The following is the actual more extensive matrix that I am actually interested in. The first example was just to try and figure out what was going wrong on a simpler case (using just the highest degree term in each element).

Also what I am interested in about the determinant polynomial is the set of roots. (I am actually doing an errors-in-variable estimate of the coefficients of an approximating algebraic curve.) Thus if I cannot get a good interpretable set of roots, I am lost. Thus the overall error in the determinant coefficients is not relevant here.

Below I give the definition of the matrix used, and then I give the results for the minor method, the default method, and the unifloat method. Yes, the default method is the unifloat method because my matrix elements are in R[x], BUT the help page says it uses gaussian elimination. The fracfree method gives an error, no doubt because the help page says it is for matrices with elements in Z[x]. I notice that one responder first turns the polynomial coefficients into exact rationals. I already pointed out that that works because Maple does exact calculations with rationals.

You will see that only the "minor" method below gives usable, interpretable roots. Also, as I previously noted, with rational coefficients the results are exactly the same as for the "minor" method. For the default, unifloat, method, all but one of the roots are solidly complex. If they were even close to the correct roots, I might know how to interpret them, but since they are not the correct roots, they only serve to obscure things.

So here's the example:

`ψ_data_ex` := Matrix(6, 6, [[180*sigma2^2-53417.4875669262*sigma2+2.55312019922288*10^6, -17980.4676072929*sigma2+1.48876202145622*10^6, 60*sigma2^2-17777.2544622252*sigma2+1.26999620604100*10^6, -1792.74593023400*sigma2+1.47107749463710*10^5, -597.004097753022*sigma2+89710.8619398197, -60*sigma2+8902.91459448771], [-17980.4676072929*sigma2+1.48876202145622*10^6, 60*sigma2^2-17777.2544622252*sigma2+1.26999620604100*10^6, 1.47594266713082*10^6-17980.4676072929*sigma2, -597.004097753022*sigma2+89710.8619398197, 89455.7164588473-597.581976744666*sigma2, 5993.48920243096], [60*sigma2^2-17777.2544622252*sigma2+1.26999620604100*10^6, 1.47594266713082*10^6-17980.4676072929*sigma2, 180*sigma2^2-53246.0392064249*sigma2+2.52236958317357*10^6, 89455.7164588473-597.581976744666*sigma2, -1791.01229325907*sigma2+1.46048381658527*10^5, -60*sigma2+8874.33986773749], [-1792.74593023400*sigma2+1.47107749463710*10^5, -597.004097753022*sigma2+89710.8619398197, 89455.7164588473-597.581976744666*sigma2, -60*sigma2+8902.91459448771, 5993.48920243096, 597.581976744666], [-597.004097753022*sigma2+89710.8619398197, 89455.7164588473-597.581976744666*sigma2, -1791.01229325907*sigma2+1.46048381658527*10^5, 5993.48920243096, -60*sigma2+8874.33986773749, 597.004097753022], [-60*sigma2+8902.91459448771, 5993.48920243096, -60*sigma2+8874.33986773749, 597.581976744666, 597.004097753022, 60]])

" method = minor";

`Det_ψ_data_ex` := sort(Determinant(`ψ_data_ex`, method = minor));

norm_fac := max(seq(abs(coeff(`Det_ψ_data_ex`, sigma2, i)), i = 0 .. 9));

relative_p := sort(`Det_ψ_data_ex`/norm_fac);

roots_p := fsolve(relative_p, complex)

"method = default";

`Det_ψ_data_ex` := sort(Determinant(`ψ_data_ex`));

norm_fac := max(seq(abs(coeff(`Det_ψ_data_ex`, sigma2, i)), i = 0 .. 9));

relative_p := sort(`Det_ψ_data_ex`/norm_fac);

roots_p := fsolve(relative_p, complex);

" method = unifloat[sigma2]";

`Det_ψ_data_ex` := sort(Determinant(`ψ_data_ex`, method = unifloat[sigma2]));

norm_fac := max(seq(abs(coeff(`Det_ψ_data_ex`, sigma2, i)), i = 0 .. 9));

relative_p := sort(`Det_ψ_data_ex`/norm_fac);

roots_p := fsolve(relative_p, complex)

Here are the results:

" method = minor"

`Det_ψ_data_ex` := 186624000000*sigma2^8-7.322237352*10^13*sigma2^7+1.167816239*10^16*sigma2^6-9.707497494*10^17*sigma2^5+4.477166594*10^19*sigma2^4-1.124489089*10^21*sigma2^3+1.395557199*10^22*sigma2^2-6.72105102*10^22*sigma2+1.2667708*10^22

norm_fac := 6.72105102*10^22

relative_p := 2.776708575*10^(-12)*sigma2^8-1.089448262*10^(-9)*sigma2^7+1.737550029*10^(-7)*sigma2^6-0.1444342182e-4*sigma2^5+0.6661408434e-3*sigma2^4-0.1673085185e-1*sigma2^3+.2076397270*sigma2^2-1.000000000*sigma2+.1884780812

roots_p := .196358275828771, 14.2552568223045, 14.7363162314595, 48.2900528956757, 49.8699195146855, 83.2511169784422, 83.7567600395173, 97.9966150998332

"method = default"

`Det_ψ_data_ex` := -8.09527159898091*10^14*sigma2^9-3.71919218887944*10^14*sigma2^8+4.68051527145545*10^14*sigma2^7+1.16457363073298*10^16*sigma2^6-9.69868249773796*10^17*sigma2^5+4.47716413947149*10^19*sigma2^4-1.12448839673387*10^21*sigma2^3+1.39555702965615*10^22*sigma2^2-6.72104791815170*10^22*sigma2+1.26677278632965*10^22

norm_fac := 6.72104791815170*10^22

relative_p := -1.20446568713159*10^(-8)*sigma2^9-5.53364926745266*10^(-9)*sigma2^8+6.96396652494423*10^(-9)*sigma2^7+1.73272627262154*10^(-7)*sigma2^6-0.144303129747736e-4*sigma2^5+0.666140785483749e-3*sigma2^4-0.167308492727293e-1*sigma2^3+.207639797640356*sigma2^2-1.*sigma2+.188478463739031

roots_p := -11.0014855758297-5.90889388930547*I, -11.0014855758297+5.90889388930547*I, -1.71291902117475-10.5657205273659*I, -1.71291902117475+10.5657205273659*I, .196358694448831, 5.10515019731233-7.13344289603453*I, 5.10515019731233+7.13344289603453*I, 7.28136119151786-2.22432765697437*I, 7.28136119151786+2.22432765697437*I

" method = unifloat[sigma2]"

`Det_ψ_data_ex` := -8.09527159898091*10^14*sigma2^9-3.71919218887944*10^14*sigma2^8+4.68051527145545*10^14*sigma2^7+1.16457363073298*10^16*sigma2^6-9.69868249773796*10^17*sigma2^5+4.47716413947149*10^19*sigma2^4-1.12448839673387*10^21*sigma2^3+1.39555702965615*10^22*sigma2^2-6.72104791815170*10^22*sigma2+1.26677278632965*10^22

norm_fac := 6.72104791815170*10^22

relative_p := -1.20446568713159*10^(-8)*sigma2^9-5.53364926745266*10^(-9)*sigma2^8+6.96396652494423*10^(-9)*sigma2^7+1.73272627262154*10^(-7)*sigma2^6-0.144303129747736e-4*sigma2^5+0.666140785483749e-3*sigma2^4-0.167308492727293e-1*sigma2^3+.207639797640356*sigma2^2-1.*sigma2+.188478463739031

roots_p := -11.0014855758297-5.90889388930547*I, -11.0014855758297+5.90889388930547*I, -1.71291902117475-10.5657205273659*I, -1.71291902117475+10.5657205273659*I, .196358694448831, 5.10515019731233-7.13344289603453*I, 5.10515019731233+7.13344289603453*I, 7.28136119151786-2.22432765697437*I, 7.28136119151786+2.22432765697437*I