Welcome, guest | Sign In | My Account | Store | Cart
 #!/usr/bin/env python
# A. Pletzer Tue Mar 20 11:42:05 EST 2001

"""
Gauss Integration
"""


_ngmax = 12
_ngmin = 1


_nodes  =(
(0.,),
(-0.5773502691896257,
 0.5773502691896257,),
(-0.7745966692414834,
 0.,
 0.7745966692414834,),
(-0.861136311594053,
 -0.3399810435848562,
 0.3399810435848562,
 0.861136311594053,),
(-0.906179845938664,
 -0.5384693101056829,
 0.,
 0.5384693101056829,
 0.906179845938664,),
(-0.932469514203152,
 -0.6612093864662646,
 -0.2386191860831968,
 0.2386191860831968,
 0.6612093864662646,
 0.932469514203152,),
(-0.949107912342759,
 -0.7415311855993937,
 -0.4058451513773972,
 0.,
 0.4058451513773971,
 0.7415311855993945,
 0.949107912342759,),
(-0.960289856497537,
 -0.7966664774136262,
 -0.5255324099163289,
 -0.1834346424956498,
 0.1834346424956498,
 0.5255324099163289,
 0.7966664774136262,
 0.960289856497537,),
(-0.968160239507626,
 -0.836031107326637,
 -0.6133714327005903,
 -0.3242534234038088,
 0.,
 0.3242534234038088,
 0.6133714327005908,
 0.836031107326635,
 0.968160239507627,),
(-0.973906528517172,
 -0.865063366688984,
 -0.6794095682990246,
 -0.433395394129247,
 -0.1488743389816312,
 0.1488743389816312,
 0.433395394129247,
 0.6794095682990246,
 0.865063366688984,
 0.973906528517172,),
(-0.97822865814604,
 -0.88706259976812,
 -0.7301520055740422,
 -0.5190961292068116,
 -0.2695431559523449,
 0.,
 0.2695431559523449,
 0.5190961292068117,
 0.73015200557405,
 0.887062599768093,
 0.978228658146058,),
(-0.981560634246732,
 -0.904117256370452,
 -0.7699026741943177,
 -0.5873179542866143,
 -0.3678314989981804,
 -0.1252334085114688,
 0.1252334085114688,
 0.3678314989981804,
 0.5873179542866143,
 0.7699026741943177,
 0.904117256370452,
 0.981560634246732,),
)

_weights=(
(2.,),
(1.,
 1.,),
(0.5555555555555553,
 0.888888888888889,
 0.5555555555555553,),
(0.3478548451374539,
 0.6521451548625462,
 0.6521451548625462,
 0.3478548451374539,),
(0.2369268850561887,
 0.4786286704993665,
 0.5688888888888889,
 0.4786286704993665,
 0.2369268850561887,),
(0.1713244923791709,
 0.3607615730481379,
 0.4679139345726913,
 0.4679139345726913,
 0.3607615730481379,
 0.1713244923791709,),
(0.129484966168868,
 0.2797053914892783,
 0.3818300505051186,
 0.4179591836734694,
 0.3818300505051188,
 0.279705391489276,
 0.1294849661688697,),
(0.1012285362903738,
 0.2223810344533786,
 0.3137066458778874,
 0.3626837833783619,
 0.3626837833783619,
 0.3137066458778874,
 0.2223810344533786,
 0.1012285362903738,),
(0.0812743883615759,
 0.1806481606948543,
 0.2606106964029356,
 0.3123470770400029,
 0.3302393550012597,
 0.3123470770400025,
 0.2606106964029353,
 0.1806481606948577,
 0.0812743883615721,),
(0.06667134430868681,
 0.149451349150573,
 0.2190863625159832,
 0.2692667193099968,
 0.2955242247147529,
 0.2955242247147529,
 0.2692667193099968,
 0.2190863625159832,
 0.149451349150573,
 0.06667134430868681,),
(0.05566856711621584,
 0.1255803694648743,
 0.1862902109277404,
 0.2331937645919927,
 0.2628045445102466,
 0.2729250867779006,
 0.2628045445102466,
 0.2331937645919933,
 0.1862902109277339,
 0.1255803694649132,
 0.05566856711616958,),
(0.04717533638647547,
 0.1069393259953637,
 0.1600783285433586,
 0.2031674267230672,
 0.2334925365383534,
 0.2491470458134027,
 0.2491470458134027,
 0.2334925365383534,
 0.2031674267230672,
 0.1600783285433586,
 0.1069393259953637,
 0.04717533638647547,),
)

_nodesLog  =(
(0.3333333333333333,),
(0.1120088061669761,
 0.6022769081187381,),
(0.06389079308732544,
 0.3689970637156184,
 0.766880303938942,),
(0.04144848019938324,
 0.2452749143206022,
 0.5561654535602751,
 0.848982394532986,),
(0.02913447215197205,
 0.1739772133208974,
 0.4117025202849029,
 0.6773141745828183,
 0.89477136103101,),
(0.02163400584411693,
 0.1295833911549506,
 0.3140204499147661,
 0.5386572173517997,
 0.7569153373774084,
 0.922668851372116,),
(0.0167193554082585,
 0.100185677915675,
 0.2462942462079286,
 0.4334634932570557,
 0.6323509880476823,
 0.81111862674023,
 0.940848166743287,),
(0.01332024416089244,
 0.07975042901389491,
 0.1978710293261864,
 0.354153994351925,
 0.5294585752348643,
 0.7018145299391673,
 0.849379320441094,
 0.953326450056343,),
(0.01086933608417545,
 0.06498366633800794,
 0.1622293980238825,
 0.2937499039716641,
 0.4466318819056009,
 0.6054816627755208,
 0.7541101371585467,
 0.877265828834263,
 0.96225055941096,),
(0.00904263096219963,
 0.05397126622250072,
 0.1353118246392511,
 0.2470524162871565,
 0.3802125396092744,
 0.5237923179723384,
 0.6657752055148032,
 0.7941904160147613,
 0.898161091216429,
 0.9688479887196,),
(0.007643941174637681,
 0.04554182825657903,
 0.1145222974551244,
 0.2103785812270227,
 0.3266955532217897,
 0.4554532469286375,
 0.5876483563573721,
 0.7139638500230458,
 0.825453217777127,
 0.914193921640008,
 0.973860256264123,),
(0.006548722279080035,
 0.03894680956045022,
 0.0981502631060046,
 0.1811385815906331,
 0.2832200676673157,
 0.398434435164983,
 0.5199526267791299,
 0.6405109167754819,
 0.7528650118926111,
 0.850240024421055,
 0.926749682988251,
 0.977756129778486,),
)

_weightsLog=(
(-1.,),
(-0.7185393190303845,
 -0.2814606809696154,),
(-0.5134045522323634,
 -0.3919800412014877,
 -0.0946154065661483,),
(-0.3834640681451353,
 -0.3868753177747627,
 -0.1904351269501432,
 -0.03922548712995894,),
(-0.2978934717828955,
 -0.3497762265132236,
 -0.234488290044052,
 -0.0989304595166356,
 -0.01891155214319462,),
(-0.2387636625785478,
 -0.3082865732739458,
 -0.2453174265632108,
 -0.1420087565664786,
 -0.05545462232488041,
 -0.01016895869293513,),
(-0.1961693894252476,
 -0.2703026442472726,
 -0.239681873007687,
 -0.1657757748104267,
 -0.0889432271377365,
 -0.03319430435645653,
 -0.005932787015162054,),
(-0.164416604728002,
 -0.2375256100233057,
 -0.2268419844319134,
 -0.1757540790060772,
 -0.1129240302467932,
 -0.05787221071771947,
 -0.02097907374214317,
 -0.003686407104036044,),
(-0.1400684387481339,
 -0.2097722052010308,
 -0.211427149896601,
 -0.1771562339380667,
 -0.1277992280331758,
 -0.07847890261203835,
 -0.0390225049841783,
 -0.01386729555074604,
 -0.002408041036090773,),
(-0.12095513195457,
 -0.1863635425640733,
 -0.1956608732777627,
 -0.1735771421828997,
 -0.135695672995467,
 -0.0936467585378491,
 -0.05578772735275126,
 -0.02715981089692378,
 -0.00951518260454442,
 -0.001638157633217673,),
(-0.1056522560990997,
 -0.1665716806006314,
 -0.1805632182877528,
 -0.1672787367737502,
 -0.1386970574017174,
 -0.1038334333650771,
 -0.06953669788988512,
 -0.04054160079499477,
 -0.01943540249522013,
 -0.006737429326043388,
 -0.001152486965101561,),
(-0.09319269144393,
 -0.1497518275763289,
 -0.166557454364573,
 -0.1596335594369941,
 -0.1384248318647479,
 -0.1100165706360573,
 -0.07996182177673273,
 -0.0524069547809709,
 -0.03007108900074863,
 -0.01424924540252916,
 -0.004899924710875609,
 -0.000834029009809656,),
)


def sum(a):
	return reduce(lambda x, y: x+y, a)

def mult(a, b):
	return map(lambda x, y: x*y, a, b)

# make global 
xmin, xmax, =0., 1.
dx = xmax - xmin

def gauss(xmin, xmax, funct, ng=10):
	"""
	Gauss quadature (weight function = 1.0):
	xmin, xmax: boundaries of integration domain
	funct: integrand function
	ng: Gauss integration order
	"""
	ng = max((min((ng, _ngmax)), _ngmin))
	ns = _nodes[ng-1]
	ws = _weights[ng-1];
	dx = xmax - xmin
	x = map(lambda y: (dx*y + xmin + xmax)/2., ns)
	return 0.5*dx*sum(mult(funct(x), ws))

def gaussLog(xmin, xmax, funct, ng=10):
	"""
	Gauss quadature with Log singularity at x=xmin:
	xmin, xmax: boundaries of integration domain
	funct: integrand function
	ng: Gauss integration order
	"""
	ng = max((min((ng, _ngmax)), _ngmin))
	ns = _nodesLog[ng-1]
	ws = _weightsLog[ng-1];
	dx = xmax - xmin
	x = map(lambda y: (dx*y + xmin), ns)
	return dx*sum(mult(funct(x), ws))


####

if __name__ == '__main__':

	from math import *


	def f2(x):
		return map(lambda y: y**2, x)
	def f3(x):
		return map(lambda y: y**4, x)
	def f4(x):
		return map(lambda y: cos(2.*pi*(y-0.128726465)), x)
	def f5(x):
		return map(lambda y: 2.*cos(2.*pi*(y-0.128726465))**2, x)

	print '-'*80
	print 'Gauss (weight function = 1)'
	print '-'*80

	# simple tests
	print 'gauss(0., 1., f3, 1)=', gauss(0., 1., f3, 1)
	print 'gauss(0., 1., f3, 2)=', gauss(0., 1., f3, 2)
	print 'gauss(0., 1., f4, 3)=', gauss(0., 1., f3, 3)
	print 'gauss(0., 1., f3   )=', gauss(0., 1., f3   )

	# convergence test 
	ng = range(_ngmin, _ngmax+1)

	print """\n
	Integrate[Cos[2.*Pi*(x-0.128726465)], {x, 0, 1}]
	\n"""

	error = []
	for n in ng:
		error.append(gauss(0., 10.0, f4, n))
	print '    n = ', '%8d'*len(ng)    % tuple(ng)
	print 'error = ','%8.0e'*len(error) % tuple(error) 
		
	print """\n
	Integrate[2.*Cos[2.*Pi*(x-0.128726465)]^2, {x, 0, 1}]
	\n"""

	error = []
	for n in ng:
		error.append(gauss(0., 1.0, f5, n)-1.0)
	print '    n = ', '%8d'*len(ng)    % tuple(ng)
	print 'error = ','%8.0e'*len(error) % tuple(error) 


	print '-'*80
	print 'Gauss with Log singularity at left boundary'
	print '-'*80

	a, b = 0., 1.

	print """\n
	Integrate[Log[x]*x^2, {x, 0, 1}]
	\n"""

	exact = -1./9.
	error = []
	for n in ng:
		error.append(gaussLog(a, b, f2, n) - exact)
	print '    n = ', '%8d'*len(ng)    % tuple(ng)
	print 'error = ','%8.0e'*len(error) % tuple(error) 

	print """\n
	Integrate[Log[x]*2.*Cos[2.*Pi*(x-0.128726465)]^2, {x, 0, 1}]
	\n"""

	exact = -1.242002481967963
	error = []
	for n in ng:
		error.append(gaussLog(a, b, f5, n) - exact)
	print '    n = ', '%8d'*len(ng)    % tuple(ng)
	print 'error = ','%8.0e'*len(error) % tuple(error) 

History

  • revision 2 (21 years ago)
  • previous revisions are not available