# Inverse and backslash \\ division

**URL:** https://discourse.julialang.org/t/inverse-and-backslash-division/44680
**Category:** Statistics
**Tags:** question, optimization
**Created:** [August 10, 2020, 1:19pm UTC](https://discourse.julialang.org/t/inverse-and-backslash-division/44680 "2020-08-10T13:19:31Z")
**Posts on this page:** 1
**Showing post:** 6

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### Author: ![chakravala](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chakravala/32/6832_2.png) [@chakravala](https://discourse.julialang.org/u/chakravala)
#### Post date: [August 14, 2020, 8:18pm UTC](https://discourse.julialang.org/t/inverse-and-backslash-division/44680/6 "2020-08-14T20:18:06Z")

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I have come up with a more efficient method than the SaticArrays method for an underdetermined system:

> [@Grassmann.jl A\\b 3x faster than Julia's StaticArrays.jl](https://discourse.julialang.org/t/grassmann-jl-a-b-3x-faster-than-julias-staticarrays-jl/41451/35):
>
> This was because of a special explicit case for dimensions 1 and 2 and 3, which are now accounted for in Grassmann.jl also. Also, support has been added for Moore-Penrose inverses for underdetermined and overdetermined linear systems. For underdetermined cases, the exterior product algorithm works ~20x faster than the SMatrix algorithm, and it is numerically stable. For overdetermined equations, the method used is based on the traditional normal equations, and this is prone to more numerical i…

For underdetermined systems, my new method is about ~20x faster than the `SMatrix` and it is numerically stable. However, for overdetermined cases, I am falling back on the traditional normal equations, which are less numerically stable.

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