# 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)

and

 $$u_{ij}=\frac{a_{ij}-\displaystyle\sum_{p=1}^{i-1}l_{ip}u_{pj}}{l_{ii}},\mbox{ where }i \lt j$$ (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 $i=j,\cdots,n$ $\alpha=a_{ij}$ for $=1,\cdots,j-1$ $\alpha=\alpha-a_{ip}a_{pj}$ $a_{ij}=\alpha$ for $j=j+1,\cdots,n$ $\alpha=a_{ij}$ for $p=1,\cdots,i-1$ $\alpha=\alpha-a_{jp}a_{pi}$ $a_{ji}=\displaystyle\frac{\alpha}{a_{jj}}$

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

Figure 2: Computational Sequence of 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] .