Class: Integration

Inherits:
Object
  • Object
show all
Defined in:
lib/integration.rb,
lib/integration/methods.rb,
lib/integration/version.rb

Overview

Diverse integration methods Use Integration.integrate as wrapper to direct access to methods

Constant Summary collapse

MInfinity =

Minus Infinity

:minfinity
Infinity =

Infinity

:infinity
RUBY_METHODS =

Pure Ruby methods available.

[:rectangle, :trapezoid, :simpson, :adaptive_quadrature,
:gauss, :romberg, :monte_carlo, :gauss_kronrod,
:simpson3by8, :boole, :open_trapezoid, :milne]
GSL_METHODS =

Methods available when using the rb-gsl gem.

[:qng, :qag]
VERSION = 
'0.1.4'

Class Method Summary collapse

Class Method Details

.adaptive_quadrature(a, b, tolerance) ⇒ Object

Adaptive Quadrature Calls the Simpson's rule recursively on subintervals in case the error exceeds the desired tolerance +tolerance+ is the desired tolerance of error



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# File 'lib/integration/methods.rb', line 98

def adaptive_quadrature(a, b, tolerance)
  h = (b.to_f - a) / 2
  fa = yield(a)
  fc = yield(a + h)
  fb = yield(b)
  s = h * (fa + (4 * fc) + fb) / 3

  helper = proc do |_a, _b, _fa, _fb, _fc, _h, _s, level|
    if level < 1 / tolerance.to_f
      fd = yield(_a + (_h / 2))
      fe = yield(_a + (3 * (_h / 2)))
      s1 = _h * (_fa + (4.0 * fd) + _fc) / 6
      s2 = _h * (_fc + (4.0 * fe) + _fb) / 6
      if ((s1 + s2) - _s).abs <= tolerance
        s1 + s2
      else
        helper.call(_a, _a + _h, _fa, _fc, fd, _h / 2, s1, level + 1) +
        helper.call(_a + _h, _b, _fc, _fb, fe, _h / 2, s2, level + 1)
      end
    else
      fail 'Integral did not converge'
    end

  end
  helper.call(a, b, fa, fb, fc, h, s, 1)
end

.boole(t1, t2, n, &f) ⇒ Object

Boole's Rule +n+ implies number of subdivisions Source: Weisstein, Eric W. "Boole's Rule." From MathWorld—A Wolfram Web Resource



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# File 'lib/integration/methods.rb', line 58

def boole(t1, t2, n, &f)
  d = (t2 - t1) / n.to_f
  ac = 0
  (0..n - 1).each do |i|
    ac += (d / 90.0) * (7 * f[t1 + i * d] + 32 * f[t1 + i * d + d / 4] + 12 * f[t1 + i * d + d / 2] + 32 * f[t1 + i * d + 3 * d / 4] + 7 * f[t1 + (i + 1) * d])
  end
  ac
end

.gauss(t1, t2, n) ⇒ Object

Gaussian Quadrature n-point Gaussian quadrature rule gives an exact result for polynomials of degree 2n − 1 or less



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# File 'lib/integration/methods.rb', line 127

def gauss(t1, t2, n)
  case n
    when 1
      z = [0.0]
      w = [2.0]
    when 2
      z = [-0.57735026919, 0.57735026919]
      w = [1.0, 1.0]
    when 3
      z = [-0.774596669241, 0.0, 0.774596669241]
      w = [0.555555555556, 0.888888888889, 0.555555555556]
    when 4
      z = [-0.861136311594, -0.339981043585, 0.339981043585, 0.861136311594]
      w = [0.347854845137, 0.652145154863, 0.652145154863, 0.347854845137]
    when 5
      z = [-0.906179845939, -0.538469310106, 0.0, 0.538469310106, 0.906179845939]
      w = [0.236926885056, 0.478628670499, 0.568888888889, 0.478628670499, 0.236926885056]
    when 6
      z = [-0.932469514203, -0.661209386466, -0.238619186083, 0.238619186083, 0.661209386466, 0.932469514203]
      w = [0.171324492379, 0.360761573048, 0.467913934573, 0.467913934573, 0.360761573048, 0.171324492379]
    when 7
      z = [-0.949107912343, -0.741531185599, -0.405845151377, 0.0, 0.405845151377, 0.741531185599, 0.949107912343]
      w = [0.129484966169, 0.279705391489, 0.381830050505, 0.417959183673, 0.381830050505, 0.279705391489, 0.129484966169]
    when 8
      z = [-0.960289856498, -0.796666477414, -0.525532409916, -0.183434642496, 0.183434642496, 0.525532409916, 0.796666477414, 0.960289856498]
      w = [0.10122853629, 0.222381034453, 0.313706645878, 0.362683783378, 0.362683783378, 0.313706645878, 0.222381034453, 0.10122853629]
    when 9
      z = [-0.968160239508, -0.836031107327, -0.613371432701, -0.324253423404, 0.0, 0.324253423404, 0.613371432701, 0.836031107327, 0.968160239508]
      w = [0.0812743883616, 0.180648160695, 0.260610696403, 0.31234707704, 0.330239355001, 0.31234707704, 0.260610696403, 0.180648160695, 0.0812743883616]
    when 10
      z = [-0.973906528517, -0.865063366689, -0.679409568299, -0.433395394129, -0.148874338982, 0.148874338982, 0.433395394129, 0.679409568299, 0.865063366689, 0.973906528517]
      w = [0.0666713443087, 0.149451349151, 0.219086362516, 0.26926671931, 0.295524224715, 0.295524224715, 0.26926671931, 0.219086362516, 0.149451349151, 0.0666713443087]
    else
      fail "Invalid number of spaced abscissas #{n}, should be 1-10"
  end

  sum = 0
  (0...n).each do |i|
    t = ((t1.to_f + t2) / 2.0) + (((t2 - t1) / 2.0) * z[i])
    sum += w[i] * yield(t)
  end
  ((t2 - t1) / 2.0) * sum
end

.gauss_kronrod(t1, t2, n, points) ⇒ Object

Gauss Kronrod Rule: Provides a 3n+1 order estimate while re-using the function values of a lower-order(n order) estimate Source: "Gauss–Kronrod quadrature formula", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4



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# File 'lib/integration/methods.rb', line 175

def gauss_kronrod(t1, t2, n, points)
  # g7k15
  case points
    when 15

      z = [-0.9914553711208126, -0.9491079123427585, -0.8648644233597691,
           -0.7415311855993945, -0.5860872354676911, -0.4058451513773972,
           -0.20778495500789848, 0.0, 0.20778495500789848,
           0.4058451513773972, 0.5860872354676911, 0.7415311855993945,
           0.8648644233597691, 0.9491079123427585, 0.9914553711208126]

      w = [0.022935322010529224, 0.06309209262997856, 0.10479001032225019,
           0.14065325971552592, 0.1690047266392679, 0.19035057806478542,
           0.20443294007529889, 0.20948214108472782, 0.20443294007529889,
           0.19035057806478542, 0.1690047266392679, 0.14065325971552592,
           0.10479001032225019, 0.06309209262997856, 0.022935322010529224]

    when 21
      # g10k21

      z = [-0.9956571630258081, -0.9739065285171717, -0.9301574913557082,
           -0.8650633666889845, -0.7808177265864169, -0.6794095682990244,
           -0.5627571346686047, -0.4333953941292472, -0.2943928627014602,
           -0.14887433898163122, 0.0, 0.14887433898163122,
           0.2943928627014602, 0.4333953941292472, 0.5627571346686047,
           0.6794095682990244, 0.7808177265864169, 0.8650633666889845,
           0.9301574913557082, 0.9739065285171717, 0.9956571630258081]

      w = [0.011694638867371874, 0.032558162307964725,
           0.054755896574351995, 0.07503967481091996, 0.0931254545836976,
           0.10938715880229764, 0.12349197626206584, 0.13470921731147334,
           0.14277593857706009, 0.14773910490133849, 0.1494455540029169,
           0.14773910490133849, 0.14277593857706009, 0.13470921731147334,
           0.12349197626206584, 0.10938715880229764, 0.0931254545836976,
           0.07503967481091996, 0.054755896574351995, 0.032558162307964725,
           0.011694638867371874]

    when 31
      # g15k31

      z = [-0.9980022986933971, -0.9879925180204854, -0.9677390756791391,
           -0.937273392400706, -0.8972645323440819, -0.8482065834104272,
           -0.790418501442466, -0.7244177313601701, -0.650996741297417,
           -0.5709721726085388, -0.4850818636402397, -0.3941513470775634,
           -0.29918000715316884, -0.20119409399743451, -0.1011420669187175,
           0.0, 0.1011420669187175, 0.20119409399743451,
           0.29918000715316884, 0.3941513470775634, 0.4850818636402397,
           0.5709721726085388, 0.650996741297417, 0.7244177313601701,
           0.790418501442466, 0.8482065834104272, 0.8972645323440819,
           0.937273392400706, 0.9677390756791391, 0.9879925180204854,
           0.9980022986933971]

      w = [0.005377479872923349, 0.015007947329316122, 0.02546084732671532,
           0.03534636079137585, 0.04458975132476488, 0.05348152469092809,
           0.06200956780067064, 0.06985412131872826, 0.07684968075772038,
           0.08308050282313302, 0.08856444305621176, 0.09312659817082532,
           0.09664272698362368, 0.09917359872179196, 0.10076984552387559,
           0.10133000701479154, 0.10076984552387559, 0.09917359872179196,
           0.09664272698362368, 0.09312659817082532, 0.08856444305621176,
           0.08308050282313302, 0.07684968075772038, 0.06985412131872826,
           0.06200956780067064, 0.05348152469092809, 0.04458975132476488,
           0.03534636079137585, 0.02546084732671532, 0.015007947329316122,
           0.005377479872923349]

    when 41
      # g20k41

      z = [-0.9988590315882777, -0.9931285991850949, -0.9815078774502503,
           -0.9639719272779138, -0.9408226338317548, -0.912234428251326,
           -0.878276811252282, -0.8391169718222188, -0.7950414288375512,
           -0.7463319064601508, -0.6932376563347514, -0.636053680726515,
           -0.5751404468197103, -0.5108670019508271, -0.4435931752387251,
           -0.37370608871541955, -0.301627868114913, -0.22778585114164507,
           -0.15260546524092267, -0.07652652113349734, 0.0,
           0.07652652113349734, 0.15260546524092267, 0.22778585114164507,
           0.301627868114913, 0.37370608871541955, 0.4435931752387251,
           0.5108670019508271, 0.5751404468197103, 0.636053680726515,
           0.6932376563347514, 0.7463319064601508, 0.7950414288375512,
           0.8391169718222188, 0.878276811252282, 0.912234428251326,
           0.9408226338317548, 0.9639719272779138, 0.9815078774502503,
           0.9931285991850949, 0.9988590315882777]

      w = [0.0030735837185205317, 0.008600269855642943,
           0.014626169256971253, 0.020388373461266523, 0.02588213360495116,
           0.0312873067770328, 0.036600169758200796, 0.041668873327973685,
           0.04643482186749767, 0.05094457392372869, 0.05519510534828599,
           0.05911140088063957, 0.06265323755478117, 0.06583459713361842,
           0.06864867292852161, 0.07105442355344407, 0.07303069033278667,
           0.07458287540049918, 0.07570449768455667, 0.07637786767208074,
           0.07660071191799965, 0.07637786767208074, 0.07570449768455667,
           0.07458287540049918, 0.07303069033278667, 0.07105442355344407,
           0.06864867292852161, 0.06583459713361842, 0.06265323755478117,
           0.05911140088063957, 0.05519510534828599, 0.05094457392372869,
           0.04643482186749767, 0.041668873327973685, 0.036600169758200796,
           0.0312873067770328, 0.02588213360495116, 0.020388373461266523,
           0.014626169256971253, 0.008600269855642943,
           0.0030735837185205317]

    when 61
      # g30k61

      z = [-0.9994844100504906, -0.9968934840746495, -0.9916309968704046,
           -0.9836681232797472, -0.9731163225011262, -0.9600218649683075,
           -0.94437444474856, -0.9262000474292743, -0.9055733076999078,
           -0.8825605357920527, -0.8572052335460612, -0.8295657623827684,
           -0.799727835821839, -0.7677774321048262, -0.7337900624532268,
           -0.6978504947933158, -0.6600610641266269, -0.6205261829892429,
           -0.5793452358263617, -0.5366241481420199, -0.49248046786177857,
           -0.44703376953808915, -0.4004012548303944, -0.3527047255308781,
           -0.30407320227362505, -0.25463692616788985,
           -0.20452511668230988, -0.15386991360858354,
           -0.10280693796673702, -0.0514718425553177, 0.0,
           0.0514718425553177, 0.10280693796673702, 0.15386991360858354,
           0.20452511668230988, 0.25463692616788985, 0.30407320227362505,
           0.3527047255308781, 0.4004012548303944, 0.44703376953808915,
           0.49248046786177857, 0.5366241481420199, 0.5793452358263617,
           0.6205261829892429, 0.6600610641266269, 0.6978504947933158,
           0.7337900624532268, 0.7677774321048262, 0.799727835821839,
           0.8295657623827684, 0.8572052335460612, 0.8825605357920527,
           0.9055733076999078, 0.9262000474292743, 0.94437444474856,
           0.9600218649683075, 0.9731163225011262, 0.9836681232797472,
           0.9916309968704046, 0.9968934840746495, 0.9994844100504906]

      w = [0.0013890136986770077, 0.003890461127099884,
           0.0066307039159312926, 0.009273279659517764,
           0.011823015253496341, 0.014369729507045804, 0.01692088918905327,
           0.019414141193942382, 0.021828035821609193, 0.0241911620780806,
           0.0265099548823331, 0.02875404876504129, 0.030907257562387762,
           0.03298144705748372, 0.034979338028060025, 0.03688236465182123,
           0.038678945624727595, 0.040374538951535956,
           0.041969810215164244, 0.04345253970135607, 0.04481480013316266,
           0.04605923827100699, 0.04718554656929915, 0.04818586175708713,
           0.04905543455502978, 0.04979568342707421, 0.05040592140278235,
           0.05088179589874961, 0.051221547849258774, 0.05142612853745902,
           0.05149472942945157, 0.05142612853745902, 0.051221547849258774,
           0.05088179589874961, 0.05040592140278235, 0.04979568342707421,
           0.04905543455502978, 0.04818586175708713, 0.04718554656929915,
           0.04605923827100699, 0.04481480013316266, 0.04345253970135607,
           0.041969810215164244, 0.040374538951535956,
           0.038678945624727595, 0.03688236465182123, 0.034979338028060025,
           0.03298144705748372, 0.030907257562387762, 0.02875404876504129,
           0.0265099548823331, 0.0241911620780806, 0.021828035821609193,
           0.019414141193942382, 0.01692088918905327, 0.014369729507045804,
           0.011823015253496341, 0.009273279659517764,
           0.0066307039159312926, 0.003890461127099884,
           0.0013890136986770077]

    else # using 15 point quadrature

      n = 15

      z = [-0.9914553711208126, -0.9491079123427585, -0.8648644233597691,
           -0.7415311855993945, -0.5860872354676911, -0.4058451513773972,
           -0.20778495500789848, 0.0, 0.20778495500789848,
           0.4058451513773972, 0.5860872354676911, 0.7415311855993945,
           0.8648644233597691, 0.9491079123427585, 0.9914553711208126]

      w = [0.022935322010529224, 0.06309209262997856, 0.10479001032225019,
           0.14065325971552592, 0.1690047266392679, 0.19035057806478542,
           0.20443294007529889, 0.20948214108472782, 0.20443294007529889,
           0.19035057806478542, 0.1690047266392679, 0.14065325971552592,
           0.10479001032225019, 0.06309209262997856, 0.022935322010529224]

  end

  sum = 0
  (0...n).each do |i|
    t = ((t1.to_f + t2) / 2.0) + (((t2 - t1) / 2.0) * z[i])
    sum += w[i] * yield(t)
  end

  ((t2 - t1) / 2.0) * sum
end

.has_gsl? ⇒ Boolean

Check if GSL is available. Require the library if it is present. Return a boolean indicating its availability.

Returns:

  • (Boolean) —

    Whether GSL is available.



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# File 'lib/integration.rb', line 186

def has_gsl?
  gsl_available = '@@gsl'
  if class_variable_defined? gsl_available
    class_variable_get(gsl_available)
  else
    begin
      require 'gsl'
      class_variable_set(gsl_available, true)
    rescue LoadError
      class_variable_set(gsl_available, false)
    end
  end
end

.infinite?(value) ⇒ Boolean

Check if value is plus or minus infinity.

Parameters:

  • value —

    Value to be tested.

Returns:

  • (Boolean) 


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# File 'lib/integration.rb', line 52

def infinite?(value)
  value == Integration::Infinity || value == Integration::MInfinity
end

.integrate(t1, t2, options = {}, &f) ⇒ Object

Get the integral for a function +f+, with bounds +t1+ and +t2+ given a hash of +options+. If Ruby/GSL is available, you can use +Integration::Minfinity+ and +Integration::Infinity+ as bounds. Method

Options are: [:tolerance] Maximum difference between real and calculated integral. Default: 1e-10. [:initial_step] Initial number of subdivisions. [:step] Subdivition increment on each iteration. [:method] Integration method.

Available methods are:

[:rectangle] for [:initial_step+:step*iteration] quadrilateral subdivisions. [:trapezoid] for [:initial_step+:step*iteration] trapezoid-al subdivisions. [:simpson] for [:initial_step+:step*iteration] parabolic subdivisions. [:adaptive_quadrature] for recursive appoximations until error [tolerance]. [:gauss] [:initial_step+:step*iteration] weighted subdivisons using translated -1 -> +1 endpoints. [:romberg] extrapolation of recursion approximation until error < [tolerance]. [:monte_carlo] make [:initial_step+:step*iteration] random samples, and check for above/below curve. [:qng] GSL QNG non-adaptive Gauss-Kronrod integration. [:qag] GSL QAG adaptive integration, with support for infinite bounds.



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# File 'lib/integration.rb', line 80

def integrate(t1, t2, options = {}, &f)
  inf_bounds = (infinite?(t1) || infinite?(t2))

  fail 'No function passed' unless block_given?
  fail 'Non-numeric bounds' unless ((t1.is_a? Numeric) && (t2.is_a? Numeric)) || inf_bounds

  if inf_bounds
    lower_bound = t1
    upper_bound = t2
    options[:method] = :qag if options[:method].nil?
  else
    lower_bound = [t1, t2].min
    upper_bound = [t1, t2].max
  end

  def_method = (Integration.has_gsl?) ? :qag : :simpson
  default_opts = { tolerance: 1e-10, initial_step: 16, step: 16, method: def_method }
  options = default_opts.merge(options)

  if RUBY_METHODS.include? options[:method]
    fail "Ruby methods doesn't support infinity bounds" if inf_bounds
    integrate_ruby(lower_bound, upper_bound, options, &f)
  elsif GSL_METHODS.include? options[:method]
    integrate_gsl(lower_bound, upper_bound, options, &f)
  else
    fail "Unknown integration method \"#{options[:method]}\""
  end
end

.integrate_gsl(lower_bound, upper_bound, options, &f) ⇒ Object

Integrate using the GSL bindings.



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# File 'lib/integration.rb', line 110

def integrate_gsl(lower_bound, upper_bound, options, &f)
  f = GSL::Function.alloc(&f)
  method = options[:method]
  tolerance = options[:tolerance]

  if (method == :qag)
    w = GSL::Integration::Workspace.alloc

    val = if infinite?(lower_bound) && infinite?(upper_bound)
      f.qagi([tolerance, 0.0], 1000, w)
    elsif infinite?(lower_bound)
      f.qagil(upper_bound, [tolerance, 0], w)
    elsif infinite?(upper_bound)
      f.qagiu(lower_bound, [tolerance, 0], w)
    else
      f.qag([lower_bound, upper_bound], [tolerance, 0.0], GSL::Integration::GAUSS61, w)
    end

  elsif (method == :qng)
    val = f.qng([lower_bound, upper_bound], [tolerance, 0.0])

  else
    fail "Unknown integration method \"#{method}\""
  end

  val[0]
end

.integrate_ruby(lower_bound, upper_bound, options, &f) ⇒ Object 



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# File 'lib/integration.rb', line 138

def integrate_ruby(lower_bound, upper_bound, options, &f)
  method = options[:method]
  tolerance = options[:tolerance]
  initial_step = options[:initial_step]
  step = options[:step]
  points = options[:points]

  begin
    method_obj = Integration.method(method.to_s.downcase)
  rescue
    raise "Unknown integration method \"#{method}\""
  end

  current_step = initial_step

  if [:adaptive_quadrature, :romberg, :gauss, :gauss_kronrod].include? method
    if (method == :gauss)
      initial_step = 10 if initial_step > 10
      tolerance = initial_step
      method_obj.call(lower_bound, upper_bound, tolerance, &f)
    elsif (method == :gauss_kronrod)
      initial_step = 10 if initial_step > 10
      tolerance = initial_step
      points = points unless points.nil?
      method_obj.call(lower_bound, upper_bound, tolerance, points, &f)
    else
      method_obj.call(lower_bound, upper_bound, tolerance, &f)
    end
  else
    value = method_obj.call(lower_bound, upper_bound, current_step, &f)
    previous = value + (tolerance * 2)
    diffs = []

    while (previous - value).abs > tolerance
      diffs.push((previous - value).abs)
      current_step += step
      previous = value
      value = method_obj.call(lower_bound, upper_bound, current_step, &f)
    end

    value
  end
end

.milne(t1, t2, n, &f) ⇒ Object

Milne's Method +n+ implies number of subdivisions Source: Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 896-897, 1972.



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# File 'lib/integration/methods.rb', line 85

def milne(t1, t2, n, &f)
  d = (t2 - t1) / n.to_f
  ac = 0
  (0..n - 1).each do |i|
    ac += (d / 3.0) * (2 * f[t1 + i * d + d / 4] - f[t1 + i * d + d / 2] + 2 * f[t1 + i * d + 3 * d / 4])
  end
  ac
end

.monte_carlo(t1, t2, n) ⇒ Object

Monte Carlo

Uses a non-deterministic approach for calculation of definite integrals. Estimates the integral by randomly choosing points in a set and then calculating the number of points that fall in the desired area.



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# File 'lib/integration/methods.rb', line 376

def monte_carlo(t1, t2, n)
  width = (t2 - t1).to_f
  height = nil
  vals = []
  n.times do
    t = t1 + (rand * width)
    ft = yield(t)
    height = ft if height.nil? || ft > height
    vals << ft
  end
  area_ratio = 0
  vals.each do |ft|
    area_ratio += (ft / height.to_f) / n.to_f
  end
  (width * height) * area_ratio
end

.open_trapezoid(t1, t2, n, &f) ⇒ Object

Open Trapezoid method +n+ implies number of subdivisions Values computed at mid point and end point instead of starting points



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# File 'lib/integration/methods.rb', line 70

def open_trapezoid(t1, t2, n, &f)
  d = (t2 - t1) / n.to_f
  ac = 0
  (0..n - 1).each do |i|
    ac += (d / 2.0) * (f[t1 + i * d + d / 3] + f[t1 + i * d + 2 * d / 3])
  end
  ac
end

.rectangle(t1, t2, n, &f) ⇒ Object Also known as: midpoint

Rectangle method +n+ implies number of subdivisions Source:

  • Ayres : Outline of calculus


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# File 'lib/integration/methods.rb', line 8

def rectangle(t1, t2, n, &f)
  d = (t2 - t1) / n.to_f
  n.times.inject(0) do|ac, i|
    ac + f[t1 + d * (i + 0.5)]
  end * d
end

.romberg(a, b, tolerance, max_iter = 20) ⇒ Object

Romberg Method: It is obtained by applying extrapolation techniques repeatedly on the trapezoidal rule



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# File 'lib/integration/methods.rb', line 351

def romberg(a, b, tolerance, max_iter = 20)
  # NOTE one-based arrays are used for convenience
  h = b.to_f - a
  close = 1
  r = [[(h / 2) * (yield(a) + yield(b))]]
  j = 0
  hn = ->(n) { h / (2**n) }
  while j <= max_iter && tolerance < close
    j += 1
    r.push((j + 1).times.map { [] })
    ul = 2**(j - 1)
    r[j][0] = r[j - 1][0] / 2.0 + hn[j] * (1..ul).inject(0) { |ac, k| ac + yield(a + (2 * k - 1) * hn[j]) }
    (1..j).each do |k|
      r[j][k] = ((4**k) * r[j][k - 1] - r[j - 1][k - 1]) / ((4**k) - 1)
    end
    close = (r[j][j] - r[j - 1][j - 1])
  end
  r[j][j]
end

.simpson(t1, t2, n, &f) ⇒ Object

Simpson's rule +n+ implies number of subdivisions Source:

  • Ayres : Outline of calculus


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# File 'lib/integration/methods.rb', line 32

def simpson(t1, t2, n, &f)
  n += 1 unless n.even?
  d = (t2 - t1) / n.to_f
  out = (d / 3.0) * (f[t1.to_f].to_f + ((1..(n - 1)).inject(0) do|ac, i|
    ac + ((i.even?) ? 2 : 4) * f[t1 + d * i]
  end) + f[t2.to_f].to_f)
  out
end

.simpson3by8(t1, t2, n, &f) ⇒ Object

Simpson's 3/8 Rule +n+ implies number of subdivisions Source:

  • Burden, Richard L. and Faires, J. Douglas (2000): Numerical Analysis (7th ed.). Brooks/Cole


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# File 'lib/integration/methods.rb', line 45

def simpson3by8(t1, t2, n, &f)
  d = (t2 - t1) / n.to_f
  ac = 0
  (0..n - 1).each do |i|
    ac += (d / 8.0) * (f[t1 + i * d] + 3 * f[t1 + i * d + d / 3] + 3 * f[t1 + i * d + 2 * d / 3] + f[t1 + (i + 1) * d])
  end
  ac
end

.trapezoid(t1, t2, n, &f) ⇒ Object

Trapezoid method +n+ implies number of subdivisions Source:

  • Ayres : Outline of calculus


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# File 'lib/integration/methods.rb', line 21

def trapezoid(t1, t2, n, &f)
  d = (t2 - t1) / n.to_f
  (d / 2.0) * (f[t1] + 2 * (1..(n - 1)).inject(0) do|ac, i|
    ac + f[t1 + d * i]
  end + f[t2])
end