Multivariate Derivative

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Wed, 11/02/2016 - 08:55 am

I have the exponential rise and decay function

f(t) = Amplitude*(1 - exp(-t/Rise))*exp(-t/Fall) + Offset


with Rise and Fall time constants.

I need to find the numerical derivative of this function with respect to the following parameters


1) Rise
2) Fall
3) Offset
4) Amplitude

Is there a way Igor can do this?
If I understand, you want to do partial derivatives numerically? I cannot address this. I can however offer a different idea.

The partial derivatives exist analytically. By example: (df/dR) = -A*t*exp(-t/R)*exp(-t/F)/R^2. Fit your measured data to the equation and obtain A, R, F, and O. Generate a wave to show the derivative equation using the fit coefficients.

make/N=101 dfdR
dfdR =  -A*x*exp(-x/R)*exp(-x/F)/R^2
display dfdR


BTW, (df/dO) = 1 and (df/dA) = 1. Why would you want these?

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAH

November 2, 2016 at 09:44 am - Permalink

Yes numerical partial derivatives is the goal. I plan to use them all to weight a fitting function. Since MATLAB has a way to do this, I wonder if Igor does.

November 2, 2016 at 09:59 am - Permalink

It's called Differentiate. To do the partials you want, you would need a wave full of values of f(t), and a wave full of values of each of the independent variables. Then use Differentiate with your f(t) wave as the Y wave and one of your independent variable waves as the X wave.

John Weeks
WaveMetrics, Inc.
support@wavemetrics.com

November 3, 2016 at 12:04 pm - Permalink

To follow on John Week's comment, you will have to

* fit your data to the model
* extract the fit parameters
* create a new wave called the "x-wave", one for each parameter
* numerically differentiate the y-data with respect to each x-wave

So, let's suppose that you have fit the data to your equation and have determined A, R, F, and O. You have stored the values in variables vA, vR, vF, and vO. You have your data in a wave called fwave. Here's how to get dfdA numerically.

duplicate/O fwave, xAwave, dfdA
xAwave = vA
differentiate fwave /X=xAwave /D=dfdA
...


John please correct me if I am wrong in any of this.

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAH

November 3, 2016 at 05:48 pm - Permalink