Sastre code gen

[jarlebring]

Although generators/sastre.jl does contain some code corresponding to the paper https://doi.org/10.1016/j.laa.2017.11.010 it is not yet complete.

I don't think the MEP-code should go into the package / paper - too complicated to make it complete. If possible we can provide the coefficients that were not reported in the paper by using the MEP-approach. This is currently done for the exponential with 6 multiplications.

Details to be specified.

[eringh]

I started on this by introducing tests, and I cannot get gen_sastre_basic_exp(6) to work. I have looked, and the coefficients match those in Table 7, but the output is not a Taylor approximation of the exponential.

(graph,cref) = gen_sastre_basic_exp(6);
using Polynomials
x=Polynomial("x");
coeffs(eval_graph(Compgraph(Any,graph),x))
21-element Vector{Float64}:
 2.755731922398589e-7
 2.505210838544172e-8
 2.08767569878681e-9
 1.6059043836821616e-10
 1.1470745597729725e-11
 7.647163731819817e-13
 4.7794773323873865e-14
 2.8114572543455214e-15
 1.561920696858623e-16
 8.220635246624331e-18
 4.1103176233121653e-19
 1.9572941063391266e-20
 8.896791392450576e-22
 3.868170170630684e-23
 1.6117375710961186e-24
 6.446950284384475e-26
 2.4795962632247983e-27
 9.183689863795547e-29
 3.2798892370698385e-30
 1.1309962886447718e-31
 3.769987628815906e-33

[eringh]

However, it seems to be the evaluation of equation (52) in the paper that is missing:

1 ./ factorial.(Int128.(10:30))
21-element Vector{Float64}:
 2.755731922398589e-7
 2.505210838544172e-8
 2.08767569878681e-9
 1.6059043836821613e-10
 1.1470745597729725e-11
 7.647163731819816e-13
 4.779477332387385e-14
 2.8114572543455206e-15
 1.5619206968586225e-16
 8.22063524662433e-18
 4.110317623312165e-19
 1.9572941063391263e-20
 8.896791392450574e-22
 3.8681701706306835e-23
 1.6117375710961184e-24
 6.446950284384474e-26
 2.4795962632247972e-27
 9.183689863795546e-29
 3.279889237069838e-30
 1.1309962886447718e-31
 3.7699876288159054e-33

[jarlebring]

It might be that that the data in table 7 was not correctly used. It should be combined with (52) which it is not.

I thought there were also other procedures in that paper with numbers which were not tabulated.

[eringh]

I thought there were also other procedures in that paper with numbers which were not tabulated.

There seems to be a lot of versions where the coefficients are not tabulated. And a lot of versions that are based on slightly different evaluation schemes. Currently the function gen_sastre_basic_exp(k) is differentiating methods depending on the number of multiplications. But that is not uniquely determining which of the methods described in the paper that is used (although, at least for now, it is uniquely determining method combined with coefficients described in the paper).

• De we want to implement equations (62)-(65)? And then also Table 9?
• Do we want to compute unreported coefficients somehow? (Maybe with the MEP approach.) (I think k=4 is something we have computed, because I cannot find it in the paper.)

How much is it wort to have all aspects of the paper implemented? It can get quite time consuming.

[jarlebring]

Currently the function gen_sastre_basic_exp(k) is differentiating methods depending on the number of multiplications

Then add a kwarg?

How much is it wort to have all aspects of the paper implemented? It can get quite time consuming.

It's better if you decide this yourself since you have the best overview right now. Start somewhere and see how far you want to go.

[jarlebring]

Just to be clear here: I think it's okay if the set of graphs is only a subset of everything in that paper considering that many values are not tabulated. We already have some graphs which are not tabulated (yes, I added the coefficients from an MEP-computation).

[eringh]

Sounds good. I was actually thinking of calling it done with possibly one exception: Do we want a conversion of the general framework (62)-(65) into Degopt? It might look good in the remark about related notation if we have a automatic conversion from y_ks to degopt.

[jarlebring]

Indeed, if we implement (62)-(65) it would be good to have it on degopt form.

[eringh]

Now the SID-paper can also be evaluated with the function gen_sastre_yks_degopt. Test for higher values of k gives further confidence in the implementation.