# Calculating the \`k\` smallest positive eigenvalues of a matrix with ARPACK

**URL:** <https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864>\
**Category:** General Usage\
**Created:** [September 17, 2019, 7:37pm UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864 "2019-09-17T19:37:42Z")\
**Posts on this page:** 6\
**Page:** 2

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**Author:** ![Tamas\_Papp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tamas_papp/32/25949_2.png) [@Tamas\_Papp](https://discourse.julialang.org/u/Tamas_Papp)\
**Post date:** [September 19, 2019, 5:43am UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864/21 "2019-09-19T05:43:00Z")

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Please consider writing a blog post about Hessenberg factorizations: what they are useful for, and how to use them in Julia. Your PR adds a very important capability to LinearAlgebra, but I think Hessenberg is one of the lesser-known factorizations.

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**Author:** ![juthohaegeman](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/juthohaegeman/32/8620_2.png) [@juthohaegeman](https://discourse.julialang.org/u/juthohaegeman)\
**Post date:** [September 19, 2019, 6:53am UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864/22 "2019-09-19T06:53:55Z")

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Also, if it is symmetric and of size 10000, you should also be able to get away with computing all eigenvalues. For symmetric matrices, the dense algorithms can be pretty fast, and it is possible to select to compute only a range of eigenvalues. I have little to no experience with this myself, but I know this methods exist in LAPACK and are exposed in Julia.

Also, if you do want to continue with shift and invert, for a symmetric matrix the hessenberg factorization suggested by @stevengj amounts to bringing it into a `SymTridiagonal` form, and it is the first step into computing the dense eigenvalue decomposition. I am not sure if that step is also explicitly exposed in Julia, I guess @stevengj knows better?

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [September 19, 2019, 11:47am UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864/23 "2019-09-19T11:47:49Z")

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> [@juthohaegeman](#):
>
> Also, if you do want to continue with shift and invert, for a symmetric matrix the hessenberg factorization suggested by @stevengj amounts to bringing it into a `SymTridiagonal` form, and it is the first step into computing the dense eigenvalue decomposition. I am not sure if that step is also explicitly exposed in Julia, I guess @stevengj knows better?

Yes, in Julia 1.3 if you do `hessenberg(Hermitian(A))` it will form a real `SymTridiagonal` Hessenberg factor explicitly:

```julia
julia> hessenberg(Hermitian(rand(5,5)))
Hessenberg{Float64,SymTridiagonal{Float64,Array{Float64,1}},Array{Float64,2},Array{Float64,1},Bool}
Q factor:
5×5 LinearAlgebra.HessenbergQ{Float64,Array{Float64,2},Array{Float64,1},true}:
  0.527746 0.453729 -0.641028 -0.323568 0.0
  0.425025 0.126814 0.713474 -0.542427 0.0
  0.0689248 -0.820645 -0.27317 -0.497162 0.0
 -0.73218 0.323405 -0.0735948 -0.5949 0.0
  0.0 0.0 0.0 0.0 1.0
H factor:
5×5 SymTridiagonal{Float64,Array{Float64,1}}:
 -0.34423 -0.361886 ⋅ ⋅ ⋅      
 -0.361886 -0.654238 -0.29319 ⋅ ⋅      
   ⋅ -0.29319 0.460663 0.698886 ⋅      
   ⋅ ⋅ 0.698886 2.33577 -1.54195 
   ⋅ ⋅ ⋅ -1.54195 0.361714

julia> hessenberg(Hermitian(rand(ComplexF64, 5,5)))
Hessenberg{Complex{Float64},SymTridiagonal{Float64,Array{Float64,1}},Array{Complex{Float64},2},Array{Complex{Float64},1},Bool}
Q factor:
5×5 LinearAlgebra.HessenbergQ{Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1},true}:
  0.136178-0.149696im … -0.455647-0.203191im 0.0+0.0im
 -0.530702+0.110264im -0.285712-0.428174im 0.0+0.0im
 0.0588643-0.625142im -0.136137-0.367904im 0.0+0.0im
  0.453133+0.256218im -0.439867-0.372513im 0.0+0.0im
       0.0+0.0im 0.0+0.0im 1.0+0.0im
H factor:
5×5 SymTridiagonal{Float64,Array{Float64,1}}:
 -0.434709 -0.181771 ⋅ ⋅ ⋅      
 -0.181771 0.278323 -0.758955 ⋅ ⋅      
   ⋅ -0.758955 1.01864 1.61074 ⋅      
   ⋅ ⋅ 1.61074 1.07538 -1.98639 
   ⋅ ⋅ ⋅ -1.98639 0.958969

```

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<div class="post-metadata">

**Author:** ![sergevic](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sergevic/32/9958_2.png) [@sergevic](https://discourse.julialang.org/u/sergevic)\
**Post date:** [September 28, 2019, 4:26pm UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864/24 "2019-09-28T16:26:06Z")

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Thank you for your comments, guys 🙂 Sorry for the delay but I was on vacation.

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**Author:** ![denis](https://avatars.discourse-cdn.com/v4/letter/d/5e9695/32.png) [@denis](https://discourse.julialang.org/u/denis)\
**Post date:** [November 21, 2019, 3:30pm UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864/25 "2019-11-21T15:30:05Z")

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Thanks for posting your data.  
Forwhatitsworth, the eigenvalues of M^-1 A all lie ~ on the imaginary axis, |re| \< 2e-10;  
the imaginary parts are ± 2.5e2 … 2.7e6, i.e. condition number 1e4.  
A stem-leaf plot of the exponents:

```
2 2
3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

```

(I work in python, not Julia, sorry. Running the ARPACK wrapper scipy.sparse.linalg.eigs on M^-1 A dense,  
both SM and shift-reduce, i.e. LM with sigma=0, work with tol 1e-4;  
with tol 1e-6, SM doesn’t converge.)

Now it’s easy to generate test matrices with whatever spectrum you like –  
but what’s reasonable ?  
A wiki for test cases / plots of real spectra would be welcome.

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [November 21, 2019, 7:13pm UTC](https://discourse.julialang.org/t/calculating-the-k-smallest-positive-eigenvalues-of-a-matrix-with-arpack/28864/26 "2019-11-21T19:13:20Z")

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> [@denis](#):
>
> (I work in python, not Julia, sorry.)

Note that [discourse.julialang.org](http://discourse.julialang.org) is really focused on using Julia — it is a not a general help forum for numerical linear algebra. Also, you generally shouldn’t resurrect old discussion threads to raise entirely new questions.

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