{a_{22}} a_{21} & a_{42} = a_{42} - \frac{a_{42}}{a_{22}} a_{22} & a_{43} = a_{43} - \frac{a_{42}}{a_{22}} a_{23} & a_{44} = a_{44} - \frac{a_{42}}{a_{22}} a_{24} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{32} = l_{32} & 1 & 0 & a_{31} = a_{31} & a_{32} = a_{32} & a_{33} = a_{33} & a_{34} = a_{34} \\
l_{41} = l_{41} & l_{42} = l_{42} & l_{43} = \frac{a_{43}}{a_{33}} & 1 & a_{41} = a_{41} - \frac{a_{43}}{a_{33}} a_{31} & a_{42} = a_{42} - \frac{a_{43}}{a_{33}} a_{32} & a_{43} = a_{43} - \frac{a_{43}}{a_{33}} a_{33} & a_{44} = a_{44} - \frac{a_{43}}{a_{33}} a_{34}
\end{array}
\end{equation*}
$$
Patrón de cálculo
Primer patrón
$$
\begin{equation*}
\begin{array}{llll:llll}
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = \frac{a_{21}}{a_{11}} & 1 & 0 & 0 & a_{2\color{blue}{1}} = a_{2\color{blue}{1}} - \frac{a_{21}}{a_{11}} a_{1\color{blue}{1}} & a_{2\color{green}{2}} = a_{2\color{green}{2}} - \frac{a_{21}}{a_{11}} a_{1\color{green}{2}} & a_{2\color{red}{3}} = a_{2\color{red}{3}} - \frac{a_{21}}{a_{11}} a_{1\color{red}{3}} & a_{2\color{fuchsia}{4}} = a_{2\color{fuchsia}{4}} - \frac{a_{21}}{a_{11}} a_{1\color{fuchsia}{4}} \\
l_{31} = \frac{a_{31}}{a_{11}} & l_{32} = 0 & 1 & 0 & a_{3\color{blue}{1}} = a_{3\color{blue}{1}} - \frac{a_{31}}{a_{11}} a_{1\color{blue}{1}} & a_{3\color{green}{2}} = a_{3\color{green}{2}} - \frac{a_{31}}{a_{11}} a_{1\color{green}{2}} & a_{3\color{red}{3}} = a_{3\color{red}{3}} - \frac{a_{31}}{a_{11}} a_{1\color{red}{3}} & a_{3\color{fuchsia}{4}} = a_{3\color{fuchsia}{4}} - \frac{a_{31}}{a_{11}} a_{1\color{fuchsia}{4}} \\
l_{41} = \frac{a_{41}}{a_{11}} & l_{42} = 0 & l_{43} = 0 & 1 & a_{4\color{blue}{1}} = a_{4\color{blue}{1}} - \frac{a_{41}}{a_{11}} a_{1\color{blue}{1}} & a_{4\color{green}{2}} = a_{4\color{green}{2}} - \frac{a_{41}}{a_{11}} a_{1\color{green}{2}} & a_{4\color{red}{3}} = a_{4\color{red}{3}} - \frac{a_{41}}{a_{11}} a_{1\color{red}{3}} & a_{4\color{fuchsia}{4}} = a_{4\color{fuchsia}{4}} - \frac{a_{41}}{a_{11}} a_{1\color{fuchsia}{4}} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{32} = \frac{a_{32}}{a_{22}} & 1 & 0 & a_{3\color{blue}{1}} = a_{3\color{blue}{1}} - \frac{a_{32}}{a_{22}} a_{2\color{blue}{1}} & a_{3\color{green}{2}} = a_{3\color{green}{2}} - \frac{a_{32}}{a_{22}} a_{2\color{green}{2}} & a_{3\color{red}{3}} = a_{3\color{red}{3}} - \frac{a_{32}}{a_{22}} a_{2\color{red}{3}} & a_{3\color{fuchsia}{4}} = a_{3\color{fuchsia}{4}} - \frac{a_{32}}{a_{22}} a_{2\color{fuchsia}{4}} \\
l_{41} = l_{41} & l_{42} = \frac{a_{42}}{a_{22}} & l_{43} = 0 & 1 & a_{4\color{blue}{1}} = a_{4\color{blue}{1}} - \frac{a_{42}}{a_{22}} a_{2\color{blue}{1}} & a_{4\color{green}{2}} = a_{4\color{green}{2}} - \frac{a_{42}}{a_{22}} a_{2\color{green}{2}} & a_{4\color{red}{3}} = a_{4\color{red}{3}} - \frac{a_{42}}{a_{22}} a_{2\color{red}{3}} & a_{4\color{fuchsia}{4}} = a_{4\color{fuchsia}{4}} - \frac{a_{42}}{a_{22}} a_{2\color{fuchsia}{4}} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{32} = l_{32} & 1 & 0 & a_{31} = a_{31} & a_{32} = a_{32} & a_{33} = a_{33} & a_{34} = a_{34} \\
l_{41} = l_{41} & l_{42} = l_{42} & l_{43} = \frac{a_{43}}{a_{33}} & 1 & a_{4\color{blue}{1}} = a_{4\color{blue}{1}} - \frac{a_{43}}{a_{33}} a_{3\color{blue}{1}} & a_{4\color{green}{2}} = a_{4\color{green}{2}} - \frac{a_{43}}{a_{33}} a_{3\color{green}{2}} & a_{4\color{red}{3}} = a_{4\color{red}{3}} - \frac{a_{43}}{a_{33}} a_{3\color{red}{3}} & a_{4\color{fuchsia}{4}} = a_{4\color{fuchsia}{4}} - \frac{a_{43}}{a_{33}} a_{3\color{fuchsia}{4}}
\end{array}
\end{equation*}
$$
lo anterior puede escribirse como
$$
\begin{equation*}
a_{?j} = a_{?j} - \frac{a_{??}}{a_{??}} a_{?j}
\end{equation*}
$$
para \( j = 1, 2, 3, 4 = 1 , \dots, n \)
Segundo patrón
$$
\begin{equation*}
\begin{array}{llll:llll}
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{\color{blue}{2}1} = \frac{a_{\color{blue}{2}1}}{a_{11}} & 1 & 0 & 0 & a_{\color{blue}{2}1} = a_{\color{blue}{2}1} - \frac{a_{\color{blue}{2}1}}{a_{11}} a_{11} & a_{\color{blue}{2}2} = a_{\color{blue}{2}2} - \frac{a_{\color{blue}{2}1}}{a_{11}} a_{12} & a_{\color{blue}{2}3} = a_{\color{blue}{2}3} - \frac{a_{\color{blue}{2}1}}{a_{11}} a_{13} & a_{\color{blue}{2}4} = a_{\color{blue}{2}4} - \frac{a_{\color{blue}{2}1}}{a_{11}} a_{14} \\
l_{\color{green}{3}1} = \frac{a_{\color{green}{3}1}}{a_{11}} & l_{32} = 0 & 1 & 0 & a_{\color{green}{3}1} = a_{\color{green}{3}1} - \frac{a_{\color{green}{3}1}}{a_{11}} a_{11} & a_{\color{green}{3}2} = a_{\color{green}{3}2} - \frac{a_{\color{green}{3}1}}{a_{11}} a_{12} & a_{\color{green}{3}3} = a_{\color{green}{3}3} - \frac{a_{\color{green}{3}1}}{a_{11}} a_{13} & a_{\color{green}{3}4} = a_{\color{green}{3}4} - \frac{a_{\color{green}{3}1}}{a_{11}} a_{14} \\
l_{\color{red}{4}1} = \frac{a_{\color{red}{4}1}}{a_{11}} & l_{42} = 0 & l_{43} = 0 & 1 & a_{\color{red}{4}1} = a_{\color{red}{4}1} - \frac{a_{\color{red}{4}1}}{a_{11}} a_{11} & a_{\color{red}{4}2} = a_{\color{red}{4}2} - \frac{a_{\color{red}{4}1}}{a_{11}} a_{12} & a_{\color{red}{4}3} = a_{\color{red}{4}3} - \frac{a_{\color{red}{4}1}}{a_{11}} a_{13} & a_{\color{red}{4}4} = a_{\color{red}{4}4} - \frac{a_{\color{red}{4}1}}{a_{11}} a_{14} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{\color{green}{3}2} = \frac{a_{\color{green}{3}2}}{a_{22}} & 1 & 0 & a_{\color{green}{3}1} = a_{\color{green}{3}1} - \frac{a_{\color{green}{3}2}}{a_{22}} a_{21} & a_{\color{green}{3}2} = a_{\color{green}{3}2} - \frac{a_{\color{green}{3}2}}{a_{22}} a_{22} & a_{\color{green}{3}3} = a_{\color{green}{3}3} - \frac{a_{\color{green}{3}2}}{a_{22}} a_{23} & a_{\color{green}{3}4} = a_{\color{green}{3}4} - \frac{a_{\color{green}{3}2}}{a_{22}} a_{24} \\
l_{41} = l_{41} & l_{\color{red}{4}2} = \frac{a_{\color{red}{4}2}}{a_{22}} & l_{43} = 0 & 1 & a_{\color{red}{4}1} = a_{\color{red}{4}1} - \frac{a_{\color{red}{4}2}}{a_{22}} a_{21} & a_{\color{red}{4}2} = a_{\color{red}{4}2} - \frac{a_{\color{red}{4}2}}{a_{22}} a_{22} & a_{\color{red}{4}3} = a_{\color{red}{4}3} - \frac{a_{\color{red}{4}2}}{a_{22}} a_{23} & a_{\color{red}{4}4} = a_{\color{red}{4}4} - \frac{a_{\color{red}{4}2}}{a_{22}} a_{24} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{32} = l_{32} & 1 & 0 & a_{31} = a_{31} & a_{32} = a_{32} & a_{33} = a_{33} & a_{34} = a_{34} \\
l_{41} = l_{41} & l_{42} = l_{42} & l_{\color{red}{4}3} = \frac{a_{\color{red}{4}3}}{a_{33}} & 1 & a_{\color{red}{4}1} = a_{\color{red}{4}1} - \frac{a_{\color{red}{4}3}}{a_{33}} a_{31} & a_{\color{red}{4}2} = a_{\color{red}{4}2} - \frac{a_{\color{red}{4}3}}{a_{33}} a_{32} & a_{\color{red}{4}3} = a_{\color{red}{4}3} - \frac{a_{\color{red}{4}3}}{a_{33}} a_{33} & a_{\color{red}{4}4} = a_{\color{red}{4}4} - \frac{a_{\color{red}{4}3}}{a_{33}} a_{34}
\end{array}
\end{equation*}
$$
lo anterior puede escribirse como
$$
\begin{align*}
a_{ij} &= a_{ij} - \frac{a_{i?}}{a_{??}} a_{?j} \\
l_{i?} &= \frac{a_{i?}}{a_{??}}
\end{align*}
$$
para
$$
\begin{align*}
i &= 2, 3, 4 = 2, \dots, m \\
&= 3, 4 = 3, \dots, m \\
&= 4 = 4 , \dots, m
\end{align*}
$$
Tercer patrón
$$
\begin{equation*}
\begin{array}{llll:llll}
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{2\color{blue}{1}} = \frac{a_{2\color{blue}{1}}}{a_{\color{blue}{11}}} & 1 & 0 & 0 & a_{21} = a_{21} - \frac{a_{2\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}1} & a_{22} = a_{22} - \frac{a_{2\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}2} & a_{23} = a_{23} - \frac{a_{2\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}3} & a_{24} = a_{24} - \frac{a_{2\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}4} \\
l_{3\color{blue}{1}} = \frac{a_{3\color{blue}{1}}}{a_{\color{blue}{11}}} & l_{32} = 0 & 1 & 0 & a_{31} = a_{31} - \frac{a_{3\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}1} & a_{32} = a_{32} - \frac{a_{3\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}2} & a_{33} = a_{33} - \frac{a_{3\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}3} & a_{34} = a_{34} - \frac{a_{3\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}4} \\
l_{4\color{blue}{1}} = \frac{a_{4\color{blue}{1}}}{a_{\color{blue}{11}}} & l_{42} = 0 & l_{43} = 0 & 1 & a_{41} = a_{41} - \frac{a_{4\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}1} & a_{42} = a_{42} - \frac{a_{4\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}2} & a_{43} = a_{43} - \frac{a_{4\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}3} & a_{44} = a_{44} - \frac{a_{4\color{blue}{1}}}{a_{\color{blue}{11}}} a_{\color{blue}{1}4} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{3\color{green}{2}} = \frac{a_{3\color{green}{2}}}{a_{\color{green}{22}}} & 1 & 0 & a_{31} = a_{31} - \frac{a_{3\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}1} & a_{32} = a_{32} - \frac{a_{3\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}2} & a_{33} = a_{33} - \frac{a_{3\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}3} & a_{34} = a_{34} - \frac{a_{3\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}4} \\
l_{41} = l_{41} & l_{4\color{green}{2}} = \frac{a_{4\color{green}{2}}}{a_{\color{green}{22}}} & l_{43} = 0 & 1 & a_{41} = a_{41} - \frac{a_{4\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}1} & a_{42} = a_{42} - \frac{a_{4\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}2} & a_{43} = a_{43} - \frac{a_{4\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}3} & a_{44} = a_{44} - \frac{a_{4\color{green}{2}}}{a_{\color{green}{22}}} a_{\color{green}{2}4} \\
\hline
1 & 0 & 0 & 0 & a_{11} = a_{11} & a_{12} = a_{12} & a_{13} = a_{13} & a_{14} = a_{14} \\
l_{21} = l_{21} & 1 & 0 & 0 & a_{21} = a_{21} & a_{22} = a_{22} & a_{23} = a_{23} & a_{24} = a_{24} \\
l_{31} = l_{31} & l_{32} = l_{32} & 1 & 0 & a_{31} = a_{31} & a_{32} = a_{32} & a_{33} = a_{33} & a_{34} = a_{34} \\
l_{41} = l_{41} & l_{42} = l_{42} & l_{4\color{red}{3}} = \frac{a_{4\color{red}{3}}}{a_{\color{red}{33}}} & 1 & a_{41} = a_{41} - \frac{a_{4\color{red}{3}}}{a_{\color{red}{33}}} a_{\color{red}{3}1} & a_{42} = a_{42} - \frac{a_{4\color{red}{3}}}{a_{\color{red}{33}}} a_{\color{red}{3}2} & a_{43} = a_{43} - \frac{a_{4\color{red}{3}}}{a_{\color{red}{33}}} a_{\color{red}{3}3} & a_{44} = a_{44} - \frac{a_{4\color{red}{3}}}{a_{\color{red}{33}}} a_{\color{red}{3}4}
\end{array}
\end{equation*}
$$
lo anterior puede escribirse como
$$
\begin{align*}
a_{ij} &= a_{ij} - \frac{a_{ik}}{a_{kk}} a_{kj} \\
l_{ik} &= \frac{a_{ik}}{a_{kk}}
\end{align*}
$$
para \( k = 1, 2, 3 = 1, \dots, m - 1 \)
Fórmula matemática
$$
\begin{align*}
k &= 1, \dots, m - 1 \\
& \quad i = 1 + k, \dots, m \\
& \quad \quad j = 1, \dots, n \\
& \quad \quad \quad a_{ij} = a_{ij} - \frac{a_{ik}}{a_{kk}} a_{kj} \\
& \quad \quad l_{ik} = \frac{a_{ik}}{a_{kk}}
\end{align*}
$$
Seudocódigo
function lu_doolittle(a)
m, n = tamaño(a)
for k=1 to m-1 do
for i=1+k to m do
for j=1 to n do
a(i,j) = a(i,j) - a(i,k)*a(k,j)/a(k,k)
end for
l(i,k) = a(i,k)/a(k,k)
end for
end for
end function
otra alternativa para reducir tiempo de cálculo
function lu_doolittle(a)
m, n = tamaño(a)
for k=1 to m-1 do
for i=1+k to m do
factor = a(i,k)/a(k,k)
for j=1 to n do
a(i,j) = a(i,j) - factor*a(k,j)
end for
l(i,k) = factor
end for
end for
end function
Implementación descomposición de Doolittle
import numpy as np
def lu_doolittle(A):
a = np.copy(A)
m, n = a.shape
l = np.eye(m)
for k in range(0,m-1):
for i in range(1+k,m):
factor = a[i,k]/a[k,k]
for j in range(0,n):
a[i,j] = a[i,j] - factor*a[k,j]
l[i,k] = factor
return l,a
A = np.array([[1,1,2,3],
[2,1,-1,1],
[3,-1,-1,2],
[-1,2,3,-1]],float)
print(A)
[[ 1. 1. 2. 3.]
[ 2. 1. -1. 1.]
[ 3. -1. -1. 2.]
[-1. 2. 3. -1.]]
l, u = lu_doolittle(A)
print(l)
print(u)
[[ 1. 0. 0. 0. ]
[ 2. 1. 0. 0. ]
[ 3. 4. 1. 0. ]
[-1. -3. -0.76923077 1. ]]
[[ 1. 1. 2. 3.]
[ 0. -1. -5. -5.]
[ 0. 0. 13. 13.]
[ 0. 0. 0. -3.]]
array([[ 1., 1., 2., 3.],
[ 2., 1., -1., 1.],
[ 3., -1., -1., 2.],
[-1., 2., 3., -1.]])