# Laplacian for a minimized function

**URL:** <https://discourse.julialang.org/t/laplacian-for-a-minimized-function/103836>\
**Category:** General Usage\
**Tags:** optimization, forwarddiff\
**Created:** [September 14, 2023, 5:10am UTC](https://discourse.julialang.org/t/laplacian-for-a-minimized-function/103836 "2023-09-14T05:10:36Z")\
**Posts on this page:** 1\
**Showing post:** 7

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**Author:** ![yosuga](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/yosuga/32/52638_2.png) [@yosuga](https://discourse.julialang.org/u/yosuga)\
**Post date:** [September 14, 2023, 5:57pm UTC](https://discourse.julialang.org/t/laplacian-for-a-minimized-function/103836/7 "2023-09-14T17:57:44Z")

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FYI, it may relate to the nature of `R(μ,E)`. Namely, `R(μ,E)` could take a minimum in a certain region (not at a single point) as a function of `(μ[1],μ[2])` for a fixed `E`, say `E=E0`. See [this topic](https://discourse.julialang.org/t/finding-the-minimizer-region-of-a-function/103819) as an example.  
The important features are

1. When `R(μ,E0)` has a minimizer region, the integrand of `R(μ,E0)` is well-defined in there.
2. When `R(μ,E0)` has a minimizer point, the integrand of `R(μ,E0)` shows a logarithmic singularity at that point.

although, in both cases, `R(μ,E0)` itself is well-defined and takes a finite value.

My observation tells that `DOS1(E)` doesn’t work when a minimizer point exists. For instance,

1. At `E0 = [0.0,0.0]`, which provides a minimizer point,  
`(DOS1(E0),DOS2(E0))=(2.8271597168564594e-16, 0.1418357989357821)`.

2. At `E0 = [-3.0,3.0]`, which gives a minimizer region,  
`(DOS1(E0),DOS2(E0))=(0.0, -0.00018425790497728805)`.

For a given `E0`, you can see whether `R(μ,E0)` has a minimizer point or a minimizer region by calculating [Amoeba](https://en.wikipedia.org/wiki/Amoeba_(mathematics)) of the polynomial, `p(z,w) = t*T+td*Td+γ*Γ-E0[1]-im*E0[2]`, where `z = exp(μ[1]+im*θ[1])` and `w = exp(μ[2]+im*θ[2])`.

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_[View the full topic](https://discourse.julialang.org/t/laplacian-for-a-minimized-function/103836)._