Doesn't Calculu's Hessian just do finite difference? There will naturally
be floating point errors.
isposdef returns false if the matrix is not hermitian, under the hood,
potrf from Lapack is used which work with hermitian matrices.
On Tuesday, May 31, 2016 at 7:04:04 AM UTC+2, Boylan, Ross wrote:
>
> 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
>
>
