Peakpvoigt1

Purpose

Outputs a pseudo-Voigt function, Jacobian, and Hessian for a given set of input parameters and axis.

Synopsis

[y,y1,y2] = peakpvoigt1(x,ax)

Description

Given a 4-element vector of parameters (x) and a vector of independent variables e.g. a wavelength or frequency axis (ax). PEAKPVOIGT1 outputs a pseudo-voigt peak (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 = 4 element vector with parameters
• x(1): coefficient,
• x(2): mean,
• x(3): spread, and
• x(4): fraction Gaussian.
• ax = 1 by N vector of independent variables e.g,. a wavelength or frequency axis.

Outputs

• y = 1 by N vector with the pseudo-voigt function evaluated at (x) and given by

z = p-x(2)
y = x(1)*( x(4)*exp(-4*ln(2)*z.^2/x(3)^2) + (1-x(4))*x(3)^2./(x(3)^2+z.^2) );

• y1 = dy/dxi, 4 by N matrix of the Jacobian of evaluated at (x).
• y2 = d2y/dxi^2, 4 by 4 by N matrix of the Hessian of evaluated at (x).

Algorithm

TEST $\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$

wikipedia $f\left({{a}_{i}},\mathbf {x} \right)={{x}_{1}}\left[{{x}_{4}}{{\operatorname {e} }^{\frac {-4\ln \left(2\right){{\left({{a}_{i}}-{{x}_{2}}\right)}^{2}}}{x_{3}^{2}}}}+\left(1-{{x}_{4}}\right)\left[{\frac {x_{3}^{2}}{{{\left({{a}_{i}}-{{x}_{2}}\right)}^{2}}+x_{3}^{2}}}\right]\right]$

texvc Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \[f\left( {{a}_{i}},\mathbf{x} \right)={{x}_{1}}\left[ {{x}_{4}}{{\operatorname{e}}^{\frac{-4\ln \left( 2 \right){{\left( {{a}_{i}}-{{x}_{2}} \right)}^{2}}}{x_{3}^{2}}}}+\left( 1-{{x}_{4}} \right)\left[ \frac{x_{3}^{2}}{{{\left( {{a}_{i}}-{{x}_{2}} \right)}^{2}}+x_{3}^{2}} \right] \right]\]}

Example

Make a single known peak

  ax = 0:0.1:100;
  y  = peakpvoigt1([2 51 8 0.5],ax);
  figure, plot(ax,y)

See Also

peakfunction, peakgaussian, peaklorentzian, peakpvoigt2, peakstruct