```% Boyd & Vandenberghe "Convex Optimization"
% Almir Mutapcic - Jan 2006
%
% Given a symmetric matrix A in R^(n-by-n) with some entries unspecified
% we find its completion such that A is positive semidefinite and
% it has a maximum determinant out of all possible completions.
% This problem can be formulated as a log det (and det_rootn) problem.
%
% This is a numerical instance of the specified book exercise.

% problem size
n = 4;

% create and solve the problem
cvx_begin sdp
% A is a PSD symmetric matrix (n-by-n)
variable A(n,n) symmetric;
A >= 0;

% constrained matrix entries.
A(1,1) == 3;
A(2,2) == 2;
A(3,3) == 1;
A(4,4) == 5;
% Note that because A is symmetric, these off-diagonal
% constraints affect the corresponding element on the
% opposite side of the diagonal.
A(1,2) == .5;
A(1,4) == .25;
A(2,3) == .75;

% find the solution to the problem
maximize( log_det( A ) )
% maximize( det_rootn( A ) )
cvx_end

% display solution
disp(['Matrix A with maximum determinant (' num2str(det(A)) ') is:'])
A
disp(['Its eigenvalues are:'])
eigs = eig(A)
```
```
Successive approximation method to be employed.
For improved efficiency, SDPT3 is solving the dual problem.
SDPT3 will be called several times to refine the solution.
Original size: 59 variables, 18 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
1/  1 | 2.753e-01  4.915e-03  0.000e+00 | Solved
1/  1 | 3.497e-02  8.018e-05  0.000e+00 | Solved
1/  1 | 4.030e-03  1.064e-06  0.000e+00 | Solved
0/  1 | 4.704e-04  1.396e-08  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +3.02422

Matrix A with maximum determinant (20.578) is:

A =

3.0000    0.5000    0.1875    0.2500
0.5000    2.0000    0.7500    0.0417
0.1875    0.7500    1.0000    0.0156
0.2500    0.0417    0.0156    5.0000

Its eigenvalues are:

eigs =

0.5964
2.0908
3.2773
5.0355

```