7.5 Crout’s Method for Symmetric Matrices

If A is a symmetric n × n matrix, Algorithm 12 computes – one column at a time – the lower triangular matrix L that results from Crout's LU decomposition. The lower triangular portion of A is overwritten by L.

Algorithm 12: Symmetric Crout’s Method
for $j=1,\cdots,n$
   for $i=1,\cdots,j-1$
   for $i=j,\cdots,n$
      for $p=1,\cdots,j-1$

The algorithm uses a working vector w of length n to construct the relevant portion of column j from U at each stage of the factorization. Elements from U are derived using Equation 79.

If the lower triangular portion of A is stored in a linear array, Algorithm 13 results in the same factorization (assuming zero based subscripting). The algorithm overwrites A with L.

Algorithm 13: Symmetric Crout’s Method - Array Based
for $j=0,\cdots,n-1$
   for $i=0,\cdots,j-1$
      $\mathbf{w}[j]=\displaystyle\frac{\mathbf{a}{[jn-j(j+1)/2+i]}}{% \mathbf{a}{[jn-j(j+1)/2+j]}}$
   for $i=j,\cdots,n-1$
      for $p=1,\cdots,i-1$
      $\mathbf{a}[i(i+1)/2+j] = \alpha$

For the general implementation of Crout’s method, see Section 4.3.