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chi2fit

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Provides functions for fast matrix inversion, creation of empirical CDF from sample data including handling of asymmetric errors, and fitting to a funtion using chi-squared. The fitting procedure return the full covariance matrix describing the fitted parameters.

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chi2fit lib roots.ex
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lib/roots.ex 

defmodule Chi2fit.Roots do



  # Copyright 2015-2017 Pieter Rijken

  #

  # Licensed under the Apache License, Version 2.0 (the "License");

  # you may not use this file except in compliance with the License.

  # You may obtain a copy of the License at

  #

  #     http://www.apache.org/licenses/LICENSE-2.0

  #

  # Unless required by applicable law or agreed to in writing, software

  # distributed under the License is distributed on an "AS IS" BASIS,

  # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.

  # See the License for the specific language governing permissions and

  # limitations under the License.



  @moduledoc """

  Solves roots for linear, quadratic, and cubic equations.

  """



  @doc """

  Returns the real roots of polynoms of order 1, 2 and 3 as a list.

  

  ## Examples

  

      Solve `2.0*x + 5.0 = 0`

      iex> solve [2.0,5.0]

      [-2.5]

      

      iex> solve [2.0,-14.0,24.0]

      [4.0,3.0]

      

      iex> solve [1.0,0.0,5.0,6.0]

      [-0.9999999999999999]

  """

  @spec solve([float]) :: [float]

  def solve([0.0|rest]), do: solve rest



  def solve([a1,a0]), do: [-a0/a1]



  def solve([a2,a1,a0]) do

    sqr = a1*a1-4*a2*a0

    cond do

      sqr == 0 -> -a1/2/a2

      sqr > 0 -> [(-a1+:math.sqrt(sqr))/2/a2,(-a1-:math.sqrt(sqr))/2/a2]

      true -> []

    end

  end



  def solve([1.0,0.0,p,q]) do

    ## For details see equations (83) and (84) in http://mathworld.wolfram.com/CubicFormula.html

    c = -0.5*q*:math.pow(3/abs(p),1.5)

    cond do

      p>0 ->

        [:math.sinh(1.0/3.0*:math.asinh(c))]

      c>=1 ->

        [:math.cosh(1.0/3.0*:math.acosh(c))]

      c<=-1 ->

        [-:math.cosh(1.0/3.0*:math.acosh(abs(c)))]

      true ->

        ## Three real solutions

        [:math.cos(1.0/3.0*:math.acos(c)),:math.cos(1.0/3.0*:math.acos(c) + 2*:math.pi()/3.0),:math.cos(1.0/3.0*:math.acos(c) + 4*:math.pi()/3.0)]

    end

    |> Enum.map(&(&1*2*:math.sqrt(abs(p)/3.0)))

  end

  def solve([1.0,a2,a1,a0]), do: solve([1.0,0.0,(3*a1-a2*a2)/3.0,(2*a2*a2*a2-9*a1*a2+27*a0)/27.0]) |> Enum.map(&(&1-a2/3.0))

  def solve([a3,a2,a1,a0]), do: solve([1.0,a2/a3,a1/a3,a0/a3])



end