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Rational Function Interpolation in .NET Display data matrix barcodes in .NET Rational Function Interpolation

Rational Function Interpolation use visual studio .net gs1 datamatrix barcode drawer topaint 2d data matrix barcode with .net ASP.NET Some data Data Matrix for .NET are better interpolated by rational functions rather than polynomials. A rational function R(x) is the quotient of two polynomials: R(x) = Pm (x) a 1 x m + a 2 x m 1 + + a m x + a m+1 = Qn (x) b1 x n + b2 x n 1 + + bn x + bn+1.

Interpolation and Curve Fitting Because R( Visual Studio .NET data matrix barcodes x) is a ratio, it can be scaled so that one of the coef cients (usually bn+1 ) is unity. That leaves m + n + 1 undetermined coef cients that must be computed by forcing R(x) through m + n + 1 data points.

A popular version of R(x) is the so-called diagonal rational function, where the degree of the numerator is equal to that of the denominator (m = n) if m + n is even, or less by one (m = n 1) if m + n is odd. The advantage of using the diagonal form is that the interpolation can be carried out with a Neville-type algorithm, similar to that outlined in Table 3.2.

The recursive formula that is the basis of the algorithm is due to Stoer and Bulirsch4 . It is somewhat more complex than Eq. (3.

8) used in Neville s method: R[xi , xi+1 , . . .

, xi+k ] = R[xi+1 , xi+2 , . . .

, xi+k ] + where S= x xi x xi+k 1 R[xi+1 , xi+2 , . . .

, xi+k ] R[xi , xi+1 , . . .

, xi+k 1 ] R[xi+1 , xi+2 , . . .

, xi+k ] R[xi+1 , xi+2 , . . .

, xi+k 1 ] 1 (3.9b) R[xi+1 , xi+2 , . .

. , xi+k ] R[xi , xi+1 , . .

. , xi+k 1 ] S (3.9a).

In Eqs. (3 .9) R[xi , xi+1 , .

. . , xi+k ] denotes the diagonal rational function that passes through the data points (xi , yi ), (xi+1 , yi+1 ), .

. . , (xi+k , yi+k ).

It is understood that R[xi , xi+1 , . . .

, xi 1 ] = 0 (corresponding to the case k = 1) and R[xi ] = yi (the case k = 0). The computations can be carried out in a tableau, similar to Table 3.2 used for Neville s method.

An example of the tableau for four data points is shown in Table 3.3. We start by lling the column k = 1 with zeros and entering the values of yi in the column k = 0.

The remaining entries are computed by applying Eqs. (3.9).

k = 1 0 0 0 0 k=0 R[x1 ] = y1 R[x2 ] = y2 R[x3 ] = y3 R[x4 ] = y4 k=1 R[x1 , x2 ] R[x2 , x3 ] R[x3 , x4 ] k=2 R[x1 , x2 , x3 ] R[x2 , x3 , x4 ] k=3 R[x1 , x2 , x3 , x4 ]. x1 x2 x3 x4 Table 3.3 rational W .net framework Data Matrix 2d barcode e managed to implement Neville s algorithm with the tableau compressed to a one-dimensional array. This will not work with the rational function interpolation, where the formula for computing an R in the kth column involves entries in columns k 1 as well as k 2.

However, we can work with two one-dimensional arrays, one array (called r in the program) containing the latest values of R while the other array. Stoer, J., and Bulirsch, R., Introduction to Numerical Analysis, Springer, New York, 1980.

. 3.2 Polynomial Interpolation (rOld) sav es the previous entries. Here is the algorithm for diagonal rational function interpolation:. function y datamatrix 2d barcode for .NET Interp = rational(xData,yData,x) % Rational function interpolation; % returns value of the interpolant at x. % USAGE: yInterp = rational(xData,yData,x) % xData = x-coordinates of data points.

% yData = y-coordinates of data points.. m = length(xData);. r = yData; datamatrix 2d barcode for .NET rOld = zeros(1,m); for k = 1:m-1 for i = 1:m-k if x == xData(i+k) yInterp = yData(i+k); return else c1 = r(i+1) - r(i); c2 = r(i+1) - rOld(i+1); c3 = (x - xData(i))/(x - xData(i+k)); r(i) = r(i+1)+ c1/(c3*(1 - c1/c2) - 1); rOld(i+1) = r(i+1); end end end yInterp = r(1);.
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