# Hessenberg factorization

**URL:** https://discourse.julialang.org/t/hessenberg-factorization/20125
**Category:** Numerics
**Tags:** question
**Created:** [January 26, 2019, 11:47am UTC](https://discourse.julialang.org/t/hessenberg-factorization/20125 "2019-01-26T11:47:58Z")
**Posts on this page:** 3
**Page:** 1

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### Author: ![olof3](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/olof3/32/23875_2.png) [@olof3](https://discourse.julialang.org/u/olof3)
#### Post date: [January 26, 2019, 11:47am UTC](https://discourse.julialang.org/t/hessenberg-factorization/20125/1 "2019-01-26T11:47:58Z")

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Hello,

To compute the frequency response of a linear system

\dot{x} = Ax + Bu  
y = Cx

where A is a square matrix, one should compute G(s) = C(sI - A)\backslash B for several complex values s.

This can be done efficiently [1] with a Hessenberg factorization A = QHQ^\*, since then  
G(s) = CQ(sI - H)\backslash (Q^\*B), where (sI - H) is Hessenberg.

There is the `LinearAlgebra.Hessenberg` object but I can’t find any method for an efficient \ with the Hessenberg matrix that it contains. I guess I’m missing something? Thanks

[1] Laub, Alan. “Efficient multivariable frequency response computations.” IEEE Transactions on Automatic Control 26.2 (1981)

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### Author: ![mohamed82008](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mohamed82008/32/18171_2.png) [@mohamed82008](https://discourse.julialang.org/u/mohamed82008)
#### Post date: [January 26, 2019, 11:57am UTC](https://discourse.julialang.org/t/hessenberg-factorization/20125/2 "2019-01-26T11:57:32Z")

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It seems to be in GenericLinearAlgebra [GenericLinearAlgebra.jl/eigenGeneral.jl at dcb6b7512c66a5cfab472721e519bbdb70856f18 · JuliaLinearAlgebra/GenericLinearAlgebra.jl · GitHub](https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl/blob/dcb6b7512c66a5cfab472721e519bbdb70856f18/src/eigenGeneral.jl#L28)

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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: [April 27, 2019, 9:27pm UTC](https://discourse.julialang.org/t/hessenberg-factorization/20125/3 "2019-04-27T21:27:56Z")

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Update: more efficient solvers for shifted Hessenberg systems are coming:

[https://github.com/JuliaLang/julia/pull/31853](https://github.com/JuliaLang/julia/pull/31853)