Computations for Widom 601 sequence

In this page we show the minimizations results for each initial guess. In the table we show the energy, the norm of the gradient of the energy function with the obtained configuration, some data of the eigenvalues and the Hessian condition number. In particular what we can observe in this case some solutions have only positive eigenvalues and others have only one negative eigenvalue.

Solution for initial guess A:

We used fminunc Matlab function with Trust Region algorithm and tolerance for the first order optimality of 1.0e −06. The absolute difference of the two configurations obtained with the two different initial guesses is of the order of 10e −02.

bBDNA bBDNA with gd-intras
Energy 38.5508 38.6636
Norm of the gradient 0.7167 0.6347
Min Hess eigenvalue -1.3285e-05 3.4011e-05
Max Hess eigenvalue 6.4146e+04 6.4122e+04
Median Hess eigenvalue 11.6359 11.6346
Number of negative eigenvalues 1 0
Hessian condition number 1.2426e+10 4.8016e+09
Number of links 9 9

3D plot of the configuration obtained with the bBDNA initial guess.


3D plot of the configuration obtained with the bBDNA with gd intras initial guess.


2D plot of the inter variables of the two solutions.
MATLAB figures saved variables:
W-3D-GUESS1-bBDNA.fig
W-3D-GUESS1-bBDNAgd.fig
W-2D-GUESS4.fig

Solution for initial guess B:

We run fminunc Matlab function with Quasi-Newton algorithm and tolerance for the first order optimality of 1.0e −08. The absolute difference of the two configurations obtained with the two different initial guesses is of the order of 10e −03. We then run the fminunc Matlab function again with Trust Region algorithm and first order optimality of 1.0e −06, but having as starting point the solutions obtained with the first Quasi-Newton run. Doing that we obtained an absolute difference of the two configurations with the two different initial guesses of the order of 10e −04.

bBDNA bBDNA with gd-intras
Energy 124.6879 124.6760
Norm of the gradient 0.2698 0.2954
Min Hess eigenvalue 7.1000e-06 2.3600e-05
Max Hess eigenvalue 6.2960e+04 6.2959e+04
Median Hess eigenvalue 11.5905 11.5906
Number of negative eigenvalues 0 0
Hessian condition number 2.9683e+10 8.4643e+10
Number of links 8 8

3D plot of the configuration obtained with the bBDNA initial guess.


3D plot of the configuration obtained with the bBDNA with gd intras initial guess.


2D plot of the inter variables of the two solutions.
MATLAB figures and saved variables:
W-3D-GUESS2-bBDNA.fig
W-3D-GUESS2-bBDNAgd.fig
W-2D-GUESS2.fig

Solution for initial guess C:

We used fminunc Matlab function with Trust Region algorithm and tolerance for the first order optimality of 1.0e −06. The absolute difference of the two configurations obtained with the two different initial guesses is of the order of 10e −02.

bBDNA bBDNA with gd-intras
Energy 47.3628 47.6631
Norm of the gradient 0.2870 0.0535
Min Hess eigenvalue -8.6882e-06 -1.4128e-06
Max Hess eigenvalue 6.4387e+04 6.4394e+04
Median Hess eigenvalue 11.6271 11.6276
Number of negative eigenvalues 1 1
Hessian condition number 2.2151e+10 3.8152e+10
Number of links 9 9

3D plot of the configuration obtained with the bBDNA initial guess.


3D plot of the configuration obtained with the bBDNA with gd intras initial guess.


2D plot of the inter variables of the two solutions.
MATLAB figures and saved variables:
W-3D-GUESS3-bBDNA.fig
W-3D-GUESS3-bBDNAgd.fig
W-2D-GUESS3.fig

Solution for initial guess D:

We run fminunc Matlab function with Quasi-Newton algorithm and tolerance for the first order optimality of 1.0e −08. The absolute difference of the two configurations obtained with the two different initial guesses is of the order of 10e −03. We then run the fminunc Matlab function again with Trust Region algorithm and first order optimality of 1.0e −06, but having as starting point the solutions obtained with the first Quasi-Newton run. Doing that we obtained an absolute difference of the two configurations with the two different initial guesses of the order of 10e −04.

bBDNA bBDNA with gd-intras
Energy 107.1669 106.9373
Norm of the gradient 0.4399 0.4376
Min Hess eigenvalue 8.9821e-05 6.3080e-05
Max Hess eigenvalue 6.5538e+04 6.5598e+04
Median Hess eigenvalue 11.5856 11.5857
Number of negative eigenvalues 0 0
Hessian condition number 1.9035e+09 2.7651e+09
Number of links 10 10

3D plot of the configuration obtained with the bBDNA initial guess.


3D plot of the configuration obtained with the bBDNA with gd intras initial guess.


2D plot of the inter variables of the two solutions.
MATLAB figures and saved variables:
W-3D-GUESS4-bBDNA.fig
W-3D-GUESS4-bBDNAgd.fig
W-2D-GUESS4.fig