Solving systems of matrix ODEs — textbook recommendations

Question

This is in reference to the works of Trendafilov whose approaches to multivariate statistical problems boil down to solving a dynamical system involving matrices.

Question: Can anyone suggest a book/article presenting algorithms with numerical examples that solves initial value problems (IVP) like the following system of gradient flows?

Here $\mathbf{Q}$, $\mathbf{D}$, etc are (appropriate) matrices. Most of my attempts in looking arrive with theoretical results (existence, uniqueness, etc..)

Update 3-20-2019: Just for an additional information to anyone in need, I just encountered, in addition to Rodrigo de Azevedo's suggestion, a numerical analysis book on ODEs and PDEs using the free software R. It's Griffiths' Numerical Analysis Using R: Solutions to ODEs and PDEs (2016,CUP).

Update 3-21-2019: It may look like I am writing a blog for this portion but for the love of the math.stack community and, again, to those who earnestly seek for solutions to this problem, the following resources might be of use:

• Numerical methods for ordinary differential equations on matrix manifolds by Lopez (2007)
• Solving Differential Equations on Manifolds by Ernst Hairer (2011)

Answer 1

A book on gradient flows is the following one:

• Uwe Helmke, John B. Moore, Optimization and Dynamical Systems, 2nd edition, March 1996.