Jan 29th

Action Items

• Since SCS solver failed to verify the A- and B-stability for the known GLMs, Yousef will try other SDP solvers available in JuMP.

Feb 5th

Action Items

• Yousef tried additional SDP solvers available on JuMP (ProxSDP, CSDP)

Both were able to verify the Algebraic stability definition, but not with high accuracy (although I asked for tighter tolerance). It gave something close to the actual solution of G and D given in the reference book (i.e. G = [1,0;0 11/12] and D = [1/2 0, 0 1/2]). This was the case when adding the constraint G_11 == 1, otherwise, we get zero solutions.

Here are the outputs from CSDP

\[ G = [1.0000000000084073, -1.5277187743705936e-6; -1.5277187743705936e-6, 0.9166666665946092]\] \[ D = [0.5000007635518395, -1.3781973548239023e-11; -1.3781973548239023e-11, 0.4999992363128975]\]

and the eigenvalues of the G, D, and M are respectively

\[0.9166666694761793 \] \[0.4999956700567722\] \[-7.020405003333115e-11\]

Both solvers declared that the problem of A-stability is infeasible (had to impose G_11 ==1 to avoid zero solutions)

Feb 14th

I wrote BDF2 as GLM using eq (3.4) in Butcher 2006 notes with

A = 2/3 B = [8/9;-2/9] U = [4/3 , -1/3] V = [4/3 , 1; -1/3 , 0];

The solvers (using the conjectured definition of A-stability) declared the problem infeasible again.

Feb 16th

Action items

• Try reducing the A-stability definition for GLMs to give an equivalent definition for LMMs.
• Look into The Schur-Cohn Algorithm in HW vol 1 page 388.

Feb 22nd

• A good approach is to first come up with an algebraic condition for A-stability condition for LMMs.

• Progress so far is an understanding of how different multistep methods is written as GLMs.