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Lagrange Method
X =
Y =

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

[ Initial ArrayValues X: {1,2,3,4,5} ]
[ Initial ArrayValues Y: {1,4,9,16,25} ]
[ Initial LagrangeX Values specified: {2,3,1} ]

Lagrange Interpolation

The Lagrange interpolation is a classical technique for performing interpolation. Sometimes this interpolation is also called the polynomial interpolation. In the first order approximation, it reduces to a linear interpolation, a concept also applied in (see Cubic Spline Method).

Algorithm Creation

For a given set of n + 1 data points (x0,y0),(x1,y1),..., (xn,yn), where no two xi are the same, the interpolation polynomial in the Lagrange form is a linear combination:

y = f(x) = ∑i=0n li(x) f(xi)

Testing the Lagrange Method

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

           static void TestLagrange();
                 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 };
                 double[] y = Interpolation.Lagrange(xarray, yarray, x);
                 VectorR vx = new VectorR(x);
                 VectorR vy = new VectorR(y);
                 ListBox1.Items.Add(" " + vx.ToString());
                 ListBox2.Items.Add(" " + vy.ToString());

We first defined a set of data points as xarray and yarray. We then compute the y values at the xLagrange values specified. The user can manipulate all values and try variations on the arrays themselves as well as specifying new xLagrange values.

Other Implementations...

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Numerical Integration
Numerical Differentiation
Function Evaluation

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