Theory Behind the Approximation Functions for One Equation


Derivation of the Approximation Function for Real Solutions of One Equation

In this section, we will use definitions of the slope of a function to approximate the difference between a solution of the equation, and an estimate of the solution. From this, we will derive the the approximation functions for a single equation.

Suppose that we have an equation describing a quantity named w, and that this equation is restricted to equal zero. Because of the restriction, only certain values of w will make the equation true, and these values are the solutions of the equation:

w = Solution of equation

If the equation were not restricted to equal zero, but were free to equal any number, then it would be called a "function." In general, the quantity described by this function would not be restricted, but could vary freely. Let us define z as this variable, and F(z) as the function:

z = Unrestricted variable

F(z) = Unrestricted function

If z equals a solution of the restricted equation, then the function would return the number zero. Thus, we could express the equation to be solved as follows:

Equation: F(w) = 0

Example 1. If we were solving w4 - 5w3 + 11w - 78 = 0, then the unrestricted function would be F(z) = z4 - 5z3 + 11z - 78. Variable z can have any value, but solution w is restricted to four values.

We can assign a value to z as an estimate of solution w. Most likely, our estimate will not be perfect, so F(z) will not equal zero. We will define the difference between z and w as Dz, and the difference between F(z) and F(w) as DF:

Dz = z - w

DF = F(z) - F(w)

Since F(w) = 0, the increment DF is reduced to F(z):

DF = F(z)

While F(z) is easy to compute, we cannot compute Dz because we don't know solution w. However, we can approximate Dz by considering the slope of F(z). The slope of F(z) is defined by the first derivative dF/dz:

S(z) = dF/dz = limit as Dz®0   [F(z + Dz) - F(z)]/Dz

If we replace dF and dz with the increments F(z) and Dz, then we will have an approximation of S(z):

S(z) ~ F(z)/Dz

Multiplying both sides of the above by Dz/S(z) gives an approximation of Dz:

Dz ~ F(z)/S(z)

Earlier we defined Dz = z - w. We can use this definition to express w as

w = z - Dz

Substituting the approximation of Dz into the above gives us an approximation of w:

w ~ z - F(z)/S(z)

This is the approximation function for computing real solutions of one equation. The function returns a new approximation of w that will be used as the next estimate z, and the computations repeated until z converges.


Example 2. The equation and function defined in Example 1 were

w4 - 5w3 + 11w - 78 = 0

F(z) = z4 - 5z3 + 11z - 78

The slope of F(z) is defined by the derivative: S(z) = dF/dz = 4 z3 - 15 z2 + 11

The approximation function for computing real solutions of the equation is

w ~ z - [z4 - 5z3 + 11z - 78]/[4z3 - 15z2 + 11]

Comment: The formula for dF/dz can be found in this Table of Common Derivatives.


Graphic Representation of the Real-Valued Approximation Function

In this graph, the blue curve represents a function that crosses the z-axis at point [w, 0]. This point is a solution of F(w) = 0.

The green line represents a tangent touching the graph at point [z, F(z)]. This point represents an approximation of solution [w, 0]. The slope of the tangent is S(z).

The tangent crosses the z-axis at the point [z - F(z)/S(z), 0]. This point represents the next approximation of solution [w, 0].


Derivation of the Approximation Functions for Complex Solutions of One Equation

In the previous section, we found an approximation function for real solutions of one equation,

w ~ z - F(z)/S(z)

where w is a solution of the equation, z is an estimate, F(z) is a function based on the equation, and S(z) is the slope of F(z). We will now derive complex approximation functions by allowing w, z, F(z) and S(z) to have complex values:

w = u + iv

z = x + iy

F(z) = Fr + iFi

S(z) = Sr + iSi

Allowing functions F(z) and S(z) to be complex requires that we resolve them into their real and imaginary parts. Substituting the complex quantities into the approximation of w gives us

u + iv ~ x + iy - (Fr + iFi)/(Sr + iSi)

The last term, (Fr + iFi)/(Sr + iSi), is a fraction with a complex denominator. To resolve this fraction, we will multiply both numerator and denominator by the denominator's conjugate, (Sr - iSi):

(Fr + iFi)/(Sr + iSi) * (Sr - iSi)/(Sr - iSi)

= (Fr*Sr + Fi*Si)/(Sr2 + Si2) + i(Fi*Sr - Fr*Si)/(Sr2 + Si2)

We will substitute the resolved fraction into the approximation of u + iv to get

u + iv ~ x + iy - (Fr*Sr + Fi*Si)/(Sr2 + Si2) - i(Fi*Sr - Fr*Si)/(Sr2 + Si2)

Finally, we will separate the real and imaginary terms to get approximations for u and v:

u ~ x - (Fr*Sr + Fi*Si)/(Sr2 + Si2)

v ~ y - (Fi*Sr - Fr*Si)/(Sr2 + Si2)

These are the approximation functions for computing complex solutions of one equation. The functions return new approximations of u and v to be used as the next estimates x and y. The computations are repeated until x and y converge.


Example 3. The equation, function and slope defined in Examples 1 and 2 were

w4 - 5w3 + 11w - 78 = 0

F(z) = z4 - 5z3 + 11z - 78

S(z) = 4 z3 - 15 z2 + 11

The functions F(z) and S(z) can be resolved into Fr + iFi and Sr + iSi, where

Fr = R4 Cos(4q) - 5 R3 Cos(3q) + 11 R Cos(q) - 78

Fi = R4 Sin(4q) - 5 R3 Sin(3q) + 11 R Sin(q)

Sr = 4 R3 Cos(3q) - 15 R2 Cos(2q) + 11

Si = 4 R3 Sin(3q) - 15 R2 Sin(2q)

These functions can be substituted into the approximation functions, u ~ x - (Fr*Sr + Fi*Si)/(Sr2 + Si2)

v ~ y - (Fi*Sr - Fr*Si)/(Sr2 + Si2)

to compute better approximations of u and v from estimates x and y.

Comment: Variables R and q are the absolute value and imaginary argument of z.



Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.