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

• 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.

• Boole's Rule +n+ implies number of subdivisions Source: Weisstein, Eric W.

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

• 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.

• Check if GSL is available.

• Check if `value` is plus or minus infinity.

• Get the integral for a function +f+, with bounds +t1+ and +t2+ given a hash of +options+.

• Integrate using the GSL bindings.

• Milne's Method +n+ implies number of subdivisions Source: Abramowitz, M.

• Monte Carlo.

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

• .rectangle(t1, t2, n, &f) ⇒ Object (also: midpoint)

Rectangle method +n+ implies number of subdivisions Source: * Ayres : Outline of calculus.

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

• Simpson's rule +n+ implies number of subdivisions Source: * Ayres : Outline of calculus.

• Simpson's 3/8 Rule +n+ implies number of subdivisions Source: * Burden, Richard L.

• Trapezoid method +n+ implies number of subdivisions Source: * Ayres : Outline of calculus.

Class Method Details

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

 ``` 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123``` ```# 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

 ``` 58 59 60 61 62 63 64 65``` ```# 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

 ``` 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169``` ```# 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

 ``` 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347``` ```# 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.

 ``` 186 187 188 189 190 191 192 193 194 195 196 197 198``` ```# 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)
 ``` 52 53 54``` ```# 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.

 ``` 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107``` ```# 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.

 ``` 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136``` ```# 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

 ``` 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180``` ```# 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.

 ``` 85 86 87 88 89 90 91 92``` ```# 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.

 ``` 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391``` ```# 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

 ``` 70 71 72 73 74 75 76 77``` ```# 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) ⇒ ObjectAlso known as: midpoint

Rectangle method +n+ implies number of subdivisions Source:

• Ayres : Outline of calculus
 ``` 8 9 10 11 12 13``` ```# 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

 ``` 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369``` ```# 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
 ``` 32 33 34 35 36 37 38 39``` ```# 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
 ``` 45 46 47 48 49 50 51 52``` ```# 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
 ``` 21 22 23 24 25 26``` ```# 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```