# 7.4 Doolittle’s Method for Symmetric Matrices

If A is a symmetric n × n matrix, Algorithm 10 computes – one row at a time – the upper triangular matrix U that results from a Doolittle decomposition. The upper triangular portion of A is overwritten by U.

Algorithm 10: Doolittle’s Method - Symmetric Implementation
 for $i=1,\cdots,n$ for $j=1,\cdots,i-1$ $w_{j}=\displaystyle\frac{a_{ji}}{a_{jj}}$ for $j=i,\cdots,n$ $\alpha=a_{ij}$ for $p=1,\cdots,i-1$ $\alpha=\alpha-a_{ip}w_{p}$ $a_{ij}=\alpha$

The algorithm uses a working vector w of length n to construct the relevant portion of row i from L at each stage of the factorization. Elements from L are derived using Equation 65.

If the upper triangular portion of A is stored in a linear array, Algorithm 11 results in the same factorization (assuming zero based subscripting). The upper triangular portion ofA is overwritten by U.

Algorithm 11: Doolittle’s Method - Symmetric, Array Based
 for $i=0,\cdots,n-1$ for $j=0,\cdots,i-1$ $\mathbf{w}\text{[j]}=\displaystyle\frac{\mathbf{a}\text{[jn-j(j+1)/2+i]}}{% \mathbf{a}\text{[jn-j(j+1)/2+j]}}$ for $j=i,\cdots,n-1$ $\alpha=\mathbf{a}$[in-i(i+1)/2+j] for $p=1,\cdots,i-1$ $\alpha=\alpha-\mathbf{a}$[in-i(i+1)/2+p]$\cdot\mathbf{w}$[p] $\mathbf{a}$[in-i(i+1)/2+j] = $\alpha$

For the general implementation of Doolittle’s method, see Section 4.2.