```% Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 01/29/06
% Updated to use GP mode by Almir Mutapcic 02/08/06
%
% Given a square matrix M, the goal is to find a vector (with dii > 0)
% such that ||DMD^{-1}||_F is minimized, where D = diag(d).
% The problem can be cast as an unconstrained geometric program:
%           minimize sqrt( sum_{i,j=1}^{n} Mij^2*di^2/dj^2 )
%

rs = randn( 'state' );
randn( 'state', 0 );

% matrix size (M is an n-by-n matrix)
n = 4;
M = randn(n,n);

% formulating the problem as a GP
cvx_begin gp
variable d(n)
minimize( sqrt( sum( sum( diag(d.^2)*(M.^2)*diag(d.^-2) ) ) ) )
% Alternate formulation: norm( diag(d)*abs(M)*diag(1./d), 'fro' )
cvx_end

% displaying results
D = diag(d);
disp('The matrix D that minimizes ||DMD^{-1}||_F is: ');
disp(D);
disp('The minimium Frobenius norm achieved is: ');
disp(norm(D*M*inv(D),'fro'));
disp('while the Frobunius norm of the original matrix M is: ');
disp(norm(M,'fro'));
```
```
Successive approximation method to be employed.
SDPT3 will be called several times to refine the solution.
Original size: 82 variables, 32 equality constraints
16 exponentials add 128 variables, 80 equality constraints
-----------------------------------------------------------------
Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
16/ 16 | 8.000e+00  1.276e+01  0.000e+00 | Solved
16/ 16 | 1.590e+00  1.546e-01  0.000e+00 | Solved
15/ 15 | 7.950e-02  4.018e-04  0.000e+00 | Solved
2/  9 | 1.198e-03  7.548e-08  0.000e+00 | Solved
0/  0 | 2.616e-05  0.000e+00  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +3.25231

The matrix D that minimizes ||DMD^{-1}||_F is:
1.0885         0         0         0
0    0.9138         0         0
0         0    0.9754         0
0         0         0    1.6294

The minimium Frobenius norm achieved is:
3.2523

while the Frobunius norm of the original matrix M is:
3.6126

```