Client Account:   Login
Home Site Statistics   Contact   About Us   Sunday, March 26, 2017

users on-line: 2 | Forum entries: 6   
j0110924 - Back to Home
   Skip Navigation LinksHOME › AREAS OF EXPERTISE › Interpolation Applications › ~ Newton Divided Difference Method

"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} ]

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)
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();
                 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.

Other Implementations...

Object-Oriented Implementation
Graphics and Animation
Sample Applications
Ore Extraction Optimization
Vectors and Matrices
Complex Numbers and Functions
Ordinary Differential Equations - Euler Method
Ordinary Differential Equations 2nd-Order Runge-Kutta
Ordinary Differential Equations 4th-Order Runge-Kutta
Higher Order Differential Equations
Nonlinear Systems
Numerical Integration
Numerical Differentiation
Function Evaluation

Skip Navigation Links.

Home Skip Navigation Links
   Algorithms, Graphics, Vectors,
            implementation techniques.
   Mineral Transactions info,
            sales, agreements...
   Numerical Modeling services
            mineral environment.
   Want to know about Mining?
            basic knowledge here...
   What are Mineral Commodities?
            our elementary charts.
   Math, Analysis and More...
            our expertise in the matter.


Platform Implementation

Home Algorithm Implementation
We design applications for different environments and platforms...
        Home Graphics and Animation
The graphics classes in Smalltalk were designed...
        Home Optimization Algorithms
An optimization problem is a numerical problem...
        Home Vectors and Matrices
The concise notation introduced in linear algebra for vector...

2017 © Keystone Mining Post  |   2461 E. Orangethorpe Av., Fullerton, CA 92631 USA  |