After diagonalising the small-oscillation energy of a triatomic molecule
PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists
Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness
Problem
After diagonalising the small-oscillation energy of a triatomic molecule, two of its normal coordinates (x,y) contribute a quadratic energy f(x,y)=450x2+0.5y2(arbitrary energy units).
((a)) Predict, from the condition number of the Hessian, the number of steepest-descent steps (with exact line search) needed to reduce f−f∗ by four orders of magnitude from the worst starting direction, a count no other start can exceed.
((b)) Predict, without doing the arithmetic of a full run, how many iterations Newton's method needs to reach the minimum exactly from any starting point, and explain why the answer does not depend on κ.