# Writing a Plugin Model

## Overview

In addition to the models provided with the sasmodels package, you are free to create your own models.

Models can be of three types:

- A pure python model : Example - broadpeak.py
- A python model with embedded C : Example - sphere.py
- A python wrapper with separate C code : Example - cylinder.py, cylinder.c

When using SasView, plugin models should be saved to the SasView
*plugin_models* folder *C:\Users\{username}\.sasview\plugin_models*
(on Windows) or */Users/{username}/.sasview\plugin_models* (on Mac).
The next time SasView is started it will compile the plugin and add
it to the list of *Plugin Models* in a FitPage. Scripts can load
the models from anywhere.

The built-in modules are available in the *models* subdirectory
of the sasmodels package. For SasView on Windows, these will
be found in *C:\Program Files (x86)\SasView\sasmodels-data\models*.
On Mac OSX, these will be within the application bundle as
*/Applications/SasView 4.0.app/Contents/Resources/sasmodels-data/models*.

Other models are available for download from the Model Marketplace. You can contribute your own models to the Marketplace as well.

## Create New Model Files

Copy the appropriate files to your plugin models directory (we recommend using the examples above as templates) as mymodel.py (and mymodel.c, etc) as required, where “mymodel” is the name for the model you are creating.

*Please follow these naming rules:*

- No capitalization and thus no CamelCase
- If necessary use underscore to separate words (i.e. barbell not BarBell or broad_peak not BroadPeak)
- Do not include “model” in the name (i.e. barbell not BarBellModel)

## Edit New Model Files

### 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*

- this is the

- a
- 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

- this is the

- a
- 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

- this is the

- a
- a
**parameter table**: - this will be auto-generated from the
*parameters*in the*.py*file

- this will be auto-generated from the

- a
- 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!

- this is ReStructuredText enclosed between the r”“” and “”” delimiters
at the top of the

- a
- a
**definition**of the model: - as part of the
**long description**

- as part of the

- a
- a
**formula**defining the function the model calculates: - as part of the
**long description**

- as part of the

- a
- an
**explanation of the parameters**: - as part of the
**long description** - explaining how the symbols in the formula map to the model parameters

- as part of the

- an
- a
**plot**of the function, with a**figure caption**: - this is automatically generated from your default parameters

- a
- at least one
**reference**: - as part of the
**long description** - specifying where the reader can obtain more information about the model

- as part of the

- at least one
- 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

- as part of the

- the

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, :math:`V` is the volume of the scatterer,
:math:`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 :math:`\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
`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 `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 Iq(),
Iqac(), Iqabc() and form_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 nameevery 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.

- new units can be added using the macros defined in

**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.

- lower and upper limits can be any number, or
**“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 Iq(), Iqac(), Iqabc() and form_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. If
*form_volume* is not defined, then the default *form_volume = 1.0* will be
used.

### 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:

```
#include <math.h>
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.

**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*.

### 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
Oriented particles, 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 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 \(u = \cos(\alpha)\),
\(du = -\sin(\alpha)\,d\alpha\) 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, ...)*.

### 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:
- \(\frac{\pi}{180}\), \(\frac{4\pi}{3}\)
- SINCOS(x, s, c):
- Macro which sets s=sin(x) and c=cos(x). The variables
candsmust 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\((x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,dt\) and complementary error function sas_erfc\((x) = \frac{2}{\sqrt\pi}\int_x^{\infty} e^{-t^2}\,dt\).

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 \(J_0(x) = \frac{1}{\pi}\int_0^\pi \cos(x\sin(\tau))\,d\tau\).

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 \(J_1(x) = \frac{1}{\pi}\int_0^\pi \cos(\tau - x\sin(\tau))\,d\tau\).

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 \(J_n(x) = \frac{1}{\pi}\int_0^\pi \cos(n\tau - x\sin(\tau))\,d\tau\). 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,

\[\text{Si}(x) \sim \frac{\pi}{2} - \frac{\cos(x)}{x}\left(1 - \frac{2!}{x^2} + \frac{4!}{x^4} - \frac{6!}{x^6} \right) - \frac{\sin(x)}{x}\left(\frac{1}{x} - \frac{3!}{x^3} + \frac{5!}{x^5} - \frac{7!}{x^7}\right)\]For small arguments,

\[\text{Si}(x) \sim x - \frac{x^3}{3\times 3!} + \frac{x^5}{5 \times 5!} - \frac{x^7}{7 \times 7!} + \frac{x^9}{9\times 9!} - \frac{x^{11}}{11\times 11!}\]

`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 average radius of the
form averaged over all the polydispersity values.

```
def ER(radius, thickness):
"""Effective radius of a core-shell sphere."""
return radius + thickness
```

Now consider the *core_shell_sphere*, which has a simple effective radius
equal to the radius of the core plus the thickness of the shell, as
shown above. Given polydispersity over *(r1, r2, ..., rm)* in radius and
*(t1, t2, ..., tn)* in thickness, *ER* is called with a mesh
grid covering all possible combinations of radius and thickness.
That is, *radius* is *(r1, r2, ..., rm, r1, r2, ..., rm, ...)*
and *thickness* is *(t1, t1, ... t1, t2, t2, ..., t2, ...)*.
The *ER* function returns one effective radius for each combination.
The effective radius calculator weights each of these according to
the polydispersity distributions and calls the structure factor
with the average *ER*.

```
def VR(radius, thickness):
"""Sphere and shell volumes for a core-shell sphere."""
whole = 4.0/3.0 * pi * (radius + thickness)**3
core = 4.0/3.0 * pi * radius**3
return whole, whole - core
```

Core-shell type models have an additional volume ratio which scales
the structure factor. The *VR* function returns the volume of
the whole sphere and the volume of the shell. Like *ER*, there is
one return value for each point in the mesh grid.

*NOTE: we may be removing or modifying this feature soon. As of the
time of writing, core-shell sphere returns (1., 1.) for VR, giving a volume
ratio of 1.0.*

### 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}, "ER", 120.],
[{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "VR", 1.],
]
```

**tests=[[{parameters}, q, result], ...]** 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
*ER*and*VR*, give the inputs as “ER” and “VR” respectively; the output for*VR*should be the sphere/shell ratio, not the individual sphere and shell values.

## Test Your New 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.

If you are not using sasmodels from SasView, skip this step.

### Recommended Testing

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, 2
*v*), 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")
```

## Check The Docs

You can get a rough idea of how the documentation will look using the following:

```
compare("-help", "~/.sasview/plugin_models/model.py")
```

This does not use the same styling as the rest of the docs, but it will allow you to check that your ReStructuredText and LaTeX formatting. Here are some tools to help with the inevitable syntax errors:

There is also a neat online WYSIWYG ReStructuredText editor at http://rst.ninjs.org.

## Clean Lint - (Developer Version Only)

**NB: For now we are not providing pylint with the installer version
of SasView; so unless you have a SasView build environment available,
you can ignore this section!**

Run the lint check with:

```
python -m pylint --rcfile=extra/pylint.rc ~/.sasview/plugin_models/model.py
```

We are not aiming for zero lint just yet, only keeping it to a minimum.
For now, don’t worry too much about *invalid-name*. If you really want a
variable name *Rg* for example because \(R_g\) is the right name for the model
parameter then ignore the lint errors. Also, ignore *missing-docstring*
for standard model functions *Iq*, *Iqac*, etc.

We will have delinting sessions at the SasView Code Camps, where we can decide on standards for model files, parameter names, etc.

For now, you can tell pylint to ignore things. For example, to align your parameters in blocks:

```
# pylint: disable=bad-whitespace,line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
["contrast_factor", "barns", 10.0, [-inf, inf], "", "Contrast factor of the polymer"],
["bjerrum_length", "Ang", 7.1, [0, inf], "", "Bjerrum length"],
["virial_param", "1/Ang^2", 12.0, [-inf, inf], "", "Virial parameter"],
["monomer_length", "Ang", 10.0, [0, inf], "", "Monomer length"],
["salt_concentration", "mol/L", 0.0, [-inf, inf], "", "Concentration of monovalent salt"],
["ionization_degree", "", 0.05, [0, inf], "", "Degree of ionization"],
["polymer_concentration", "mol/L", 0.7, [0, inf], "", "Polymer molar concentration"],
]
# pylint: enable=bad-whitespace,line-too-long
```

Don’t put in too many pylint statements, though, since they make the code ugly.