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A Sample LDA Toy Problem (Trace Minimization with Nonlinear Term)

We attempt to solve a toy LDA problem described in § for four electrons on a one-dimensional grid with 12 points and a coupling constant of 20. In this case, we use the first four eigenvectors of the linear problem as a starting point.

>> !cp examples/ldatoy/*.m .
>> K = Kinetic(12);
>> parameters(K,20);
>> Y0 = guess(4);
>> [fn,Yn] = sg_min(Y0,'newton','euclidean');
iter    grad            F(Y)              flops         step type
0       2.432521e+00    7.750104e+00         4554       none
  invdgrad: Hessian not positive definite, CG terminating early
1       4.770038e-01    7.546426e+00       367756       steepest step
2       4.664971e-02    7.531450e+00       789739       Newton step
3       3.363736e-04    7.531232e+00      1284452       Newton step
4       1.809273e-07    7.531232e+00      1976096       Newton step
5       1.843536e-07    7.531232e+00      2778377       Newton step

Figure 9.4 shows the convergence curve for this run.

Figure 9.4: LDA toy problem
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Susan Blackford 2000-11-20