4.3 Crout’s LU Factorization

An equivalent LU decomposition of A = LU may be obtained by assuming that L is lower triangular and U is unit upper triangular. This factorization scheme is referred to as Crout’s method. The defining equations for Crout’s method are

$$l_{ij}=a_{ij}-\displaystyle\sum_{p=1}^{i-1}l_{ip}u_{pj},\mbox{ where }i \geq j$$ (49)


$$u_{ij}=\frac{a_{ij}-\displaystyle\sum_{p=1}^{i-1}l_{ip}u_{pj}}{l_{ii}},\mbox{ where }i \lt j$$ (50)

Algorithm 3 implements Crout’s method. Calculations are sequenced to compute one column of L followed by the corresponding row of U until A is exhausted.

Algorithm 3: Crout’s LU Decomposition
for $j=1,\cdots,n$
   for $i=j,\cdots,n$
      for $=1,\cdots,j-1$
   for $j=j+1,\cdots,n$
      for $p=1,\cdots,i-1$

Figure 2 depicts the computational sequence associated with Crout’s method.

Figure 2: Computational Sequence of Crout’s Method
The computational sequence for used to create a LU factorization of a matrix using Crout's method

You should observe that Crout’s method, like Doolittle’s, exhibits inner product accumulation.

A good comparison of the various compact factorization schemes is found in Duff et al.[ 3] .