The hessian produced by Calculus.hessian has some properties that seem odd to
me. First, the resulting matrix is not symmetric (though it's really close).
Second, it is not positive definite even though the eigenvalues are all
positive. This might partly be the result of the first problem.
Comments?
julia> @time h6 = Calculus.hessian(RB.mylike, sol6[2])
206.988247 seconds (2.43 G allocations: 83.830 GB, 15.48% gc time)
10x10 Array{Float64,2}:
....
julia> issym(h6)
false
julia> isposdef(h6)
false
julia> eigvals(h6)
10-element Array{Float64,1}:
3.27548e5
41407.1
1873.44
809.515
385.469
336.118
269.82
64.4228
117.009
122.845
julia> [det(h6[1:i, 1:i]) for i in 1:10] # all submatrix determinants are
positive
10-element Array{Any,1}:
583.356
2.93666e5
3.76334e7
4.26776e9
3.17529e12
6.53405e14
1.12477e17
8.54011e21
3.75658e26
6.65866e29
julia> showall(h6)
[583.356378500377 -215.9519377704782 42.707753088841706 -12.046905619427504
100.26418659571075 30.907721733245413 5.1999446295105445 -3522.0797654066855
4410.401341876791 29.123944464570087
-215.95193777047822 583.3512842422355 -12.0619061856546 42.69860798380086
-82.01778794681003 -20.00373356894301 -8.643894790795585 1484.6195212852567
-3316.905502014349 -2.7622075180025965
42.7077530888417 -12.061906185654598 131.30467049554812 -44.14799847519552
16.748039679197223 -10.98960509547607 43.02830286169418 234.68906012694512
-1258.6622801992237 9.86994636336174
-12.046905619427504 42.69860798380086 -44.147998475195514 131.35743705036847
-17.498616283805696 4.724998443749094 -31.481387268916205 -95.390157895702
209.5245372890838 12.903636963625035
100.26418659571075 -82.01778794681003 16.748039679197223 -17.498616283805696
766.6651018177663 -20.766430168130857 -2.164074090378522 610.206909770971
-6927.858446549729 -200.4409796490271
30.907721733245413 -20.00373356894301 -10.98960509547607 4.724998443749093
-20.766430168130857 209.8283757716871 -60.222027409043555 280.3888402431943
-1638.62573982097 -158.60599205247166
5.199944629510544 -8.643894790795585 43.02830286169418 -31.4813872689162
-2.164074090378524 -60.222027409043555 204.34068072875016 -3.631354965653126
314.50383342119943 30.12788292183477
-3522.079765406686 1484.6195212852565 234.6890601269451 -95.39015789570199
610.2069097709712 280.3888402431943 -3.631354965653125 102596.98982032725
-117630.58331133435 267.23733374838395
4410.401341876791 -3316.9055020143487 -1258.6622801992235 209.5245372890838
-6927.858446549729 -1638.62573982097 314.5038334211995 -117630.58331133435
265600.50878023007 5508.687364627178
29.123944464570084 -2.762207518002596 9.86994636336174 12.903636963625033
-200.4409796490271 -158.6059920524717 30.12788292183477 267.23733374838395
5508.687364627178 2125.8954716959997]
The largest absolute difference between symmetrical entries is < 5e-13.
Ross Boylan
