LU decomposition: With or without pivoting?

[mathmari]

Hey!

We consider the matrix $$A=\begin{pmatrix}1 & -2 & 5 & 0 & 5\\ 1 & 0 & 2 & 0 & 3\\ 1 & 2 & 5 & 4 & 6 \\ -2 & 2 & -4 & 1 & -6 \\ 3 & 4 & 9 & 5 & 11\end{pmatrix}$$ I want to find the LU decomposition.

How do we know if we have to do the decomposition with pivoting or without? :unsure:

 

[I like Serena]

Hey mathmari!

Wiki explains that a decomposition without pivoting ($A=LU$) can fail for a non-singular matrix A, and that this is a procedural problem.
It happens when during the procedure we get a sub matrix with a $0$ at the left top.
Decomposition is still possible, but we need to reorder rows or columns to make it happen.

In other words, I believe we simply have to try the LU decomposition procedure without pivoting to find out if it is possible.
And if we find that decomposition without pivoting is not possible, we can reorder the rows and continue the algorithm.

 

[mathmari]

I found the following code for the LU-decomposition with partial pivoting:

Algorithm LUP-Decomposition(A)
Input matrix A
Output matrices L,U,P: PA=LU
n <- A.rows 
L <- new n x n matrix
U <- new n x n matrix
P <- new n x n matrix
#initialization, as before
#for L, U, and P = I
for i <- 0 to n do
       for j <- 0 to n do
               if i > j then
                     U[i][j] <- 0
                     P[i][j] <- 0
               else if i == j then
                     L[i][j] <- 1
                     P[i][j] <- 1
               else
                     L[i][j] <- 0
                     P[i][j] <- 0
for k <- 0 to n do # for each equation
       p <- 0
       for i <- k to n dο #find pivot row
                if abs(A[i][k]) > p then
                       p <- abs(A[i][k])
                       pivot_row <- i
       if p == 0 then
               error("singular matrix")
       for j <- 0 to n do #swap rows
               swap L[k][j] with L[pivot_row][j]
               swap U[k][j] with U[pivot_row][j]
               swap P[k][j] with P[pivot_row][j]
               swap A[k][j] with A[pivot_row][j]
       U[k][k] <- A[k][k]
       for i <- k + 1 to n dο
               L[i][k] <- A[i][k] / U[k][k]  # L column
               U[k][i] <- A[k][i]  # U row
       for i <- k + 1 to n do #gauss
               for j <- k + 1 to n do #elimination
                        A[i][j] <- A[i][j] – L[i][k]*U[k][j]
return L, U, P

Are the lines:

swap L[k][j] with L[pivot_row][j]
swap U[k][j] with U[pivot_row][j]

correct? Why do we have to swap also these matrices? After swapping them they are no more a lower and upper matrix respectively, are they? :unsure:I tried to apply that algorithm to a matrix, for example $A=\begin{pmatrix}1 & 2 & 4 \\ 3 & 8 & 14 \\ 2 & 6 & 13\end{pmatrix}$ :

At the initializations we get $L=\begin{pmatrix}1 & 0 & 0 \\ \ell_{21} & 1 & 0 \\ \ell_{31} & \ell_{32} & 1\end{pmatrix}$, $U=\begin{pmatrix}u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33}\end{pmatrix}$ and $P=I_{3\times 3}$.

The maximum element by absolute value at the first column is $3$, at the second row.

Now we swap the first the second lines at all the matrices $L,U,P,A$ and we get:
$$L=\begin{pmatrix} \ell_{21} & 1 & 0 \\ 1 & 0 & 0 \\ \ell_{31} & \ell_{32} & 1\end{pmatrix}, \ U=\begin{pmatrix} 0 & u_{22} & u_{23} \\ u_{11} & u_{12} & u_{13} \\ 0 & 0 & u_{33}\end{pmatrix}, \ P=\begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}, \ A=\begin{pmatrix} 3 & 8 & 14 \\ 1 & 2 & 4 \\ 2 & 6 & 13\end{pmatrix}$$

Then we get $U[1,1]=3$.

We also get $L[2,1]=\frac{a_{21}'}{3}=\frac{1}{3}$, $U[1,2]=a_{12}'=8$ and $L[3,1]=\frac{a_{31}'}{3}=\frac{2}{3}$, $U[1,3]=a_{13}'=14$Is the first loop correct?:unsure: