Theory Behind the Approximation Functions for Several Equations

In this section, we will use differential equations to approximate the difference between a solution of several equations, and an estimate of the solution. From this, we will derive the approximation functions for the system of equations. The derivation refers to a system of three equations, but it applies to any number of equations that relate an equal number of quantities.

Suppose that we have three equations describing the quantities w₁, w₂ and w₃, and that the equations are restricted to equal zero. Because of the restriction, the quantities w₁, w₂ and w₃ are also restricted to certain sets of numbers. These sets are the solutions of the equations. We'll define W as a set of numbers that is a valid solution:

If the equations were not restricted to equal zero, but were free to equal any number, then they would be called "functions." In general, the quantities described by these functions would not be restricted, but could vary independently. Let us define Z as the set of variables, and F(Z) as the set of functions:

If Z equals a solution W, then each of the functions would return the number zero. Thus, we could describe the three equations to be solved as F(W) = (0, 0, 0):

Example 1. Suppose that we need to solve this set of equations:

These equations can be described by F(W) = (0, 0, 0). The elements of unrestricted F(Z) are

(End of Example 1)

We can assign numbers to Z as an estimate of solution W. Most likely, our estimate will not be perfect, so F(Z) will not equal (0, 0, 0). We will define the difference between Z and W as DZ, and the difference between F(Z) and F(W) as DF:

Since F(W) = (0, 0, 0), the increment DF is reduced to F(Z):

This system of differential equations can be written as a matrix product:

The matrix of partial derivatives is called the Jacobian of F(Z) and Z. We will represent this matrix by J(Z), where the (Z) indicates that the elements are computed with estimate Z. If we also represent the column-vectors of differentials by dF and dZ, then we can express the matrix-equation in a single line:

Example 2. The table below shows the sets of restricted equations and corresponding unrestricted functions presented in Example 1:

The differentials of the functions are

We can write the differentials as a matrix expression, dF = J(Z)*dZ:

(End of Example 2)

Next, we will substitute the increments DZ and F(Z) for the differentials dZ and dF to get an approximation relating F(Z) and DZ:

At this stage, we'll restrict F(Z), J(Z), and DZ to real values. Let 1/J be the inverse of matrix J(Z), and left-multiply both sides of the approximation by 1/J:

This gives us an approximation of DZ:

Earlier we defined DZ = Z - W. We can use this definition to express W as

Substituting the approximation of DZ into the above gives us an approximation of W:

This is the approximation function for computing real solutions of several equations. The function returns a new approximation of W to be used as the next estimate Z. The computations are repeated until Z converges.

In the previous section, we derived an approximation function for real solutions of several equations. The derivation included the approximation