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"Interpolation Solutions"
Newton Divided Difference Method
X =
Y =

Array X = { , , , , }
Array Y = { , , , , }
Specify New X values = { , , , }



[ Initial ArrayValues X: {1970,1980,1990,2000,2010} ]
[ Initial ArrayValues Y: {150697,179323,203212,226505,249633} ]
[ Initial Newton-Diff-X Values specified: {1975,1985,1995, 2005} ]

IMPLEMENTATION
Newton Divided Difference Interpolation

The Newton divided difference interpolation is the interpolation polynomial approximation for a given set of data points in the Newton form (i.e. uses the Taylor expansion to perform the interpolation). Basically, the divided differences are used to approximately calculate the various derivatives.

Algorithm Creation

The general form of the Newton divided difference polynomial for a given set of (n + 1) data points, (x0,y0),(x1,y1),...,(xn,yn) which can be written as,

fn(x) = c0 + c1(x - x0) + ... cn(x - x0)(x - x1)... (x - xn-1)
where,
c0 = f[x0]
c1 = f[x1,x0]
.
.
.
cn = f[xn,xn-1, ...,x0]

from above definition, the Newton differences are calculated recursively.



Testing the Newton Divided Difference Method

In order to test the Spline method as defined above, a new TestNewtonDividedDifference() static method has been added and executed. Supporting code and methods are not shown.

           static void TestNewtonDividedDifference();
              {
                 ListBox1.Items.Clear();
                 ListBox2.Items.Clear();
                 double[] xarray = new double[] { t1, t2, t3, t4, t5 };
                 double[] yarray = new double[] { t6, t7, t8, t9, t10 };
                 double[] x = new double[] { t11, t12, t13, t14 };
                  double[] y = Interpolation.NewtonDividedDifference(xarray, yarray, x);
                 VectorR vx = new VectorR(x);
                 VectorR vy = new VectorR(y);
                 ListBox1.Items.Add(" " + vx.ToString());
                 ListBox2.Items.Add(" " + vy.ToString());
              }

This type of interpolation is used in the mining reserve estimation process. It is also, however, heavily used in statistics and as example we show a set of data points as xarray and yarray which represent the census years from 1970 to 2010 and their corresponding population in millions of people. We can now use the Newton divided difference method to interpolate the census data to estimate population in 1975,1985,1995 and 2005. The user can manipulate all values and try variations on the arrays themselves as well as specifying new estimate values.



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