I.4.3.1: The Cartesian Folium

The ``folium cartesii'' (letter of Descartes to Mersenne, August 23, 1638) is the function defined by

The gradient is and the Hessian is It follows that has a saddle point at and an isolated local minimum at . These are the only two stationary points. At the eigenvalues of the Hessian are and , at they are and .

The Hessian is singular if and only if is on the hyperbola . It is positive definite if and only if is above the branch of the hyperbola in the positive orthant.

See Figure 1 for contour plots of sections of on two different scales.


plot of chunk folium

Figure 1: Folium, two scales, two sections


Now apply coordinate descent [De Leeuw, 2007b]. The minimum over for fixed only exists if , in which case it is attained at . In the same way, the minimum over for fixed is attained at . Thus the algorithm is simply and the algorithmic map is The algorithm can only work if we start with . It then converges, linearly and monotonically, to with convergence rate . If we start with then is unbounded below and thus coordinate descent fails.