Module

for

Nonlinear Curve Fitting

   

Data Linearization Method for Exponential Curve Fitting.  

    Fit the curve  [Graphics:Images/NonLinearCurveFitMod_gr_1.gif]  to the data points  [Graphics:Images/NonLinearCurveFitMod_gr_2.gif].  

Taking the logarithm of both sides we obtain  [Graphics:Images/NonLinearCurveFitMod_gr_3.gif][Graphics:Images/NonLinearCurveFitMod_gr_4.gif],  thus

        [Graphics:Images/NonLinearCurveFitMod_gr_5.gif].  

Introduce the change of variable  [Graphics:Images/NonLinearCurveFitMod_gr_6.gif].  Then the previous equation becomes  

        [Graphics:Images/NonLinearCurveFitMod_gr_7.gif]  

which is a linear equation in the variables X and Y.  

    Use the change of variables  [Graphics:Images/NonLinearCurveFitMod_gr_8.gif]  on all the data points and obtain

        [Graphics:Images/NonLinearCurveFitMod_gr_9.gif]  for  [Graphics:Images/NonLinearCurveFitMod_gr_10.gif].  

Fit the points [Graphics:Images/NonLinearCurveFitMod_gr_11.gif]  with a "least squares line" of the form  [Graphics:Images/NonLinearCurveFitMod_gr_12.gif].  

Comparing the equations  [Graphics:Images/NonLinearCurveFitMod_gr_13.gif]  we see that  [Graphics:Images/NonLinearCurveFitMod_gr_14.gif].  Thus

        [Graphics:Images/NonLinearCurveFitMod_gr_15.gif]

are used to construct the coefficients which are then used to "fit the curve"  

          [Graphics:Images/NonLinearCurveFitMod_gr_16.gif]  

to the given data points  [Graphics:Images/NonLinearCurveFitMod_gr_17.gif]  in the xy-plane.

Proof  Nonlinear Curve Fitting  Nonlinear Curve Fitting  

 

Computer Programs  Nonlinear Curve Fitting  Nonlinear Curve Fitting  

 

Example 1.  Fit the curve  [Graphics:Images/NonLinearCurveFitMod_gr_18.gif]  to the data points  [Graphics:Images/NonLinearCurveFitMod_gr_19.gif].  
Solution 1.

 

 

Data Linearization Method for a Power Function Curve Fitting.  

    Fit the curve  [Graphics:Images/NonLinearCurveFitMod_gr_70.gif]  to the data points  [Graphics:Images/NonLinearCurveFitMod_gr_71.gif].  

Taking the logarithm of both sides we obtain  [Graphics:Images/NonLinearCurveFitMod_gr_72.gif][Graphics:Images/NonLinearCurveFitMod_gr_73.gif],  thus

        [Graphics:Images/NonLinearCurveFitMod_gr_74.gif].  

Introduce the change of variable  [Graphics:Images/NonLinearCurveFitMod_gr_75.gif].  Then the previous equation becomes  

        [Graphics:Images/NonLinearCurveFitMod_gr_76.gif]  

which is a linear equation in the variables X and Y.  

    Use the change of variables  [Graphics:Images/NonLinearCurveFitMod_gr_77.gif]  on all the data points and obtain

        [Graphics:Images/NonLinearCurveFitMod_gr_78.gif]  for  [Graphics:Images/NonLinearCurveFitMod_gr_79.gif].  

Fit the points [Graphics:Images/NonLinearCurveFitMod_gr_80.gif]  with a "least squares line" of the form  [Graphics:Images/NonLinearCurveFitMod_gr_81.gif].  

Comparing the equations  [Graphics:Images/NonLinearCurveFitMod_gr_82.gif]  we see that  [Graphics:Images/NonLinearCurveFitMod_gr_83.gif].  Thus

        [Graphics:Images/NonLinearCurveFitMod_gr_84.gif]

are used to construct the coefficients which are then used to "fit the curve"  

          [Graphics:Images/NonLinearCurveFitMod_gr_85.gif]  

to the given data points  [Graphics:Images/NonLinearCurveFitMod_gr_86.gif]  in the xy-plane.

Proof  Nonlinear Curve Fitting  Nonlinear Curve Fitting  

 

Example 2.  Fit the curve  [Graphics:Images/NonLinearCurveFitMod_gr_87.gif]  to the data points  [Graphics:Images/NonLinearCurveFitMod_gr_88.gif].  
Solution 2.

 

Example  3.  Often times a  scientist must decide which formula "fits" the data best.
3 (a).  Find both the "exponential fit" and "power fit" for the data points  [Graphics:Images/NonLinearCurveFitMod_gr_138.gif].  
3 (b).  Discuss how this was accomplished and what transformations were used in the process.
3 (c).  Determine which curve fits the data best.
Solution 3.

 

Old Lab Project (Nonlinear Curve Fitting  Nonlinear Curve Fitting).  Internet hyperlinks to an old lab project.  

 

Research Experience for Undergraduates

Nonlinear Curve Fitting  Nonlinear Curve Fitting  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Nonlinear Curve Fitting

 

Return to Numerical Methods - Numerical Analysis

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004