Adjusting the third algorithm of pdsp

[FanwangM]

As @banrovegrie mentioned that the third algorithm is not peer-reviewed. Do you think we should adjust a new one? I find this one might be interesting. Thank you. @PaulWAyers
Gillis, N., & Sharma, P. (2018). A semi-analytical approach for the positive semidefinite Procrustes problem. Linear Algebra and its Applications, 540, 112-137.

If you want to find the Matlab codes, you will need to navigate to this tab. @banrovegrie

[banrovegrie]

Yee, meanwhile I will test the code and confirm the accuracy.

[PaulWAyers]

The original "algorithm 3" has been peer reviewed (I think; it is published in a conference proceedings and that is never a sure thing).
http://www.scielo.org.co/pdf/rcm/v55n1/0034-7426-rcm-55-01-109.pdf

However, I really like Fanwang's suggestion. The Oviedo paper (algorithm 3) performs similarly to what they call PARATAN and what the above reference calls PARTAN. The above reference seems to have good performance, and has four closely related algorithms which could be implemented simulataneously, switching between the algorithms as specified by the user by a keyword. This is explained at the top of Section 5.

Specifically, the

• projected gradient method
• fast gradient method
• parallel tangents method
• semi-analytic fast gradient method

This gives a nice alternative to the first two algorithms, which focus on a more analytic approach (more like the other methods we have considered). This is a numerical approach, and perhaps might be faster (hard to know until we test).