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F x a,( )

f x a0, a1,( )

dfd x a0, a1,( )

dfd x a0, a1,( )

:=

Formulate a vector function that will bethe argument to the "genfit" function.

dfd x , ,( ) x e x:=

Next, write down the partial derivatives of

the function f with respect to the twoparameters of interest, & .

dfd x , ,( ) e x:=

f x , ,( ) e x:= This is the fitting function. The coefficient will be negative and |-1/| = is the lifetimefor the radioactive decay.

Always plot the data before attempting a fit.

0 20 40 60100

200

300

400

500

y

x

i 0 n 1..:= we will want to use thisrange variable later

n 5=number of data pointsn rows x( ):=y

409

304

260

192

170

:=x

10

20

30

40

50

:=

Raw Data:

The data is from Taylor 2nd ed, problem 8.25. The rate at which a radioactive material emitsradiation (and number of remaining radioactive nuclei) is expected to decrease exponentiallywith time. The two data vectors are "x", the elapsed time t (in min), and "y": the number ofcounts in a 15-second interval.

This example worksheet uses a generalized least-squares fitting procedure that is built intoMathcad to find the optimal fit parameters for an arbitrary (nonlinear) model function.

Nonlinear Regression IIPhysics 258 - DS Hamilton 2004

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Initial guess for the two parameters. Thisis one reason to plot the data first.

a500

0.5

:=

Call the function "genfit" to find thebest-fit coefficients.

genfit x y, a, F,( ):=

The solution is:

505.3= 0.023= 1

43.392= These are basically identical to those

found in the "minssd" example.

t 0 0.1, 60..:= Use this dummy variable to plot the fit so that it looks like asmooth curve through 600 points.

0 10 20 30 40 50 600

100

200

300

400

500

600

y

f t , ,( )

x t,

SSD

i

yi f xi , ,( )( )2:=

This is the RMS difference between the data pints andthe fitting function.

SSD

n10.1=