Backward Difference Method - Keystone Mining Post

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Numerical Difference

~ Forward Difference

~ Backward Difference

~ Richardson Extrap

"Numerical Differentiation"

Backward Difference Method

Derivatives for f(x) = sin(x)

Y-Array =

Derivatives for Y Array values:

Specify New Y Array size:

Specify New delegate x values = { , }

[ Initial delegate function: x = {7) ]
(s/b within y-array size)
[ Initial Y index: {7} ]
(selected delegate function: x values will match Y index values)
[ Initial Y Array: {15} ]

IMPLEMENTATION
Backward Difference Method

The backward finite difference can be obtained from the equation shown under "Overview - Numerical Differentiation" tab. The first four derivatives of the function f(x) are given by the following formulas:

f'(x) = f(x + h) - f(x)
                 h
f''(x) = f(x + 2h) - 2f(x + h) + f(x)
                      h²
f'''(x) = f(x + 3h) - 3f(x + 2h) + 3(f + h) - f(x)
                            h³
f⁽⁴⁾(x) = f(x + 4h) - 4f(x + 3h) + 6f(x + 2h) - 4f(x + h) + f(x)
                                    h⁴

The above equations are usually not used to compute derivatives because they have a large truncation error (order of O(h)). The common practice is to use expressions of O(h²). To obtain backward difference formulas of this order, it is necessary to retain more terms in the Taylor series. Below are listed the results (without derivations):

f'(x) = -3f(x) + 4f(x + h) - f(x + 2h)
                     2h
f''(x) = 2f(x) - 5f(x + h) + 4f(x + 2h) - f(x +3h)
                             h²
f'''(x) = -5f(x) + 18f(x + h) - 24f(x + 2h) + 14f(x +3h) - 3f(x +4h)
                                      2h³
f⁽⁴⁾(x) = 3f(x) - 14f(x + h) + 26f(x + 2h) - 24f(x +3h) + 11f(x +4h) - 2f(x + 5h)
                                                  h⁴

Each derivative has two overloaded methods: one for the function and another for the array of data values. In order to calculate derivatives for a function, one has to pass a delegate function f(x), the value of x at which the derivative is computed, and the increment h as input parameters.

On the other hand, in order to calculate derivatives for an array of data values, each method has input parameters which pass the yarray values, the index called yindex which specifies the location at which the derivative is computed, and the increment h. Here, the increment parameter is the value used to generate the function values.

The x increment defined as h = 1 and created the Y array of the sine function using this increment. Then calculated the first four derivatives directly for the delegate function at x = 7. Derivatives for Y array also at x = 7, which corresponds to yindex = 7.

Running this example generates the results above. The reader will see that derivatives for the function and for the array values, as expected, are almost identical. Slight differences only because numerical round-off error. Users can manipulate function and array values.

Testing the Backward Difference Method

To test the Backward Difference method, a new static method has been added to show how to calculate the first four derivatives for f(x) = Sin x. The TestBackwardMethod() method has been written, added and executed:

           static void TestBackwardMethod();
              {
                 double h = 0.1;
                 double[] yarray=new double[10];
           // create yarray:
                 for (int i = 0; i < 10; i++)
                    yarray[i] = f(i * h);
           // Calculate derivatives for function:
                     double dy = Differentiation.Backward(f, 7, h);
                     listBox1.Items.Add("f'(x) = " + dy.ToString());
                     dy = Differentiation.Backward2(f, 7, h);
                     listBox1.Items.Add("f''(x) = " + dy.ToString());
                     dy = Differentiation.Backward3(f, 7, h);
                     listBox1.Items.Add("f'''(x) = " + dy.ToString());
                     dy = Differentiation.Backward4(f, 7, h);
                     listBox1.Items.Add("f''''(x) = " + dy.ToString());
           // Calculate derivatives for array values:
                 foreach (double y1 in yarray)
                     listBox2.Items.Add(" " + y1.ToString());
                     dy = Differentiation.Backward1(yarray, 7, h);
                     listBox3.Items.Add("y' = " + dy.ToString());
                     dy = Differentiation.Backward2(yarray, 7, h);
                     listBox3.Items.Add("y'' = " + dy.ToString());
                     dy = Differentiation.Backward3(yarray, 7, h);
                     listBox3.Items.Add("y''' = " + dy.ToString());
                     dy = Differentiation.Backward4(yarray, 7, h);
                     listBox3.Items.Add("y'''' = " + dy.ToString());
              }
           static double f(double x)
              {
              return Math.Sin(x);
              }