 Spectral weighting of conjugate gradient error

For a symmetric positive-definite matrix, choose an orthonormal eigenbasis and write the exact solution as $x={\sum }_i{\alpha }_i{v}_i$. Starting at zero, the conjugate gradient method chooses an admissible polynomial $p(0)=1$ of degree at most the iteration count that minimizes the displayed error. The weights depend on both the eigenvalues and the solution coefficients. For two eigenvalues $1,L$, the first polynomial is $1-\eta t$, with $\eta =({\alpha }^2+{\beta }^2{L}^2)/({\alpha }^2+{\beta }^2{L}^3)$. Comparable nonzero coefficients and large $L$ give $\eta \approx 1/L$, so the high-eigenvalue error is removed first. A sufficiently tiny high-mode coefficient changes this behavior; the spectrum alone does not determine the first step.