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Add Specialized Methods in Hessenberg decomposition for Symmetric/Hermitian Matrices. #47

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jacobdwatters opened this issue Apr 22, 2023 · 2 comments
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@jacobdwatters
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The Hessenberg decomposition of a general matrix will produce a similar matrix in upper Hessenberg form at the cost of ~ $\cfrac{10}{3}n^3$ flops. However, if the matrix is Symmetric/Hermitian, the Hessenberg decomposition will result in a similar tridiagonal matrix and the symmetry can be exploited to compute this in ~ $\cfrac{4}{3}n^3$ which is less than half that of the general case. As such, a specialized algorithm should be added to the Hessenberg decomposition for the symmetric case.

A pseudocode implementation from Fundamentals of matrix computation (David S Watkins) is provided as reference.
symmHess

@jacobdwatters jacobdwatters added the enhancement New feature or request label Jun 21, 2023
@jacobdwatters jacobdwatters self-assigned this Jun 21, 2023
@jacobdwatters
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This has been implemented for real symmetric matrices. Still need an implementation for complex Hermitian matrices.

@jacobdwatters
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Feature has been implemented in org.flag4j.linalg.decompositions.hess.SymmHess and org.flag4j.linalg.decompositions.hess.HermHess.

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