Approximation and Interpolation
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This collection approximates multivariable tabulated or continuous functions in 1 to 4 independent variables by finite multivariable power, trigonometric or mixed series.
Requirements: Windows 98 or later.
Click on any image for full size screenshot
Important Note: You need programs of the Support Collection to study these approximations.
Download Polynomial Regression
Approximates tabulated functions in 1 to 4 independent variables by a finite multivariable power series using the Least Squares method. The program determines the coefficients of the polynomial, the generalized correlation coefficient and the standard error of estimate.
mix4.gif (41695 bytes) mix4eval.gif (8553 bytes) Download Mixed Polynomial and Trigonometric Approximations (Continuous) Approximates continuous functions in 1 to 4 independent variables by finite multivariable power and/or trigonometric series. The program fits the function exactly at the grid points.
dmix4.gif (25170 bytes) Download Mixed Polynomial and Trigonometric Approximations (Tabulated) Approximates tabulated functions in 1 to 4 independent variables by finite multivariable power and/or trigonometric series. The program fits the function exactly at the grid points.
***You need MIXED SERIES EVALUATIONS  to study solutions obtained by the above***)
evalgrid.gif (36594 bytes)evgredit.gif (13307 bytes)
Download Mixed Series Evaluations
Evaluates a mixed series or any partial derivative at all points of a rectangular grid simultaneously.

Pending programs:

MIXED POLYNOMIAL AND TRIGONOMETRIC APPROXIMATIONS for CONTINUOUS FUNCTIONS (FUNCTION DEFINED BY AN EQUATION) approximates IMPLICITLY defined continuous functions in 1 to 4 independent variables by finite multivariable power and/or trigonometric series. This program is similar to the MIXED POLYNOMIAL & TRIGONOMETRIC APPROXIMATIONS for CONTINUOUS FUNCTIONS in which functions are EXPLICITLY defined, but here the function may be defined by an equation. For instance, if the function is y and the variable is x, then y may be defined by the nonlinear equation x3-4x2y+2xy2-y3=0.

FOURIER TRANSFORMS approximates tabulated functions in 1 to 4 independent variables by finite multivariable trigonometric series using fast Fourier transform methods. The program determines the coefficients of the approximation, the generalized correlation coefficient and the standard error of estimate.

In the programs of the APPROXIMATION AND INTERPOLATION collection, depending upon the program, the coefficients of the approximating expression are found using the method of Undetermined Coefficients, the method of Least Squares or Fast Fourier Transform methods.

Solutions produced can be filed and such files retrieved by support programs and studied to find roots, (multiple) integral, (partial) derivatives, maxima and minima.

The values of tabulated functions must be taken for values of the independent variables belonging to a rectangular grid which may be unevenly spaced, that is, a function of n variables must be tabulated at the nodes of an n-dimensional rectangular grid. For example, if X and Y are the independent variables, we may, for instance, have 6 values of X and 5 values of Y and 30 values of the function, one for each X-Y combination, and the grid may look like:

In the FOURIER TRANSFORMS programs, however, the grid must be evenly spaced because fast Fourier transform methods are used. In addition, one of the versions requires the number of subintervals in any direction to be a power of 2, while the other one only requires it to be even.

TYPES OF APPROXIMATIONS:

The programs of the APPROXIMATION AND INTERPOLATION collection produce series approximations which may be pure power, pure trigonometric or mixed power and trigonometric series.

In all these programs, you can arrange the independent variables in any order, from innermost to outermost.

Programs which produce MIXED APPROXIMATIONS can produce not only mixed series but also pure power series or pure trigonometric series, depending on how you select the type of approximation in each one of the variables. In all, you may independently select, for each variable, the type of series, center of expansion, number of terms and period (if applicable).

The POLYNOMIAL REGRESSION program will produce a pure power series. You may independently select, for each variable, the center of expansion and number of terms.

The FOURIER TRANSFORMS programs will produce a pure trigonometric series. You may independently select, for each variable, the number of terms.

TABULATED FUNCTIONS:

It was mentioned before that, for tabulated functions, the values of the independent variables must belong to a rectangular grid. An example of a tabulated function with one evenly spaced variable and one unevenly spaced is:

    Y:  -5     -2     2    3     6 
X: 
2      -720    -36   36   144   1260 
3      -2880   -144  144  576   5040 
4      -7200   -360  360  1440  12600
5      -14400  -720  720  2880  25200


Note that the difference between two consecutive values of X is always the same, but the difference between two consecutive Y's varies.

An example of a tabulated function in two variables which may be approximated by the FOURIER TRANSFORMS program is:

    Y: 0   p/2   p    3p/2   2p 
X:   
0      2   -1    2    -3     2  
p/2    4    1    4    -1     4  
p      2   -1    2    -3     2  
3
p/2   0   -3    0    -5     0  
2
p     2   -1    2    -3     2  


Note that, even though there are 5 values in each direction, only 4 of the function's values are independent because its value at the interval's end point must be the same as at the initial point because of the required periodicity.

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