# Changeset d0dc9a3 in sasmodels

Ignore:
Timestamp:
Dec 4, 2017 8:29:04 AM (5 years ago)
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
df69efa
Parents:
7dde87f
Message:

document GAUSS_N, GAUSS_Z, GAUSS_W and simplify use from sasmodels.special

Files:
2 edited

### Legend:

Unmodified
 r3048ec6 M_PI, M_PI_2, M_PI_4, M_SQRT1_2, M_E: $\pi$, $\pi/2$, $\pi/4$, $1/\sqrt{2}$ and Euler's constant $e$ exp, log, pow(x,y), expm1, sqrt: Power functions $e^x$, $\ln x$, $x^y$, $e^x - 1$, $\sqrt{x}$. The function expm1(x) is accurate across all $x$, including $x$ very close to zero. exp, log, pow(x,y), expm1, log1p, sqrt, cbrt: Power functions $e^x$, $\ln x$, $x^y$, $e^x - 1$, $\ln 1 + x$, $\sqrt{x}$, $\sqrt[3]{x}$. The functions expm1(x) and log1p(x) are accurate across all $x$, including $x$ very close to zero. sin, cos, tan, asin, acos, atan: Trigonometry functions and inverses, operating on radians. atan(y/x) would return a value in quadrant I. Similarly for quadrants II and IV when $x$ and $y$ have opposite sign. fmin(x,y), fmax(x,y), trunc, rint: fabs(x), fmin(x,y), fmax(x,y), trunc, rint: Floating point functions.  rint(x) returns the nearest integer. NAN: NaN, Not a Number, $0/0$.  Use isnan(x) to test for NaN.  Note that you cannot use :code:x == NAN to test for NaN values since that will always return false.  NAN does not equal NAN! will always return false.  NAN does not equal NAN!  The alternative, :code:x != x may fail if the compiler optimizes the test away. INFINITY: $\infty, 1/0$.  Use isinf(x) to test for infinity, or isfinite(x) Similar arrays are available in :code:gauss20.c for 20-point quadrature and in :code:gauss150.c for 150-point quadrature. The macros :code:GAUSS_N, :code:GAUSS_Z and :code:GAUSS_W are defined so that you can change the order of the integration by selecting an different source without touching the C code. :code:source = ["lib/gauss76.c", ...]
 re65c3ba ................. The C code follows the C99 standard, with the usual math functions, as defined in OpenCL _. This includes the following: This following standard C99 math functions are available: M_PI, M_PI_2, M_PI_4, M_SQRT1_2, M_E: $\pi$, $\pi/2$, $\pi/4$, $1/\sqrt{2}$ and Euler's constant $e$ exp, log, pow(x,y), expm1, sqrt: Power functions $e^x$, $\ln x$, $x^y$, $e^x - 1$, $\sqrt{x}$. The function expm1(x) is accurate across all $x$, including $x$ very close to zero. exp, log, pow(x,y), expm1, log1p, sqrt, cbrt: Power functions $e^x$, $\ln x$, $x^y$, $e^x - 1$, $\ln 1 + x$, $\sqrt{x}$, $\sqrt[3]{x}$. The functions expm1(x) and log1p(x) are accurate across all $x$, including $x$ very close to zero. sin, cos, tan, asin, acos, atan: quadrants II and IV when $x$ and $y$ have opposite sign. fmin(x,y), fmax(x,y), trunc, rint: fabs(x), fmin(x,y), fmax(x,y), trunc, rint: Floating point functions.  rint(x) returns the nearest integer. NaN, Not a Number, $0/0$.  Use isnan(x) to test for NaN.  Note that you cannot use :code:x == NAN to test for NaN values since that will always return false.  NAN does not equal NAN! will always return false.  NAN does not equal NAN!  The alternative, :code:x != x may fail if the compiler optimizes the test away. INFINITY: for forcing a constant to stay double precision. The following special functions and scattering calculations are defined in sasmodels/models/lib _. The following special functions and scattering calculations are defined. These functions have been tuned to be fast and numerically stable down to $q=0$ even in single precision.  In some cases they work around bugs Gauss76Z[i], Gauss76Wt[i]: gauss76.n, gauss76.z[i], gauss76.w[i]: Points $z_i$ and weights $w_i$ for 76-point Gaussian quadrature, respectively, computing $\int_{-1}^1 f(z)\,dz \approx \sum_{i=1}^{76} w_i\,f(z_i)$. Similar arrays are available in :code:gauss20.c for 20-point quadrature and in :code:gauss150.c for 150-point quadrature. When translating the model to C, include 'lib/gauss76.c' in the source and use :code:GAUSS_N, :code:GAUSS_Z, and :code:GAUSS_W. Similar arrays are available in :code:gauss20 for 20-point quadrature and :code:gauss150.c for 150-point quadrature. By using :code:import gauss76 as gauss it is easy to change the number of points in the integration. """ # pylint: disable=unused-import # C99 standard math library functions from numpy import exp, log, power as pow, expm1, sqrt from numpy import exp, log, power as pow, expm1, logp1, sqrt, cbrt from numpy import sin, cos, tan, arcsin as asin, arccos as acos, arctan as atan from numpy import sinh, cosh, tanh, arcsinh as asinh, arccosh as acosh, arctanh as atanh from numpy import arctan2 as atan2 from numpy import fmin, fmax, trunc, rint from numpy import NAN, inf as INFINITY from numpy import fabs, fmin, fmax, trunc, rint from numpy import pi, nan, inf from scipy.special import gamma as sas_gamma from scipy.special import erf as sas_erf # C99 standard math constants M_PI, M_PI_2, M_PI_4, M_SQRT1_2, M_E = np.pi, np.pi/2, np.pi/4, np.sqrt(0.5), np.e NAN = nan INFINITY = inf # non-standard constants """return sin(x), cos(x)""" return sin(x), cos(x) sincos = SINCOS def square(x): # Gaussians Gauss20Wt = np.array([ .0176140071391521, .0406014298003869, .0626720483341091, .0832767415767047, .10193011981724, .118194531961518, .131688638449177, .142096109318382, .149172986472604, .152753387130726, .152753387130726, .149172986472604, .142096109318382, .131688638449177, .118194531961518, .10193011981724, .0832767415767047, .0626720483341091, .0406014298003869, .0176140071391521 ]) Gauss20Z = np.array([ -.993128599185095, -.963971927277914, -.912234428251326, -.839116971822219, -.746331906460151, -.636053680726515, -.510867001950827, -.37370608871542, -.227785851141645, -.076526521133497, .0765265211334973, .227785851141645, .37370608871542, .510867001950827, .636053680726515, .746331906460151, .839116971822219, .912234428251326, .963971927277914, .993128599185095 ]) Gauss76Wt = np.array([ .00126779163408536,         #0 .00294910295364247, .00462793522803742, .00629918049732845, .00795984747723973, .00960710541471375, .0112381685696677, .0128502838475101, .0144407317482767, .0160068299122486, .0175459372914742,          #10 .0190554584671906, .020532847967908, .0219756145344162, .0233813253070112, .0247476099206597, .026072164497986, .0273527555318275, .028587223650054, .029773487255905, .0309095460374916,          #20 .0319934843404216, .0330234743977917, .0339977794120564, .0349147564835508, .0357728593807139, .0365706411473296, .0373067565423816, .0379799643084053, .0385891292645067, .0391332242205184,          #30 .0396113317090621, .0400226455325968, .040366472122844, .0406422317102947, .0408494593018285, .040987805464794, .0410570369162294, .0410570369162294, .040987805464794, .0408494593018285,          #40 .0406422317102947, .040366472122844, .0400226455325968, .0396113317090621, .0391332242205184, .0385891292645067, .0379799643084053, .0373067565423816, .0365706411473296, .0357728593807139,          #50 .0349147564835508, .0339977794120564, .0330234743977917, .0319934843404216, .0309095460374916, .029773487255905, .028587223650054, .0273527555318275, .026072164497986, .0247476099206597,          #60 .0233813253070112, .0219756145344162, .020532847967908, .0190554584671906, .0175459372914742, .0160068299122486, .0144407317482767, .0128502838475101, .0112381685696677, .00960710541471375,         #70 .00795984747723973, .00629918049732845, .00462793522803742, .00294910295364247, .00126779163408536          #75 (indexed from 0) ]) Gauss76Z = np.array([ -.999505948362153,          #0 -.997397786355355, -.993608772723527, -.988144453359837, -.981013938975656, -.972229228520377, -.961805126758768, -.949759207710896, -.936111781934811, -.92088586125215, -.904107119545567,          #10 -.885803849292083, -.866006913771982, -.844749694983342, -.822068037328975, -.7980001871612, -.77258672828181, -.74587051350361, -.717896592387704, -.688712135277641, -.658366353758143,          #20 -.626910417672267, -.594397368836793, -.560882031601237, -.526420920401243, -.491072144462194, -.454895309813726, -.417951418780327, -.380302767117504, -.342012838966962, -.303146199807908,          #30 -.263768387584994, -.223945802196474, -.183745593528914, -.143235548227268, -.102483975391227, -.0615595913906112, -.0205314039939986, .0205314039939986, .0615595913906112, .102483975391227,                   #40 .143235548227268, .183745593528914, .223945802196474, .263768387584994, .303146199807908, .342012838966962, .380302767117504, .417951418780327, .454895309813726, .491072144462194,           #50 .526420920401243, .560882031601237, .594397368836793, .626910417672267, .658366353758143, .688712135277641, .717896592387704, .74587051350361, .77258672828181, .7980001871612,     #60 .822068037328975, .844749694983342, .866006913771982, .885803849292083, .904107119545567, .92088586125215, .936111781934811, .949759207710896, .961805126758768, .972229228520377,           #70 .981013938975656, .988144453359837, .993608772723527, .997397786355355, .999505948362153            #75 ]) Gauss150Z = np.array([ -0.9998723404457334, -0.9993274305065947, -0.9983473449340834, -0.9969322929775997, -0.9950828645255290, -0.9927998590434373, -0.9900842691660192, -0.9869372772712794, -0.9833602541697529, -0.9793547582425894, -0.9749225346595943, -0.9700655145738374, -0.9647858142586956, -0.9590857341746905, -0.9529677579610971, -0.9464345513503147, -0.9394889610042837, -0.9321340132728527, -0.9243729128743136, -0.9162090414984952, -0.9076459563329236, -0.8986873885126239, -0.8893372414942055, -0.8795995893549102, -0.8694786750173527, -0.8589789084007133, -0.8481048644991847, -0.8368612813885015, -0.8252530581614230, -0.8132852527930605, -0.8009630799369827, -0.7882919086530552, -0.7752772600680049, -0.7619248049697269, -0.7482403613363824, -0.7342298918013638, -0.7198995010552305, -0.7052554331857488, -0.6903040689571928, -0.6750519230300931, -0.6595056411226444, -0.6436719971150083, -0.6275578900977726, -0.6111703413658551, -0.5945164913591590, -0.5776035965513142, -0.5604390262878617, -0.5430302595752546, -0.5253848818220803, -0.5075105815339176, -0.4894151469632753, -0.4711064627160663, -0.4525925063160997, -0.4338813447290861, -0.4149811308476706, -0.3959000999390257, -0.3766465660565522, -0.3572289184172501, -0.3376556177463400, -0.3179351925907259, -0.2980762356029071, -0.2780873997969574, -0.2579773947782034, -0.2377549829482451, -0.2174289756869712, -0.1970082295132342, -0.1765016422258567, -0.1559181490266516, -0.1352667186271445, -0.1145563493406956, -0.0937960651617229, -0.0729949118337358, -0.0521619529078925, -0.0313062657937972, -0.0104369378042598, 0.0104369378042598, 0.0313062657937972, 0.0521619529078925, 0.0729949118337358, 0.0937960651617229, 0.1145563493406956, 0.1352667186271445, 0.1559181490266516, 0.1765016422258567, 0.1970082295132342, 0.2174289756869712, 0.2377549829482451, 0.2579773947782034, 0.2780873997969574, 0.2980762356029071, 0.3179351925907259, 0.3376556177463400, 0.3572289184172501, 0.3766465660565522, 0.3959000999390257, 0.4149811308476706, 0.4338813447290861, 0.4525925063160997, 0.4711064627160663, 0.4894151469632753, 0.5075105815339176, 0.5253848818220803, 0.5430302595752546, 0.5604390262878617, 0.5776035965513142, 0.5945164913591590, 0.6111703413658551, 0.6275578900977726, 0.6436719971150083, 0.6595056411226444, 0.6750519230300931, 0.6903040689571928, 0.7052554331857488, 0.7198995010552305, 0.7342298918013638, 0.7482403613363824, 0.7619248049697269, 0.7752772600680049, 0.7882919086530552, 0.8009630799369827, 0.8132852527930605, 0.8252530581614230, 0.8368612813885015, 0.8481048644991847, 0.8589789084007133, 0.8694786750173527, 0.8795995893549102, 0.8893372414942055, 0.8986873885126239, 0.9076459563329236, 0.9162090414984952, 0.9243729128743136, 0.9321340132728527, 0.9394889610042837, 0.9464345513503147, 0.9529677579610971, 0.9590857341746905, 0.9647858142586956, 0.9700655145738374, 0.9749225346595943, 0.9793547582425894, 0.9833602541697529, 0.9869372772712794, 0.9900842691660192, 0.9927998590434373, 0.9950828645255290, 0.9969322929775997, 0.9983473449340834, 0.9993274305065947, 0.9998723404457334 ]) Gauss150Wt = np.array([ 0.0003276086705538, 0.0007624720924706, 0.0011976474864367, 0.0016323569986067, 0.0020663664924131, 0.0024994789888943, 0.0029315036836558, 0.0033622516236779, 0.0037915348363451, 0.0042191661429919, 0.0046449591497966, 0.0050687282939456, 0.0054902889094487, 0.0059094573005900, 0.0063260508184704, 0.0067398879387430, 0.0071507883396855, 0.0075585729801782, 0.0079630641773633, 0.0083640856838475, 0.0087614627643580, 0.0091550222717888, 0.0095445927225849, 0.0099300043714212, 0.0103110892851360, 0.0106876814158841, 0.0110596166734735, 0.0114267329968529, 0.0117888704247183, 0.0121458711652067, 0.0124975796646449, 0.0128438426753249, 0.0131845093222756, 0.0135194311690004, 0.0138484622795371, 0.0141714592928592, 0.0144882814685445, 0.0147987907597169, 0.0151028518701744, 0.0154003323133401, 0.0156911024699895, 0.0159750356447283, 0.0162520081211971, 0.0165218992159766, 0.0167845913311726, 0.0170399700056559, 0.0172879239649355, 0.0175283451696437, 0.0177611288626114, 0.0179861736145128, 0.0182033813680609, 0.0184126574807331, 0.0186139107660094, 0.0188070535331042, 0.0189920016251754, 0.0191686744559934, 0.0193369950450545, 0.0194968900511231, 0.0196482898041878, 0.0197911283358190, 0.0199253434079123, 0.0200508765398072, 0.0201676730337687, 0.0202756819988200, 0.0203748563729175, 0.0204651529434560, 0.0205465323660984, 0.0206189591819181, 0.0206824018328499, 0.0207368326754401, 0.0207822279928917, 0.0208185680053983, 0.0208458368787627, 0.0208640227312962, 0.0208731176389954, 0.0208731176389954, 0.0208640227312962, 0.0208458368787627, 0.0208185680053983, 0.0207822279928917, 0.0207368326754401, 0.0206824018328499, 0.0206189591819181, 0.0205465323660984, 0.0204651529434560, 0.0203748563729175, 0.0202756819988200, 0.0201676730337687, 0.0200508765398072, 0.0199253434079123, 0.0197911283358190, 0.0196482898041878, 0.0194968900511231, 0.0193369950450545, 0.0191686744559934, 0.0189920016251754, 0.0188070535331042, 0.0186139107660094, 0.0184126574807331, 0.0182033813680609, 0.0179861736145128, 0.0177611288626114, 0.0175283451696437, 0.0172879239649355, 0.0170399700056559, 0.0167845913311726, 0.0165218992159766, 0.0162520081211971, 0.0159750356447283, 0.0156911024699895, 0.0154003323133401, 0.0151028518701744, 0.0147987907597169, 0.0144882814685445, 0.0141714592928592, 0.0138484622795371, 0.0135194311690004, 0.0131845093222756, 0.0128438426753249, 0.0124975796646449, 0.0121458711652067, 0.0117888704247183, 0.0114267329968529, 0.0110596166734735, 0.0106876814158841, 0.0103110892851360, 0.0099300043714212, 0.0095445927225849, 0.0091550222717888, 0.0087614627643580, 0.0083640856838475, 0.0079630641773633, 0.0075585729801782, 0.0071507883396855, 0.0067398879387430, 0.0063260508184704, 0.0059094573005900, 0.0054902889094487, 0.0050687282939456, 0.0046449591497966, 0.0042191661429919, 0.0037915348363451, 0.0033622516236779, 0.0029315036836558, 0.0024994789888943, 0.0020663664924131, 0.0016323569986067, 0.0011976474864367, 0.0007624720924706, 0.0003276086705538 ]) class Gauss: def __init__(self, w, z): self.n = len(w) self.w = w self.z = z gauss20 = Gauss( w=np.array([ .0176140071391521, .0406014298003869, .0626720483341091, .0832767415767047, .10193011981724, .118194531961518, .131688638449177, .142096109318382, .149172986472604, .152753387130726, .152753387130726, .149172986472604, .142096109318382, .131688638449177, .118194531961518, .10193011981724, .0832767415767047, .0626720483341091, .0406014298003869, .0176140071391521 ]), z=np.array([ -.993128599185095, -.963971927277914, -.912234428251326, -.839116971822219, -.746331906460151, -.636053680726515, -.510867001950827, -.37370608871542, -.227785851141645, -.076526521133497, .0765265211334973, .227785851141645, .37370608871542, .510867001950827, .636053680726515, .746331906460151, .839116971822219, .912234428251326, .963971927277914, .993128599185095 ]) ) gauss76 = Gauss( w=np.array([ .00126779163408536,             #0 .00294910295364247, .00462793522803742, .00629918049732845, .00795984747723973, .00960710541471375, .0112381685696677, .0128502838475101, .0144407317482767, .0160068299122486, .0175459372914742,              #10 .0190554584671906, .020532847967908, .0219756145344162, .0233813253070112, .0247476099206597, .026072164497986, .0273527555318275, .028587223650054, .029773487255905, .0309095460374916,              #20 .0319934843404216, .0330234743977917, .0339977794120564, .0349147564835508, .0357728593807139, .0365706411473296, .0373067565423816, .0379799643084053, .0385891292645067, .0391332242205184,              #30 .0396113317090621, .0400226455325968, .040366472122844, .0406422317102947, .0408494593018285, .040987805464794, .0410570369162294, .0410570369162294, .040987805464794, .0408494593018285,              #40 .0406422317102947, .040366472122844, .0400226455325968, .0396113317090621, .0391332242205184, .0385891292645067, .0379799643084053, .0373067565423816, .0365706411473296, .0357728593807139,              #50 .0349147564835508, .0339977794120564, .0330234743977917, .0319934843404216, .0309095460374916, .029773487255905, .028587223650054, .0273527555318275, .026072164497986, .0247476099206597,              #60 .0233813253070112, .0219756145344162, .020532847967908, .0190554584671906, .0175459372914742, .0160068299122486, .0144407317482767, .0128502838475101, .0112381685696677, .00960710541471375,             #70 .00795984747723973, .00629918049732845, .00462793522803742, .00294910295364247, .00126779163408536              #75 (indexed from 0) ]), z=np.array([ -.999505948362153,              #0 -.997397786355355, -.993608772723527, -.988144453359837, -.981013938975656, -.972229228520377, -.961805126758768, -.949759207710896, -.936111781934811, -.92088586125215, -.904107119545567,              #10 -.885803849292083, -.866006913771982, -.844749694983342, -.822068037328975, -.7980001871612, -.77258672828181, -.74587051350361, -.717896592387704, -.688712135277641, -.658366353758143,              #20 -.626910417672267, -.594397368836793, -.560882031601237, -.526420920401243, -.491072144462194, -.454895309813726, -.417951418780327, -.380302767117504, -.342012838966962, -.303146199807908,              #30 -.263768387584994, -.223945802196474, -.183745593528914, -.143235548227268, -.102483975391227, -.0615595913906112, -.0205314039939986, .0205314039939986, .0615595913906112, .102483975391227,                       #40 .143235548227268, .183745593528914, .223945802196474, .263768387584994, .303146199807908, .342012838966962, .380302767117504, .417951418780327, .454895309813726, .491072144462194,               #50 .526420920401243, .560882031601237, .594397368836793, .626910417672267, .658366353758143, .688712135277641, .717896592387704, .74587051350361, .77258672828181, .7980001871612, #60 .822068037328975, .844749694983342, .866006913771982, .885803849292083, .904107119545567, .92088586125215, .936111781934811, .949759207710896, .961805126758768, .972229228520377,               #70 .981013938975656, .988144453359837, .993608772723527, .997397786355355, .999505948362153                #75 ]) ) gauss150 = Gauss( z=np.array([ -0.9998723404457334, -0.9993274305065947, -0.9983473449340834, -0.9969322929775997, -0.9950828645255290, -0.9927998590434373, -0.9900842691660192, -0.9869372772712794, -0.9833602541697529, -0.9793547582425894, -0.9749225346595943, -0.9700655145738374, -0.9647858142586956, -0.9590857341746905, -0.9529677579610971, -0.9464345513503147, -0.9394889610042837, -0.9321340132728527, -0.9243729128743136, -0.9162090414984952, -0.9076459563329236, -0.8986873885126239, -0.8893372414942055, -0.8795995893549102, -0.8694786750173527, -0.8589789084007133, -0.8481048644991847, -0.8368612813885015, -0.8252530581614230, -0.8132852527930605, -0.8009630799369827, -0.7882919086530552, -0.7752772600680049, -0.7619248049697269, -0.7482403613363824, -0.7342298918013638, -0.7198995010552305, -0.7052554331857488, -0.6903040689571928, -0.6750519230300931, 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