Preconditioners overview¶

This page illustrates some of the method(s) in the Julia package Preconditioners.jl.

This page was generated from a single Julia file: 1-overview.jl.

Setup¶

Packages needed here.

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using Preconditioners: DiagonalPreconditioner, CholeskyPreconditioner
using InteractiveUtils: versioninfo
using SparseArrays: sprand
using LinearAlgebra: I, cond

Overview¶

Preconditioning is useful for accelerating solutions to systems of equations and optimization problems.

Examples¶

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n = 1000
A = sprand(n, n, 0.01)
A = A + A' + 10I

Examine conditioning:

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cA = cond(Matrix(A), 2)

Diagonal preconditioner

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Dp = DiagonalPreconditioner(A)

Apply preconditioner

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DA = Dp \ A

Examine effect on condition number

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cd = cond(Matrix(DA), 2)

Incomplete Cholesky preconditioner with cut-off level 2

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Pc = CholeskyPreconditioner(A, 2)

Here is a quick way to handle a matrix argument. This could be done more efficiently eventually; see Issue #33.

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import Base.\
\(C::CholeskyPreconditioner, A::AbstractMatrix) = reduce(hcat, [C \ c for c in eachcol(A)])

apply preconditioner

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CA = Pc \ A

examine effect on condition number

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cc = cond(Matrix(CA), 2)

Reproducibility¶

This page was generated with the following version of Julia:

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io = IOBuffer(); versioninfo(io); split(String(take!(io)), '\n')

And with the following package versions

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import Pkg; Pkg.status()

This notebook was generated using Literate.jl.