Mbot at 13:34, 8 November 2011

2011-11-08T13:34:58Z

← Older revision		Revision as of 08:34, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: con		Problem type appears to be: con
	Time for symbolic processing: 0.~~12023~~ seconds		Time for symbolic processing: 0.11941 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
	TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31		TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
	=====================================================================================		=====================================================================================
	Problem: --- 1: Problem 1 f_k 243.~~812760223895680000~~		Problem: --- 1: Problem 1 f_k 243.812760283686030000
	sum(\|constr\|) 0.~~000000015386141472~~		sum(\|constr\|) 0.000000003100005141
	f(x_k) + sum(\|constr\|) 243.~~812760239281830000~~		f(x_k) + sum(\|constr\|) 243.812760286786040000
	f(x_0) ~~302~~.~~903800574941840000~~		f(x_0) 292.427334354468940000

	Solver: snopt. EXIT=0. INFORM=1.		Solver: snopt. EXIT=0. INFORM=1.
Line 91:		Line 91:
	Optimality conditions satisfied		Optimality conditions satisfied

	FuncEv ~~112~~ GradEv ~~110~~ ConstrEv ~~110~~ ConJacEv ~~110~~ Iter ~~107~~ MinorIter ~~159~~		FuncEv 82 GradEv 80 ConstrEv 80 ConJacEv 80 Iter 79 MinorIter 129
	CPU time: 0.~~109201~~ sec. Elapsed time: 0.~~118000~~ sec.		CPU time: 0.093601 sec. Elapsed time: 0.090000 sec.

	</pre>		</pre>

Mbot: Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} TomSym implementation of GAMS Example (ELEC,SEQ=230) Given n electrons, find the equilibrium state dist..."

2011-11-02T21:03:03Z

Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} TomSym implementation of GAMS Example (ELEC,SEQ=230) Given n electrons, find the equilibrium state dist..."

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{{Part Of Manual|title=the TomSym Manual|link=[[TomSym|TomSym Manual]]}}

TomSym implementation of GAMS Example (ELEC,SEQ=230)

Given n electrons, find the equilibrium state distribution (of minimal Coulomb potential) of the electrons positioned on a conducting sphere.

This model is from the COPS benchmarking suite.

The number of electrons can be specified using the parameter np.

Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS. Tech. rep., Mathematics and Computer Science Division, 2000.

Morris, J R, Deaven, D M, and Ho, K M, Genetic Algorithm Energy Minimization for Point Charges on a Sphere. Phys. Rev. B. 53 (1996), R1740-R1743.

Saff, E B, and Kuijlaars, A, Distributing Many Points on the Sphere. Math. Intelligencer 19 (1997), 5-11.

<source lang="matlab">
np = 25;

% Coordinates of the electrons
x = tom('x',np,1);
y = tom('y',np,1);
z = tom('z',np,1);

toms potential

% The following two methods of computing the potential energy are
% mathematically equivalent.
useSlowMethod = false; % Set to "true" to try the slower method. (Tip: Use
% a small value for np when you try this.)

if useSlowMethod
% The slow way of computing the sum
% Using nested for loops, we create a symbolic expression containing
% (np-1)*(np-2)/2 terms. Each of these terms will be symbolically
% differentiated to compute the Jacobian and Hessian matrices for use
% by the solver.
% For np=10, the autogenerated Jacobian of this function contains more
% than 15 kilobytes of Matlab code (and the Hessian more than 30 kb).
% TomSym will need several minutes to generate this code, which is
% orders of magnitude more than the fraction of a second required to
% actually solve the numeric problem.
potential = 0;
for i=1:np
for j=(i+1):np
potential = potential + 1.0/sqrt((x(i)-x(j))^2 +...
(y(i)-y(j))^2 + (z(i)-z(j))^2);
end
end
else
% Using vectorized code saves a lot of time in handling the symbolic
% expressions, because the symbolic derivative is computed for the one
% vectorized expression, not on hundreds of individual terms.
% (The autogenerated Hessian of this function contains less than 1 kb
% of Matlab code, regardless of the value of np.)
[ii,jj] = meshgrid(1:np,1:np);
[i,j] = find(ii<jj);
potential = sum(1.0./sqrt((x(i)-x(j)).^2 + (y(i)-y(j)).^2 +...
(z(i)-z(j)).^2));
end

% Ball constraint
ball = (x.^2 + y.^2 + z.^2 == 1);

% Initial guess:
theta = 2*pi*rand(np,1);
phi = pi*rand(np,1);
x0 = {x == cos(theta).*sin(phi)
y == sin(theta).*sin(phi)
z == cos(phi)};

options = struct;
options.solver = 'snopt';
solution = ezsolve(potential,ball,x0,options);
</source>

<pre>
Problem type appears to be: con
Time for symbolic processing: 0.12023 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Problem 1 f_k 243.812760223895680000
sum(|constr|) 0.000000015386141472
f(x_k) + sum(|constr|) 243.812760239281830000
f(x_0) 302.903800574941840000

Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv 112 GradEv 110 ConstrEv 110 ConJacEv 110 Iter 107 MinorIter 159
CPU time: 0.109201 sec. Elapsed time: 0.118000 sec.

</pre>

TomSym Distribution of electrons on a sphere - Revision history

Mbot at 13:34, 8 November 2011

Mbot: Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} TomSym implementation of GAMS Example (ELEC,SEQ=230) Given n electrons, find the equilibrium state dist..."