functional

class functional

Functionals handling

Introduction

A functional is a function with parameters, defined as \(f(p;x)\), where \(p\) are the parameters, and \(x\) the arguments. Methods are available to calculate the value of a function for a series of argument values for the given set of parameters, and for the automatic calculation of the derivatives with respect to the parameters.

The created functionals can be used in the fitting  or in any other  that needs to have generic function values or their derivatives.

A functional has a mask associated with it, to indicate if certain parameters have to be solved for. See masks for details.

Functionals are created in a variety of ways, in general by specifying the name of the functional, together with some necessary information like e.g. the order of a polynomial, or the code needed to compile your privately defined function. Parameters can be set at creation time or later.

- a = fs.gaussian1d()              # creates a 1D Gaussian, default arguments
- b = fs.open('gaussian1')         # creates the same one
- a.state()                        # the 'state' of the functional
[type=0, order=-1, ndim=1, npar=3, params=[1 0 1] ] 
- a.type()                         # its type
gaussian1d 
- a.ndim()                         # its dimension (number of arguments)
1 
- a.npar()                         # its number of parameters
3 
- b.state()
[type=0, order=-1, ndim=1, npar=3, params=[1 0 1] ] 
- a.f(1);                          # the value at x=1
0.0625
- a.fdf([0,0.5]);                  # value and derivatives
[[1:2,]
    1   1   0       0
    0.5 0.5 1.38629 0.693147] 

In some cases an order can be specified as well (e.g. for polynomials):

- a := dfs.poly(3)               # creates a 3rd order polynomial
- b := dfs.functional('polyn',3) # creates the same one, but with
                                 # original defaults
- a.state()
[type=5, order=3, ndim=1, npar=4, params=[1 1 1 1] ] 
- b.state()
[type=5, order=3, ndim=1, npar=4, params=[0 0 0 0] ] 

An extremely valuable aspect of the Functionals module is the ability to create a functional from a compiled string specifying an arbitrary function. For example, let us make our own polynomial \(1 + 2*x + 3*x^2\) and evaluate it at a few abcissa locations

- a := dfs.compiled ('p0 + p1*x + p2*x*x', [1,2,3])   # Define
- a.f([0,10,20])                                      # Evaluate at x=[0,10,20]
[1 321 1241]  

The functions created can also be used to specify the function to be fitted in a least squares fit (see the fitting  ).

Methods Summary

done	Free the functional’s resources.
f	Calculate the value of the functional.
functional	Create a functional tool.
gaussian1d	Create a 1-dimensional Gaussian with the specified amplitude, fwhm, and center.
gaussian2d	Create a 2-dimensional Gaussian with the specified amplitude, fwhms, and center.
ndim	Return the number of dimensions.
polynomial	Create a 1-dimensional polynomial function with the specified coefficents.
powerlogpoly	Create a 1-dimensional power log polynomial function with the specified coefficents.

done()

Free the functional’s resources.

f(x=0)

Calculate the value of the functional.

Parameters

• x ({double, doubleVec}=0) - real argument values

Returns

any

Examples

gfn = fn.gaussian1d(2, 0, 1)
#returns 0.125
gfn.f(1)              
# returns array([  1.25000000e-01,   3.05175781e-05])
gfn.f([1,2])

functional()

Create a functional tool.

gaussian1d(amplitude=1, center=0, fwhm=1)

Create a 1-dimensional Gaussian with the specified amplitude, fwhm, and center.

Parameters

• amplitude (double=1) - amplitude of Gaussian

• center (double=0) - center of Gaussian

• fwhm (double=1) - FWHM of Gaussian

Returns

functional

Examples

# get the value and derivatives of a Gaussian with
# height=2; center at x=1; a width of 1 at x=[0,1,2]
gfn = fn.gaussian1d(2,1,1)

# returns array([ 0.125,  2.   ,  0.125])
vals = gfn.f([0, 1, 2])

gaussian2d(amplitude=1, center=[-1], fwhm=[-1], pa='0')

Create a 2-dimensional Gaussian with the specified amplitude, fwhms, and center. The created functional has method f to calculate the function value at a series of x, y values, or the value.

Parameters

• amplitude (double=1) - Amplitude of Gaussian

• center (doubleVec=[-1]) - Center (x,y) position. Must have exactly 2 elements.

• fwhm (doubleVec=[-1]) - FWHM of the axes. Must have exactly 2 elements.

• pa ({string, doubleQuant}='0') - The angle between the positive y axis and the major axis, measured counterclockwise.

Returns

functional

Examples

# major axis along the y axis
g2d = fn.gaussian2d(1,[0,0],[3,2],"90deg")

# both these commands return 0.5
v = g2d([0, 1])
v = g2d([1.5, 0])

# returns array([ 0.5,  0.5])
v =  g2d.f([0,1,1.5,0])

ndim()

Return the number of dimensions.

polynomial(coefficients=[0])

Create a 1-dimensional polynomial function with the specified coefficents.

Parameters

• coefficients (doubleVec=[0]) - Array of coefficients. Number of coefficients determines order of polynomial.

Returns

functional

Examples

# get the value and derivatives of 3 + 2*x + 4*x*x
poly = fn.powerlogpoly(3, 2, 4)

# value at 3
vals = poly.f(3)

powerlogpoly(coefficients=[0])

Create a 1-dimensional power log polynomial function with the specified coefficents.

Parameters

• coefficients (doubleVec=[0]) - Array of coefficients.

Returns

functional

Examples

# get the value and derivatives of 2*x**(1+ln(x))
plp = fn.powerlogpoly(2,1,1)

# value at 3
vals = plp.f(3)