Model Contents

The model interface definition is in the .py file. This file contains:

• a model name:

  • this is the name string in the .py file
  • titles should be:

  • all in lower case
  • without spaces (use underscores to separate words instead)
  • without any capitalization or CamelCase
  • without incorporating the word "model"
  • examples: barbell not BarBell; broad_peak not BroadPeak; barbell not BarBellModel

• a model title:

  • this is the title string in the .py file
  • this is a one or two line description of the model, which will appear at the start of the model documentation and as a tooltip in the SasView GUI

• a short description:

  • this is the description string in the .py file
  • this is a medium length description which appears when you click Description on the model FitPage

• a parameter table:

  • this will be auto-generated from the parameters in the .py file

• a long description:

  • this is ReStructuredText enclosed between the r""" and """ delimiters at the top of the .py file
  • what you write here is abstracted into the SasView help documentation
  • this is what other users will refer to when they want to know what your model does; so please be helpful!

• a definition of the model:

  • as part of the long description

• a formula defining the function the model calculates:

  • as part of the long description

• an explanation of the parameters:

  • as part of the long description
  • explaining how the symbols in the formula map to the model parameters

• a plot of the function, with a figure caption:

  • this is automatically generated from your default parameters

• at least one reference:

  • as part of the long description
  • specifying where the reader can obtain more information about the model

• the name of the author

  • as part of the long description
  • the .py file should also contain a comment identifying who converted/created the model file

Models that do not conform to these requirements will never be incorporated into the built-in library.

Model Documentation

The .py file starts with an r (for raw) and three sets of quotes to start the doc string and ends with a second set of three quotes. For example:

r"""
Definition
----------
The 1D scattering intensity of the sphere is calculated in the following
way (Guinier, 1955)
.. math::
    I(q) = \frac{\text{scale}}{V} \cdot \left[
        3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3}
        \right]^2 + \text{background}
where *scale* is a volume fraction, $V$ is the volume of the scatterer,
$r$ is the radius of the sphere and *background* is the background level.
*sld* and *sld_solvent* are the scattering length densities (SLDs) of the
scatterer and the solvent respectively, whose difference is $\Delta\rho$.
You can included figures in your documentation, as in the following
figure for the cylinder model.
.. figure:: img/cylinder_angle_definition.jpg
    Definition of the angles for oriented cylinders.
References
----------
A Guinier, G Fournet, *Small-Angle Scattering of X-Rays*,
John Wiley and Sons, New York, (1955)
"""

This is where the FULL documentation for the model goes (to be picked up by the automatic documentation system). Although it feels odd, you should start the documentation immediately with the definition---the model name, a brief description and the parameter table are automatically inserted above the definition, and the a plot of the model is automatically inserted before the reference.

Figures can be included using the figure command, with the name of the .png file containing the figure and a caption to appear below the figure. Figure numbers will be added automatically.

See this Sphinx cheat sheet for a quick guide to the documentation layout commands, or the Sphinx Documentation for complete details.

The model should include a formula written using LaTeX markup. The example above uses the math command to make a displayed equation. You can also use $formula$ for an inline formula. This is handy for defining the relationship between the model parameters and formula variables, such as the phrase "$r$ is the radius" used above. The live demo MathJax page http://www.mathjax.org/ is handy for checking that the equations will look like you intend.

Math layout uses the amsmath package for aligning equations (see amsldoc.pdf on that page for complete documentation). You will automatically be in an aligned environment, with blank lines separating the lines of the equation. Place an ampersand before the operator on which to align. For example:

.. math::
  x + y &= 1 \\
  y &= x - 1

produces

If you need more control, use:

.. math::
    :nowrap:

Model Definition

Following the documentation string, there are a series of definitions:

name = "sphere"  # optional: defaults to the filename without .py
title = "Spheres with uniform scattering length density"
description = """\
P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr))
                /(qr)^3]^2 + background
    r: radius of sphere
    V: The volume of the scatter
    sld: the SLD of the sphere
    sld_solvent: the SLD of the solvent
"""
category = "shape:sphere"
single = True   # optional: defaults to True
opencl = False  # optional: defaults to False
structure_factor = False  # optional: defaults to False

name = "mymodel" defines the name of the model that is shown to the user. If it is not provided it will use the name of the model file. The name must be a valid variable name, starting with a letter and contains only letters, numbers or underscore. Spaces, dashes, and other symbols are not permitted.

title = "short description" is short description of the model which is included after the model name in the automatically generated documentation. The title can also be used for a tooltip.

description = """doc string""" is a longer description of the model. It shows up when you press the "Description" button of the SasView FitPage. It should give a brief description of the equation and the parameters without the need to read the entire model documentation. The triple quotes allow you to write the description over multiple lines. Keep the lines short since the GUI will wrap each one separately if they are too long. Make sure the parameter names in the description match the model definition!

category = "shape:sphere" defines where the model will appear in the model documentation. In this example, the model will appear alphabetically in the list of spheroid models in the Shape category.

single = True indicates that the model can be run using single precision floating point values. Set it to False if the numerical calculation for the model is unstable, which is the case for about 20 of the built in models. It is worthwhile modifying the calculation to support single precision, allowing models to run up to 10 times faster. The section Test_Your_New_Model describes how to compare model values for single vs. double precision so you can decide if you need to set single to False.

opencl = False indicates that the model should not be run using OpenCL. This may be because the model definition includes code that cannot be compiled for the GPU (for example, goto statements). It can also be used for large models which can't run on most GPUs. This flag has not been used on any of the built in models; models which were failing were streamlined so this flag was not necessary.

structure_factor = True indicates that the model can be used as a structure factor to account for interactions between particles. See Form_Factors for more details.

model_info = ... lets you define a model directly, for example, by loading and modifying existing models. This is done implicitly by :func:`sasmodels.core.load_model_info`, which can create a mixture model from a pair of existing models. For example:

from sasmodels.core import load_model_info
model_info = load_model_info('sphere+cylinder')

See :class:`sasmodels.modelinfo.ModelInfo` for details about the model attributes that are defined.

Model Parameters

Next comes the parameter table. For example:

# pylint: disable=bad-whitespace, line-too-long
#   ["name",        "units", default, [min, max], "type",    "description"],
parameters = [
    ["sld",         "1e-6/Ang^2",  1, [-inf, inf], "sld",    "Layer scattering length density"],
    ["sld_solvent", "1e-6/Ang^2",  6, [-inf, inf], "sld",    "Solvent scattering length density"],
    ["radius",      "Ang",        50, [0, inf],    "volume", "Sphere radius"],
]
# pylint: enable=bad-whitespace, line-too-long

parameters = [["name", "units", default, [min,max], "type", "tooltip"],...] defines the parameters that form the model.

Note: The order of the parameters in the definition will be the order of the parameters in the user interface and the order of the parameters in Fq(), Iq(), Iqac(), Iqabc(), radius_effective(), form_volume() and shell_volume(). And scale and background parameters are implicit to all models, so they do not need to be included in the parameter table.

• "name" is the name of the parameter shown on the FitPage.

  • the name must be a valid variable name, starting with a letter and containing only letters, numbers and underscore.

  • parameter names should follow the mathematical convention; e.g., radius_core not core_radius, or sld_solvent not solvent_sld.

  • model parameter names should be consistent between different models, so sld_solvent, for example, should have exactly the same name in every model.

  • to see all the parameter names currently in use, type the following in the python shell/editor under the Tools menu:

    import sasmodels.list_pars
    sasmodels.list_pars.list_pars()
    

    re-use as many as possible!!!

  • use "name[n]" for multiplicity parameters, where n is the name of the parameter defining the number of shells/layers/segments, etc.

• "units" are displayed along with the parameter name

  • every parameter should have units; use "None" if there are no units.

  • sld's should be given in units of 1e-6/Ang^2, and not simply 1/Ang^2 to be consistent with the builtin models. Adjust your formulas appropriately.

  • fancy units markup is available for some units, including:

    Ang, 1/Ang, 1/Ang^2, 1e-6/Ang^2, degrees, 1/cm, Ang/cm, g/cm^3, mg/m^2
    

  • the list of units is defined in the variable RST_UNITS within sasmodels/generate.py

    • new units can be added using the macros defined in doc/rst_prolog in the sasmodels source.
    • units should be properly formatted using sub-/super-scripts and using negative exponents instead of the / operator, though the unit name should use the / operator for consistency.
    • please post a message to the SasView developers mailing list with your changes.

• default is the initial value for the parameter.

  • the parameter default values are used to auto-generate a plot of the model function in the documentation.

• [min, max] are the lower and upper limits on the parameter.

  • lower and upper limits can be any number, or -inf or inf.
  • the limits will show up as the default limits for the fit making it easy, for example, to force the radius to always be greater than zero.
  • these are hard limits defining the valid range of parameter values; polydisperity distributions will be truncated at the limits.

• "type" can be one of: "", "sld", "volume", or "orientation".

  • "sld" parameters can have magnetic moments when fitting magnetic models; depending on the spin polarization of the beam and the $q$ value being examined, the effective sld for that material will be used to compute the scattered intensity.
  • "volume" parameters are passed to Fq(), Iq(), Iqac(), Iqabc(), form_volume() and shell_volume(), and have polydispersity loops generated automatically.
  • "orientation" parameters are not passed, but instead are combined with orientation dispersity to translate qx and qy to qa, qb and qc. These parameters should appear at the end of the table with the specific names theta, phi and for asymmetric shapes psi, in that order.

Some models will have integer parameters, such as number of pearls in the pearl necklace model, or number of shells in the multi-layer vesicle model. The optimizers in BUMPS treat all parameters as floating point numbers which can take arbitrary values, even for integer parameters, so your model should round the incoming parameter value to the nearest integer inside your model you should round to the nearest integer. In C code, you can do this using:

static double
Iq(double q, ..., double fp_n, ...)
{
    int n = (int)(fp_n + 0.5);
    ...
}

in python:

def Iq(q, ..., n, ...):
    n = int(n+0.5)
    ...

Derivative based optimizers such as Levenberg-Marquardt will not work for integer parameters since the partial derivative is always zero, but the remaining optimizers (DREAM, differential evolution, Nelder-Mead simplex) will still function.

Model Computation

Models can be defined as pure python models, or they can be a mixture of python and C models. C models are run on the GPU if it is available, otherwise they are compiled and run on the CPU.

Models are defined by the scattering kernel, which takes a set of parameter values defining the shape, orientation and material, and returns the expected scattering. Polydispersity and angular dispersion are defined by the computational infrastructure. Any parameters defined as "volume" parameters are polydisperse, with polydispersity defined in proportion to their value. "orientation" parameters use angular dispersion defined in degrees, and are not relative to the current angle.

Based on a weighting function $G(x)$ and a number of points $n$, the computed value is

That is, the individual models do not need to include polydispersity calculations, but instead rely on numerical integration to compute the appropriately smeared pattern.

Each .py file also contains a function:

def random():
...

This function provides a model-specific random parameter set which shows model features in the USANS to SANS range. For example, core-shell sphere sets the outer radius of the sphere logarithmically in [20, 20,000], which sets the Q value for the transition from flat to falling. It then uses a beta distribution to set the percentage of the shape which is shell, giving a preference for very thin or very thick shells (but never 0% or 100%). Using -sets=10 in sascomp should show a reasonable variety of curves over the default sascomp q range. The parameter set is returned as a dictionary of {parameter: value, ...}. Any model parameters not included in the dictionary will default according to the code in the _randomize_one() function from sasmodels/compare.py.

Python Models

For pure python models, define the Iq function:

import numpy as np
from numpy import cos, sin, ...
def Iq(q, par1, par2, ...):
    return I(q, par1, par2, ...)
Iq.vectorized = True

The parameters par1, par2, ... are the list of non-orientation parameters to the model in the order that they appear in the parameter table. Note that the auto-generated model file uses x rather than q.

The .py file should import trigonometric and exponential functions from numpy rather than from math. This lets us evaluate the model for the whole range of $q$ values at once rather than looping over each $q$ separately in python. With $q$ as a vector, you cannot use if statements, but must instead do tricks like

a = x*q*(q>0) + y*q*(q<=0)

or

a = np.empty_like(q)
index = q>0
a[index] = x*q[index]
a[~index] = y*q[~index]

which sets $a$ to $q cdot x$ if $q$ is positive or $q cdot y$ if $q$ is zero or negative. If you have not converted your function to use $q$ vectors, you can set the following and it will only receive one $q$ value at a time:

Iq.vectorized = False

Return np.NaN if the parameters are not valid (e.g., cap_radius < radius in barbell). If I(q; pars) is NaN for any $q$, then those parameters will be ignored, and not included in the calculation of the weighted polydispersity.

Models should define form_volume(par1, par2, ...) where the parameter list includes the volume parameters in order. This is used for a weighted volume normalization so that scattering is on an absolute scale. For solid shapes, the I(q) function should use form_volume squared as its scale factor. If form_volume is not defined, then the default form_volume = 1.0 will be used.

Hollow shapes, where the volume fraction of particle corresponds to the material in the shell rather than the volume enclosed by the shape, must also define a shell_volume(par1, par2, ...) function. The parameters are the same as for form_volume. Here the I(q) function should use shell_volume squared instead of form_volume squared so that the scale parameter corresponds to the volume fraction of material in the sample. The structure factor calculation needs the volume fraction of the filled shapes for its calculation, so the volume fraction parameter in the model is automatically scaled by form_volume/shell_volume prior to calling the structure factor.

Embedded C Models

Like pure python models, inline C models need to define an Iq function:

Iq = """
    return I(q, par1, par2, ...);
"""

This expands into the equivalent C code:

double Iq(double q, double par1, double par2, ...);
double Iq(double q, double par1, double par2, ...)
{
    return I(q, par1, par2, ...);
}

form_volume defines the volume of the shape. As in python models, it includes only the volume parameters.

shell_volume defines the volume of the shell for hollow shapes. As in python models, it includes only the volume parameters.

source=['fn.c', ...] includes the listed C source files in the program before Iq and form_volume are defined. This allows you to extend the library of C functions available to your model.

c_code includes arbitrary C code into your kernel, which can be handy for defining helper functions for Iq and form_volume. Note that you can put the full function definition for Iq and form_volume (include function declaration) into c_code as well, or put them into an external C file and add that file to the list of sources.

Models are defined using double precision declarations for the parameters and return values. When a model is run using single precision or long double precision, each variable is converted to the target type, depending on the precision requested.

Floating point constants must include the decimal point. This allows us to convert values such as 1.0 (double precision) to 1.0f (single precision) so that expressions that use these values are not promoted to double precision expressions. Some graphics card drivers are confused when functions that expect floating point values are passed integers, such as 4*atan(1); it is safest to not use integers in floating point expressions. Even better, use the builtin constant M_PI rather than 4*atan(1); it is faster and smaller!

The C model operates on a single $q$ value at a time. The code will be run in parallel across different $q$ values, either on the graphics card or the processor.

Rather than returning NAN from Iq, you must define the INVALID(v). The v parameter lets you access all the parameters in the model using v.par1, v.par2, etc. For example:

#define INVALID(v) (v.bell_radius < v.radius)

The INVALID define can go into Iq, or c_code, or an external C file listed in source.

Structure Factors

Structure factor calculations may need the underlying $<F(q)>$ and $<F^2(q)>$ rather than $I(q)$. This is used to compute $beta = <F(q)>^2/<F^2(q)>$ in the decoupling approximation to the structure factor.

Instead of defining the Iq function, models can define Fq as something like:

double Fq(double q, double *F1, double *F2, double par1, double par2, ...);
double Fq(double q, double *F1, double *F2, double par1, double par2, ...)
{
    // Polar integration loop over all orientations.
    ...
    *F1 = 1e-2 * total_F1 * contrast * volume;
    *F2 = 1e-4 * total_F2 * square(contrast * volume);
    return I(q, par1, par2, ...);
}

If the volume fraction scale factor is built into the model (as occurs for the vesicle model, for example), then scale F1 by $surd V_f$ so that $beta$ is computed correctly.

Structure factor calculations are not yet supported for oriented shapes.

Note: only available as a separate C file listed in source, or within a c_code block within the python model definition file.

Oriented Shapes

If the scattering is dependent on the orientation of the shape, then you will need to include orientation parameters theta, phi and psi at the end of the parameter table. As described in the section :ref:`orientation`, the individual $(q_x, q_y)$ points on the detector will be rotated into $(q_a, q_b, q_c)$ points relative to the sample in its canonical orientation with $a$-$b$-$c$ aligned with $x$-$y$-$z$ in the laboratory frame and beam travelling along $-z$.

The oriented C model (oriented pure Python models are not supported) is called using Iqabc(qa, qb, qc, par1, par2, ...) where par1, etc. are the parameters to the model. If the shape is rotationally symmetric about c then psi is not needed, and the model is called as Iqac(qab, qc, par1, par2, ...). In either case, the orientation parameters are not included in the function call.

For 1D oriented shapes, an integral over all angles is usually needed for the Iq function. Given symmetry and the substitution $u = cos(alpha)$, $du = -sin(alpha),dalpha$ this becomes

for

Using the $z, w$ values for Gauss-Legendre integration in "lib/gauss76.c", the numerical integration is then:

double outer_sum = 0.0;
for (int i = 0; i < GAUSS_N; i++) {
    const double cos_alpha = 0.5*GAUSS_Z[i] + 0.5;
    const double sin_alpha = sqrt(1.0 - cos_alpha*cos_alpha);
    const double qc = cos_alpha * q;
    double inner_sum = 0.0;
    for (int j = 0; j < GAUSS_N; j++) {
        const double beta = M_PI_4 * GAUSS_Z[j] + M_PI_4;
        double sin_beta, cos_beta;
        SINCOS(beta, sin_beta, cos_beta);
        const double qa = sin_alpha * sin_beta * q;
        const double qb = sin_alpha * cos_beta * q;
        const double form = Fq(qa, qb, qc, ...);
        inner_sum += GAUSS_W[j] * form * form;
    }
    outer_sum += GAUSS_W[i] * inner_sum;
}
outer_sum *= 0.25; // = 8/(4 pi) * outer_sum * (pi/2) / 4

The z values for the Gauss-Legendre integration extends from -1 to 1, so the double sum of w[i]w[j] explains the factor of 4. Correcting for the average dz[i]dz[j] gives $(1-0) cdot (pi/2-0) = pi/2$. The $8/(4 pi)$ factor comes from the integral over the quadrant. With less symmetry (eg., in the bcc and fcc paracrystal models), then an integral over the entire sphere may be necessary.

For simpler models which are rotationally symmetric a single integral suffices:

for

with integration loop:

double sum = 0.0;
for (int i = 0; i < GAUSS_N; i++) {
    const double cos_alpha = 0.5*GAUSS_Z[i] + 0.5;
    const double sin_alpha = sqrt(1.0 - cos_alpha*cos_alpha);
    const double qab = sin_alpha * q;
    const double qc = cos_alpha * q;
    const double form = Fq(qab, qc, ...);
    sum += GAUSS_W[j] * form * form;
}
sum *= 0.5; // = 2/pi * sum * (pi/2) / 2

Magnetism

Magnetism is supported automatically for all shapes by modifying the effective SLD of particle according to the Halpern-Johnson vector describing the interaction between neutron spin and magnetic field. All parameters marked as type sld in the parameter table are treated as possibly magnetic particles with magnitude M0 and direction mtheta and mphi. Polarization parameters are also provided automatically for magnetic models to set the spin state of the measurement.

For more complicated systems where magnetism is not uniform throughout the individual particles, you will need to write your own models. You should not mark the nuclear sld as type sld, but instead leave them unmarked and provide your own magnetism and polarization parameters. For 2D measurements you will need $(q_x, q_y)$ values for the measurement to compute the proper magnetism and orientation, which you can implement using Iqxy(qx, qy, par1, par2, ...).

Note: Magnetism is not supported in pure Python models.

Special Functions

The C code follows the C99 standard, with the usual math functions, as defined in OpenCL. This includes the following:

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, 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.

sinh, cosh, tanh, asinh, acosh, atanh:

Hyperbolic trigonometry functions.

atan2(y,x):

Angle from the $x$-axis to the point $(x,y)$, which is equal to $tan^{-1}(y/x)$ corrected for quadrant. That is, if $x$ and $y$ are both negative, then atan2(y,x) returns a value in quadrant III where atan(y/x) would return a value in quadrant I. Similarly for quadrants II and IV when $x$ and $y$ have opposite sign.

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 x == NAN to test for NaN values since that will always return false. NAN does not equal NAN! The alternative, 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) to test for finite and not NaN.

erf, erfc, tgamma, lgamma: do not use

Special functions that should be part of the standard, but are missing or inaccurate on some platforms. Use sas_erf, sas_erfc, sas_gamma and sas_lgamma instead (see below).

Some non-standard constants and functions are also provided:

M_PI_180, M_4PI_3:

$frac{pi}{180}$, $frac{4pi}{3}$

SINCOS(x, s, c):

Macro which sets s=sin(x) and c=cos(x). The variables c and s must be declared first.

square(x):

$x^2$

cube(x):

$x^3$

sas_sinx_x(x):

$sin(x)/x$, with limit $sin(0)/0 = 1$.

powr(x, y):

$x^y$ for $x ge 0$; this is faster than general $x^y$ on some GPUs.

pown(x, n):

$x^n$ for $n$ integer; this is faster than general $x^n$ on some GPUs.

FLOAT_SIZE:

The number of bytes in a floating point value. Even though all variables are declared double, they may be converted to single precision float before running. If your algorithm depends on precision (which is not uncommon for numerical algorithms), use the following:

#if FLOAT_SIZE>4
... code for double precision ...
#else
... code for single precision ...
#endif

SAS_DOUBLE:

A replacement for double so that the declared variable will stay double precision; this should generally not be used since some graphics cards do not support double precision. There is no provision for forcing a constant to stay double precision.

The following special functions and scattering calculations are defined in sasmodels/models/lib. 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 which appear on some platforms but not others, so use them where needed. Add the files listed in source = ["lib/file.c", ...] to your model.py file in the order given, otherwise these functions will not be available.

polevl(x, c, n):

Polynomial evaluation $p(x) = sum_{i=0}^n c_i x^i$ using Horner's method so it is faster and more accurate.

$c = {c_n, c_{n-1}, ldots, c_0 }$ is the table of coefficients, sorted from highest to lowest.

source = ["lib/polevl.c", ...] (link to code)

p1evl(x, c, n):

Evaluation of normalized polynomial $p(x) = x^n + sum_{i=0}^{n-1} c_i x^i$ using Horner's method so it is faster and more accurate.

$c = {c_{n-1}, c_{n-2} ldots, c_0 }$ is the table of coefficients, sorted from highest to lowest.

source = ["lib/polevl.c", ...] (polevl.c)

sas_gamma(x):

Gamma function sas_gamma$(x) = Gamma(x)$.

The standard math function, tgamma(x), is unstable for $x < 1$ on some platforms.

source = ["lib/sas_gamma.c", ...] (sas_gamma.c)

sas_gammaln(x):

log gamma function sas_gammaln$(x) = log Gamma(|x|)$.

The standard math function, lgamma(x), is incorrect for single precision on some platforms.

source = ["lib/sas_gammainc.c", ...] (sas_gammainc.c)

sas_gammainc(a, x), sas_gammaincc(a, x):

Incomplete gamma function sas_gammainc$(a, x) = int_0^x t^{a-1}e^{-t},dt / Gamma(a)$ and complementary incomplete gamma function sas_gammaincc$(a, x) = int_x^infty t^{a-1}e^{-t},dt / Gamma(a)$

source = ["lib/sas_gammainc.c", ...] (sas_gammainc.c)

sas_erf(x), sas_erfc(x):

Error function sas_erf$(x) = frac{2}{sqrtpi}int_0^x e^{-t^2},dt$ and complementary error function sas_erfc$(x) = frac{2}{sqrtpi}int_x^{infty} e^{-t^2},dt$.

The standard math functions erf(x) and erfc(x) are slower and broken on some platforms.

source = ["lib/polevl.c", "lib/sas_erf.c", ...] (sas_erf.c)

sas_J0(x):

Bessel function of the first kind sas_J0$(x)=J_0(x)$ where $J_0(x) = frac{1}{pi}int_0^pi cos(xsin(tau)),dtau$.

The standard math function j0(x) is not available on all platforms.

source = ["lib/polevl.c", "lib/sas_J0.c", ...] (sas_J0.c)

sas_J1(x):

Bessel function of the first kind sas_J1$(x)=J_1(x)$ where $J_1(x) = frac{1}{pi}int_0^pi cos(tau - xsin(tau)),dtau$.

The standard math function j1(x) is not available on all platforms.

source = ["lib/polevl.c", "lib/sas_J1.c", ...] (sas_J1.c)

sas_JN(n, x):

Bessel function of the first kind and integer order $n$, sas_JN$(n, x) =J_n(x)$ where $J_n(x) = frac{1}{pi}int_0^pi cos(ntau - xsin(tau)),dtau$. If $n$ = 0 or 1, it uses sas_J0($x$) or sas_J1($x$), respectively.

Warning: JN(n,x) can be very inaccurate (0.1%) for x not in [0.1, 100].

The standard math function jn(n, x) is not available on all platforms.

source = ["lib/polevl.c", "lib/sas_J0.c", "lib/sas_J1.c", "lib/sas_JN.c", ...] (sas_JN.c)

sas_Si(x):

Sine integral Si$(x) = int_0^x tfrac{sin t}{t},dt$.

Warning: Si(x) can be very inaccurate (0.1%) for x in [0.1, 100].

This function uses Taylor series for small and large arguments:

For large arguments use the following Taylor series,

Si(x) ~ (π)/(2) − (cos(x))/(x)⎛⎝1 − (2!)/(x²) + (4!)/(x⁴) − (6!)/(x⁶)⎞⎠ − (sin(x))/(x)⎛⎝(1)/(x) − (3!)/(x³) + (5!)/(x⁵) − (7!)/(x⁷)⎞⎠

For small arguments,

Si(x) ~ x − (x³)/(3 × 3!) + (x⁵)/(5 × 5!) − (x⁷)/(7 × 7!) + (x⁹)/(9 × 9!) − (x¹¹)/(11 × 11!)

source = ["lib/Si.c", ...] (Si.c)

sas_3j1x_x(x):

Spherical Bessel form sph_j1c$(x) = 3 j_1(x)/x = 3 (sin(x) - x cos(x))/x^3$, with a limiting value of 1 at $x=0$, where $j_1(x)$ is the spherical Bessel function of the first kind and first order.

This function uses a Taylor series for small $x$ for numerical accuracy.

source = ["lib/sas_3j1x_x.c", ...] (sas_3j1x_x.c)

sas_2J1x_x(x):

Bessel form sas_J1c$(x) = 2 J_1(x)/x$, with a limiting value of 1 at $x=0$, where $J_1(x)$ is the Bessel function of first kind and first order.

source = ["lib/polevl.c", "lib/sas_J1.c", ...] (sas_J1.c)

Gauss76Z[i], Gauss76Wt[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 gauss20.c for 20-point quadrature and in gauss150.c for 150-point quadrature. The macros GAUSS_N, GAUSS_Z and GAUSS_W are defined so that you can change the order of the integration by selecting an different source without touching the C code.

source = ["lib/gauss76.c", ...] (gauss76.c)

Problems with C models

The graphics processor (GPU) in your computer is a specialized computer tuned for certain kinds of problems. This leads to strange restrictions that you need to be aware of. Your code may work fine on some platforms or for some models, but then return bad values on other platforms. Some examples of particular problems:

(1) Code is too complex, or uses too much memory. GPU devices only have a limited amount of memory available for each processor. If you run programs which take too much memory, then rather than running multiple values in parallel as it usually does, the GPU may only run a single version of the code at a time, making it slower than running on the CPU. It may fail to run on some platforms, or worse, cause the screen to go blank or the system to reboot.

(2) Code takes too long. Because GPU devices are used for the computer display, the OpenCL drivers are very careful about the amount of time they will allow any code to run. For example, on OS X, the model will stop running after 5 seconds regardless of whether the computation is complete. You may end up with only some of your 2D array defined, with the rest containing random data. Or it may cause the screen to go blank or the system to reboot.

(3) Memory is not aligned. The GPU hardware is specialized to operate on multiple values simultaneously. To keep the GPU simple the values in memory must be aligned with the different GPU compute engines. Not following these rules can lead to unexpected values being loaded into memory, and wrong answers computed. The conclusion from a very long and strange debugging session was that any arrays that you declare in your model should be a multiple of four. For example:

double Iq(q, p1, p2, ...)
{
    double vector[8];  // Only going to use seven slots, but declare 8
    ...
}

The first step when your model is behaving strangely is to set single=False. This automatically restricts the model to only run on the CPU, or on high-end GPU cards. There can still be problems even on high-end cards, so you can force the model off the GPU by setting opencl=False. This runs the model as a normal C program without any GPU restrictions so you know that strange results are probably from your code rather than the environment. Once the code is debugged, you can compare your output to the output on the GPU.

Although it can be difficult to get your model to work on the GPU, the reward can be a model that runs 1000x faster on a good card. Even your laptop may show a 50x improvement or more over the equivalent pure python model.

Form Factors

Away from the dilute limit you can estimate scattering including particle-particle interactions using $I(q) = P(q)*S(q)$ where $P(q)$ is the form factor and $S(q)$ is the structure factor. The simplest structure factor is the hardsphere interaction, which uses the effective radius of the form factor as an input to the structure factor model. The effective radius is the weighted average over all values of the shape in polydisperse systems.

There can be many notions of effective radius, depending on the shape. For a sphere it is clearly just the radius, but for an ellipsoid of revolution we provide average curvature, equivalent sphere radius, minimum radius and maximum radius. These options are listed as radius_effective_modes in the python model defintion, and must be computed by the radius_effective function which takes the radius_effective_mode parameter as an integer, along with the various model parameters. Unlike normal C/Python arrays, the first mode is 1, the second is 2, etc. Mode 0 indicates that the effective radius from the shape is to be ignored in favour of the the effective radius parameter in the structure factor model.

Consider the core-shell sphere, which defines the following effective radius modes in the python model:

radius_effective_modes = [
    "outer radius",
    "core radius",
]

and the following function in the C-file for the model:

static double
radius_effective(int mode, double radius, double thickness)
{
    switch (mode) {
        case 0: return radius + thickness;
        case 1: return radius;
        default: return 0.;
    }
}
static double
form_volume(double radius, double thickness)
{
    return M_4PI_3 * cube(radius + thickness);
}

Given polydispersity over (r1, r2, ..., rm) in radius and (t1, t2, ..., tn) in thickness, radius_effective is called over a mesh grid covering all possible combinations of radius and thickness, with a single (ri, tj) pair in each call. The weights each of these results according to the polydispersity distributions and calls the structure factor with the average effective radius. Similarly, for form_volume.

Hollow models have an additional volume ratio which is needed to scale the structure factor. The structure factor uses the volume fraction of the filled particles as part of its density estimate, but the scale factor for the scattering intensity (as non-solvent volume fraction / volume) is determined by the shell volume only. Therefore the shell_volume function is needed to compute the form:shell volume ratio, which then scales the volfraction parameter prior to calling the structure factor calculator. In the case of a hollow sphere, this would be:

static double
shell_volume(double radius, double thickness)
{
    double whole = M_4PI_3 * cube(radius + thickness);
    double core = M_4PI_3 * cube(radius);
    return whole - core;
}

If shell_volume is not present, then form_volume and shell_volume are assumed to be equal, and the shape is considered solid.

Unit Tests

THESE ARE VERY IMPORTANT. Include at least one test for each model and PLEASE make sure that the answer value is correct (i.e. not a random number).

tests = [
    [{}, 0.2, 0.726362],
    [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1.,
      "radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
     0.2, 0.228843],
    [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45},
     0.1, None, None, 120., None, 1.],  # q, F, F^2, R_eff, V, form:shell
    [{"@S": "hardsphere"}, 0.1, None],
]

tests=[[{parameters}, q, Iq], ...] is a list of lists. Each list is one test and contains, in order:

• a dictionary of parameter values. This can be {} using the default parameters, or filled with some parameters that will be different from the default, such as {"radius":10.0, "sld":4}. Unlisted parameters will be given the default values.
• the input $q$ value or tuple of $(q_x, q_y)$ values.
• the output $I(q)$ or $I(q_x,q_y)$ expected of the model for the parameters and input value given.
• input and output values can themselves be lists if you have several $q$ values to test for the same model parameters.
• for testing effective radius, volume and form:shell volume ratio, use the extended form of the tests results, with None, None, R_eff, V, V_r instead of Iq. This calls the kernel Fq function instead of Iq.
• for testing F and F^2 (used for beta approximation) do the same as the effective radius test, but include values for the first two elements, $<F(q)>$ and $<F^2(q)>$.
• for testing interaction between form factor and structure factor, specify the structure factor name in the parameters as {"@S": "name", ...} with the remaining list of parameters defined by the P@S product model.

Minimal Testing

From SasView either open the Python shell (Tools > Python Shell/Editor) or the plugin editor (Fitting > Plugin Model Operations > Advanced Plugin Editor), load your model, and then select Run > Check Model from the menu bar. An Info box will appear with the results of the compilation and a check that the model runs.

Recommended Testing

NB: For now, this more detailed testing is only possible if you have a SasView build environment available!

If the model compiles and runs, you can next run the unit tests that you have added using the test = values.

From SasView, switch to the Shell tab and type the following:

from sasmodels.model_test import run_one
run_one("~/.sasview/plugin_models/model.py")

This should print:

test_model_python (sasmodels.model_test.ModelTestCase) ... ok

To check whether single precision is good enough, type the following:

from sasmodels.compare import main as compare
compare("~/.sasview/plugin_models/model.py")

This will pop up a plot showing the difference between single precision and double precision on a range of $q$ values.

demo = dict(scale=1, background=0,
            sld=6, sld_solvent=1,
            radius=120,
            radius_pd=.2, radius_pd_n=45)

demo={'par': value, ...} in the model file sets the default values for the comparison. You can include polydispersity parameters such as radius_pd=0.2, radius_pd_n=45 which would otherwise be zero.

These commands can also be run directly in the python interpreter:

$ python -m sasmodels.model_test -v ~/.sasview/plugin_models/model.py $ python -m sasmodels.compare ~/.sasview/plugin_models/model.py

The options to compare are quite extensive; type the following for help:

compare()

Options will need to be passed as separate strings. For example to run your model with a random set of parameters:

compare("-random", "-pars", "~/.sasview/plugin_models/model.py")

For the random models,

• sld will be in the range (-0.5,10.5),
• angles (theta, phi, psi) will be in the range (-180,180),
• angular dispersion will be in the range (0,45),
• polydispersity will be in the range (0,1)
• other values will be in the range (0, 2v), where v is the value of the parameter in demo.

Dispersion parameters n, sigma and type will be unchanged from demo so that run times are more predictable (polydispersity calculated across multiple parameters can be very slow).

If your model has 2D orientation calculation, then you should also test with:

compare("-2d", "~/.sasview/plugin_models/model.py")