29.2.5 Symmetry in the stream function equation

This section on symmetry in the stream function equation was contributed by Charles Goldberg. Conjugate gradient solvers work by transforming the system of equations

$${\bf A}{\bf x}=\b$$ (29.28)

into minimizing the quadratic form

$${\bf Q}({\bf x}) = {1\over 2} {\bf x}^T {\bf A}{\bf x}- \b^T {\bf x}$$ (29.29)

This transformation is justified as long as ${\bf A}$ is symmetric, that is, ${\bf A}_{\alpha,\beta} = {\bf A}_{\beta,\alpha}$ for all $\alpha$ and $\beta$.