Examples

 Example 1. Initial data, which were subject to approximation, were generated by the function sin(x) with changing values of its argument on the interval [-π , π] with the step π/10. This example contains data with positive and negative values; corresponding curve has maximum and minimum. These data are not good for fitting, however we reached perfect approximation of them when the number of coefficients of Pade function was equal to 6. Input file contained coordinates of 21 points: -3.1415 -2.82735 -2.5132 -2.19905 -1.8849 -1.57075 -1.2566 -0.94245 -0.6283 -0.31415 0.0 0.31415 0.6283 0.94245 1.2566 1.57075 1.8849 2.19905 2.5132 2.82735 3.1415 0. -0.309008 -0.58777 -0.809001 -0.951045 -1. -0.951045 -0.809001 -0.58777 -0.309008 0.0 0.309008 0.58777 0.809001 0.951045 1. 0.951045 0.809001 0.58777 0.309008 0. Output:
Calculation #1. Tue Jun 11 16:05:36 PDT 2002

Entered number of coefficients of Pade function is: 6
Entered maximum value of correlation coefficient is: 0.99999
Required error of approximation is: 0.03
Calculated correlation coefficient is 0.9999457224112067
Maximum error is 0.029249275540080572 at x= -2.82735
Number of iterations without significant changing of correlation coefficient is 0

Approximation formula:
(-0.0014073334+0.99473155X+4.3362376E-4X2-0.15732671X3-2.94885E-5X4+0.0057283673X5)/(1.0),
where X=x1.0

Given and calculated data:

 x-coordinates Given y-coordinates Calculated y-coordinates -3.1415 0.0 -2.220446E-16 -2.82735 -0.309008 -0.29142043 -2.5132 -0.58777 -0.57676625 -2.19905 -0.809001 -0.809001 -1.8849 -0.951045 -0.95792145 -1.57075 -1.0 -1.008054 -1.2566 -0.951045 -0.9565521 -0.94245 -0.809001 -0.8110919 -0.6283 -0.58777 -0.58777 -0.31415 -0.309008 -0.3089996 0.0 0.0 -0.0014073334 0.31415 0.309008 0.30626994 0.6283 0.58777 0.5852885 0.94245 0.809001 0.809001 1.2566 0.951045 0.95495975 1.57075 1.0 1.0070201 1.8849 0.951045 0.9574435 2.19905 0.809001 0.809001 2.5132 0.58777 0.57707644 2.82735 0.309008 0.29176965 3.1415 0.0 -6.661338E-16

 Example 2. This example shows the possibility of fitting of experimental data with dispersion. For instance there are two points with the same abscise equal to -2.5132 in this example. The corresponding values of function are -0.55777 and -0.61777. The mean value is -0.58777, which exactly corresponds to the value in the previous example. There are also several similar points at x= -1.57075; -0.6283; 0.6283; 1.57075; 2.5132. Input file contained coordinates of 27 points: -3.1415 -2.82735 -2.5132 -2.5132 -2.19905 -1.8849 -1.57075 -1.57075 -1.2566 -0.94245 -0.6283 -0.6283 -0.31415 0.0 0.31415 0.6283 0.6283 0.94245 1.2566 1.57075 1.57075 1.8849 2.19905 2.5132 2.5132 2.82735 3.1415 0. 0. -0.309008 -0.55777 -0.61777 -0.809001 -0.951045 -1.05 -0.95 -0.951045 -0.809001 -0.55777 -0.61777 -0.309008 0.0 0.309008 0.55777 0.61777 0.809001 0.951045 1.05 0.95 0.951045 0.809001 0.55777 0.61777 0.309008 0. When the number of coefficients of Pade function was equal to 8, we almost reached the accuracy of approximation corresponding to the initial accuracy of entered data, because in this case we have obtained that 'Number of iterations without significant changing of correlation coefficient is 2'. Output:

Calculation #1. Tue Jun 11 17:27:54 PDT 2002

Entered number of coefficients of Pade function is: 6
Entered maximum value of correlation coefficient is: 0.99999
Required error of approximation is: 0.03
Calculated correlation coefficient is 0.9992830817369702
Maximum error is 0.24228138422813564 at x= -1.57075
Number of iterations without significant changing of correlation coefficient is 0

Approximation formula:
(0.003972023+0.9343162X-4.0247382E-4X2-0.094671614X3)/(1.0+0.0015320582X+0.091306016X2),
where X=x1.0

Given and calculated data:

 x-coordinates Given y-coordinates Calculated y-coordinates -3.1415 0.0 0.0 -2.82735 -0.309008 -0.29043132 -2.5132 -0.55777 -0.536537 -2.5132 -0.61777 -0.536537 -2.19905 -0.809001 -0.72719085 -1.8849 -0.951045 -0.8509647 -1.57075 -1.05 -0.8976465 -1.57075 -0.95 -0.8976465 -1.2566 -0.951045 -0.8604723 -0.94245 -0.809001 -0.73883086 -0.6283 -0.55777 -0.54076564 -0.6283 -0.61777 -0.54076564 -0.31415 -0.309008 -0.28422365 0.0 0.0 0.003972023 0.31415 0.309008 0.29174328 0.6283 0.55777 0.54711586 0.6283 0.61777 0.54711586 0.94245 0.809001 0.74353784 1.2566 0.951045 0.8634039 1.57075 1.05 0.8989805 1.57075 0.95 0.8989805 1.8849 0.951045 0.85109234 2.19905 0.809001 0.7266036 2.5132 0.55777 0.53573227 2.5132 0.61777 0.53573227 2.82735 0.309008 0.2898508 3.1415 0.0 0.0

Calculation #2. Tue Jun 11 17:29:20 PDT 2002

Entered number of coefficients of Pade function is: 8
Entered maximum value of correlation coefficient is: 0.99999
Required error of approximation is: 0.03
Calculated correlation coefficient is 0.9993493393502101
Maximum error is 0.13385542721078167 at x= -1.57075
Number of iterations without significant changing of correlation coefficient is 2

Approximation formula:
(-3.5166287+11.980425X-11.974845X2+3.512666X3)/(1.0-3.0697176X+4.2422833X2-2.5445533X3+0.6677138X4),
where X=exp(-0.2x)

Given and calculated data:

 x-coordinates Given y-coordinates Calculated y-coordinates -3.1415 0.0 0.0 -2.82735 -0.309008 -0.29323664 -2.5132 -0.55777 -0.56196564 -2.5132 -0.61777 -0.56196564 -2.19905 -0.809001 -0.778094 -1.8849 -0.951045 -0.91780293 -1.57075 -1.05 -0.9658279 -1.57075 -0.95 -0.9658279 -1.2566 -0.951045 -0.9175352 -0.94245 -0.809001 -0.7786925 -0.6283 -0.55777 -0.563773 -0.6283 -0.61777 -0.563773 -0.31415 -0.309008 -0.29382867 0.0 0.0 0.0054662297 0.31415 0.309008 0.3056081 0.6283 0.55777 0.57728875 0.6283 0.61777 0.57728875 0.94245 0.809001 0.7930391 1.2566 0.951045 0.9304703 1.57075 1.05 0.975482 1.57075 0.95 0.975482 1.8849 0.951045 0.92438054 2.19905 0.809001 0.78398645 2.5132 0.55777 0.56954676 2.5132 0.61777 0.56954676 2.82735 0.309008 0.3011712 3.1415 0.0 -3.2727079E-15

 Example 3. In this example the 'exact solution' of fitting problem is function 1/x0.5. Data were generated by this function on the interval [0.1; 100]. Input file contained coordinates of 7 points: 0.01 0.25 0.16 1. 4. 25 100 10 2 2.5 1 0.5 0.2 0.1 Note that in this case x-coordinates are represented in non-increasing order. It is acceptable, if the corresponding values y are placed on right positions. We reached the exact solution at the first step of approximation. The obtained formula can be easily simplified. Output:

Calculation #1. Tue Jun 11 18:27:43 PDT 2002

Entered number of coefficients of Pade function is: 3
Entered maximum value of correlation coefficient is: 0.99999
Required error of approximation is: 0.01
Calculated correlation coefficient is 1.0000000000000002
Maximum error is 7.628526304172854E-16 at x=0.01
Number of iterations without significant changing of correlation coefficient is 0

Approximation formula:
(2.3604161E16)/(1.0+2.3604161E16X-1.6470588X2),
where X=x0.5

Given and calculated data:

 x-coordinates Given y-coordinates Calculated y-coordinates 0.01 10.0 10.0 0.16 2.5 2.5 0.25 2.0 2.0 1.0 1.0 1.0 4.0 0.5 0.5 25.0 0.2 0.2 100.0 0.1 0.1