Multiplicando
\begin{equation*} \begin{bmatrix} l_{11} & l_{11} u_{12} & l_{11} u_{13} & l_{11} u_{14} \\ l_{21} & l_{21} u_{12} + l_{22} & l_{21} u_{13} + l_{22} u_{23} & l_{21} u_{14} + l_{22} u_{24} \\ l_{31} & l_{31} u_{12} + l_{32} & l_{31} u_{13} + l_{32} u_{23} + l_{33} & l_{31} u_{14} + l_{32} u_{24} + l_{33} u_{34} \\ l_{41} & l_{41} u_{12} + l_{42} & l_{41} u_{13} + l_{42} u_{23} + l_{43} & l_{41} u_{14} + l_{42} u_{24} + l_{43} u_{34} + l_{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} a_{11} & 0 & 0 & 0 \\ a_{21} & a_{22} - l_{21} u_{12} & 0 & 0 \\ a_{31} & a_{32} - l_{31} u_{12} & a_{33} - l_{31} u_{13} - l_{32} u_{23} & 0 \\ a_{41} & a_{42} - l_{41} u_{12} & a_{43} - l_{41} u_{13} - l_{42} u_{23} & a_{44} - l_{41} u_{14} - l_{42} u_{24} - l_{43} u_{34} \end{bmatrix} \begin{bmatrix} 1 & \frac{a_{12}}{l_{11}} & \frac{a_{13}}{l_{11}} & \frac{a_{14}}{l_{11}} \\ 0 & 1 & \frac{a_{23} - l_{21} u_{13}}{l_{22}} & \frac{a_{24} - l_{21} u_{14}}{l_{22}} \\ 0 & 0 & 1 & \frac{a_{34} - l_{31} u_{14} - l_{32} u_{24}}{l_{33}} \\ 0 & 0 & 0 & 1 \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}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*}Calculando la primera columna de $\mathbf{L}$ y la primera fila $\mathbf{U}$
\begin{align*} l_{11} &= a_{11} = 1 \\ l_{21} &= a_{21} = 2 \\ l_{31} &= a_{31} = 3 \\ l_{41} &= a_{41} = -1 \\ u_{12} &= \frac{a_{12}}{l_{11}} = \frac{1}{1} = 1 \\ u_{13} &= \frac{a_{13}}{l_{11}} = \frac{2}{1} = 2 \\ u_{14} &= \frac{a_{14}}{l_{11}} = \frac{3}{1} = 3 \end{align*}Reemplazando
\begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & & 0 & 0 \\ 3 & & & 0 \\ -1 & & & \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 1 & & \\ 0 & 0 & 1 & \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{equation*}Calculando la segunda columna de $\mathbf{L}$ y la segunda fila $\mathbf{U}$
\begin{align*} l_{22} &= a_{22} - l_{21} u_{12} = 1 - 2 (1) = -1 \\ l_{32} &= a_{32} - l_{31} u_{12} = -1 - 3 (1) = -4 \\ l_{42} &= a_{42} - l_{41} u_{12} = 2 - (-1) (1) = 3 \\ u_{23} &= \frac{a_{23} - l_{21} u_{13}}{l_{22}} = \frac{-1 - 2 (2)}{-1} = 5 \\ u_{24} &= \frac{a_{24} - l_{21} u_{14}}{l_{22}} = \frac{1 - 2 (3)}{-1} = 5 \end{align*}Reemplazando
\begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 3 & -4 & & 0 \\ -1 & 3 & & \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 1 & 5 & 5 \\ 0 & 0 & 1 & \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{equation*}Calculando la tercera columna de $\mathbf{L}$ y la tercera fila $\mathbf{U}$
\begin{align*} l_{33} &= a_{33} - l_{31} u_{13} - l_{32} u_{23} = -1 - 3 (2) - (-4) (5) = 13 \\ l_{43} &= a_{43} - l_{41} u_{13} - l_{42} u_{23} = 3 - (-1) (2) - 3 (5) = -10 \\ u_{34} &= \frac{a_{34} - l_{31} u_{14} - l_{32} u_{24}}{l_{33}} = \frac{2 - 3 (3) - (-4) (5)}{13} = 1 \end{align*}Reemplazando
\begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 3 & -4 & 13 & 0 \\ -1 & 3 & -10 & \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 1 & 5 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{equation*}Calculando la cuarta columna de $\mathbf{L}$
\begin{equation*} l_{44} = a_{44} - l_{41} u_{14} - l_{42} u_{24} - l_{43} u_{34} = -1 - (-1) (3) - 3 (5) - (-10) (1) = -3 \end{equation*}Reemplazando
\begin{equation*} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 3 & -4 & 13 & 0 \\ -1 & 3 & -10 & -3 \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 1 & 5 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{equation*}Lo anterior puede escribirse como
\begin{equation*} l_{i1} = a_{i1} \end{equation*}para $i=1, 2, 3, 4 = 1, \dots, m$
Lo anterior puede escribirse como
\begin{equation*} l_{1j} = \frac{a_{1j}}{l_{11}} \end{equation*}para $j=2, 3, 4 = 2, \dots, n$
Lo anterior puede escribirse como
\begin{align*} l_{jj} &= a_{jj} - \sum l_{j?} u_{?j} \\ l_{?j} &= a_{?j} - \sum l_{??} u_{?j} \\ u_{j?} &= \frac{a_{j?} - \sum l_{j?} u_{??}}{l_{jj}} \end{align*}para $j=2, 3 = 2, \dots, n-1$
Lo anterior puede escribirse como
\begin{align*} l_{ij} &= a_{ij} - \sum l_{i?} u_{?j} \\ u_{ji} &= \frac{a_{ji} - \sum l_{j?} u_{?i}}{l_{jj}} \end{align*}para
\begin{align*} i &= 3, 4 \\ &= 4 \end{align*}Lo anterior puede escribirse como
\begin{align*} l_{jj} &= a_{jj} - \sum l_{jk} u_{kj} \\ l_{ij} &= a_{ij} - \sum l_{ik} u_{kj} \\ u_{ji} &= \frac{a_{ji} - \sum l_{jk} u_{ki}}{l_{jj}} \end{align*}para
\begin{align*} k &= 1 \\ &= 1, 2 \end{align*}Lo anterior puede escribirse como
\begin{equation*} l_{nn} = a_{nn} - \sum_{k=1}^{n-1} l_{nk} u_{kn} \end{equation*}function lu_crout(A)
for i=1 to m do
l(i,1) = a(i,1)
end for
for j=2 to n do
u(1,j) = a(1,j)/l(1,1)
end for
for j=2 to n-1 do
suma = a(j,j)
for k=1 to j-1 do
suma = suma - l(j,k)*u(k,j)
end for
l(j,j) = suma
for i=j+1 to m do
sumav = a(i,j)
sumah = a(j,i)
for k=1 to j-1 do
sumav = sumav - l(i,k)*u(k,j)
sumah = sumah - l(j,k)*u(k,i)
end for
l(i,j) = sumav
u(j,i) = sumah/l(j,j)
end for
end for
suma = a(n,n)
for k=1 to n-1 do
suma = suma - l(n,k)*u(k,n)
end for
l(n,n) = suma
return a
end function
In [5]:
import numpy as np
def lu_crout(a):
m, n = a.shape
u = np.eye(m)
l = np.zeros((m,n))
for i in range(0,m):
l[i,0] = a[i,0]
for j in range(1,n):
u[0,j] = a[0,j]/l[0,0]
for j in range(1,n-1):
suma = a[j,j]
for k in range(0,j):
suma = suma - l[j,k]*u[k,j]
l[j,j] = suma
for i in range(j+1,m):
sumav = a[i,j]
sumah = a[j,i]
for k in range(0,j):
sumav = sumav - l[i,k]*u[k,j]
sumah = sumah - l[j,k]*u[k,i]
l[i,j] = sumav
u[j,i] = sumah/l[j,j]
suma = a[n-1,n-1]
for k in range(0,n-1):
suma = suma - l[n-1,k]*u[k,n-1]
l[n-1,n-1] = suma
return l, u
In [6]:
A = np.array([[1,1,2,3],
[2,1,-1,1],
[3,-1,-1,2],
[-1,2,3,-1]],float)
print(A)
In [7]:
#resultado
L, U = lu_crout(A)
print(L)
print(U)
In [8]:
#revisando el resultado
np.dot(L,U)
Out[8]:
In [9]:
A = np.array([[3,-0.1,-0.2],
[0.1,7,-0.3],
[0.3,-0.2,10]],float)
print(A)
In [10]:
#resultado
L, U = lu_crout(A)
print(L)
print(U)
In [11]:
#revisando el resultado
np.dot(L,U)
Out[11]: