A simplified model of a stretched diatomic bond's potential energy surface

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

A simplified model of a stretched diatomic bond's potential energy surface, in reduced dimensionless coordinates xx and yy, is f(x,y)=3x2+y2f(x,y) = 3x^2 + y^2. At the point (x,y)=(1,2)(x,y) = (1,2), steepest descent proposes the direction d=−∇f\mathbf{d} = -\nabla f. A backtracking line search starts at α=1\alpha = 1 and halves α\alpha whenever the Armijo condition f(x+αd)≤f(x)+c1α ∇f⋅d,c1=10−4f(\mathbf{x} + \alpha\mathbf{d}) \le f(\mathbf{x}) + c_1\alpha\,\nabla f\cdot\mathbf{d}, \qquad c_1 = 10^{-4} fails. ((a)) Find the smallest number of halvings needed before a step is accepted. ((b)) State the accepted step length α\alpha and the resulting point (x,y)(x,y).

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