Peakexponential

Purpose

Outputs an Exponential function, Jacobian, and Hessian for a given set of input parameters and axis.

Synopsis

[y,y1,y2] = peakexponential(x,ax)

Description

Given a 2-element vector of parameters (x) and a vector of independent variables e.g., a wavelength or frequency axis (ax), PEAKEXPONENTIAL outputs an exponential function (y). If more than one output is requested, it also outputs the Jacobian (y1) and Hessian (y2). Derivatives are with respect to the parameters and are evaluated at (x). This function is called by PEAKFUNCTION.

Inputs

• x = 2-element vector with parameters

• x(1) = coefficient, and

• x(2) = decay constant.

• ax = vector of independent variables e.g., a wavelength or frequency axis with elements.

Outputs

• y = vector with the exponential function.

• y1 = matrix of the Jacobian of evaluated at (x).

• y2 = matrix of the Hessian of evaluated at (x).

Algorithm

The function is

• y(1) = ${{x}_{1}}{{\operatorname {e} }^{-{x}_{2}}}$ , 1xN

• y(2) = ${\operatorname {d} \!y \over \operatorname {d} \!{x}_{i}}$ , 3xN

• y(3) = ${\operatorname {d^{2}} \!y \over \operatorname {d} \!{{x}_{i}}^{2}}$ , 3x3xN

Examples

%Make a single exponential function

ax = 0:0.1:100;

y = peakexponential([1 0.01],ax);

plot(ax,y)

See Also

peakfunction, peakgaussian, peaklorentzian, peaksigmoid, peakpvoigt1, peakpvoigt2, peakstruct