{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# Preconditioners overview\n",
    "\n",
    "This page illustrates some of the method(s) in the Julia package\n",
    "[`Preconditioners.jl`](https://github.com/JuliaLinearAlgebra/Preconditioners.jl).\n",
    "\n",
    "This page was generated from a single Julia file:\n",
    "[1-overview.jl](https://github.com/JuliaLinearAlgebra/Preconditioners.jl/blob/master/docs/lit/examples/1-overview.jl)."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Setup"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "Packages needed here."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using Preconditioners: DiagonalPreconditioner, CholeskyPreconditioner\n",
    "using InteractiveUtils: versioninfo\n",
    "using SparseArrays: sprand\n",
    "using LinearAlgebra: I, cond"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Overview\n",
    "\n",
    "Preconditioning is useful\n",
    "for accelerating solutions\n",
    "to systems of equations\n",
    "and optimization problems.\n",
    "\n",
    "\n",
    "## Examples"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "n = 1000\n",
    "A = sprand(n, n, 0.01)\n",
    "A = A + A' + 10I"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Examine conditioning:"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "cA = cond(Matrix(A), 2)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Diagonal preconditioner"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "Dp = DiagonalPreconditioner(A)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Apply preconditioner"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "DA = Dp \\ A"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Examine effect on condition number"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "cd = cond(Matrix(DA), 2)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Incomplete Cholesky preconditioner with cut-off level 2"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "Pc = CholeskyPreconditioner(A, 2)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Here is a quick way to handle a matrix argument.\n",
    "This could be done more efficiently eventually; see Issue #33."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "import Base.\\\n",
    "\\(C::CholeskyPreconditioner, A::AbstractMatrix) = reduce(hcat, [C \\ c for c in eachcol(A)])"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "apply preconditioner"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "CA = Pc \\ A"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "examine effect on condition number"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "cc = cond(Matrix(CA), 2)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Reproducibility"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "This page was generated with the following version of Julia:"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "io = IOBuffer(); versioninfo(io); split(String(take!(io)), '\\n')"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "And with the following package versions"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "import Pkg; Pkg.status()"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "---\n",
    "\n",
    "*This notebook was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).*"
   ],
   "metadata": {}
  }
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