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 w_{1}, w_{2} and w_{3}, and that the equations are restricted to equal zero. Because of the restriction, the quantities w_{1}, w_{2} and w_{3} 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:
F(Z) = ( F_{1}(Z), F_{2}(Z), F_{3}(Z) ) = Unrestricted 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
Example 1. Suppose that we need to solve this set of equations:
7w_{1} w_{2}^{2} - 3w_{2} w_{3}^{4} + 7 = 0
w_{1}^{3}w_{2} + w_{1}^{4}w_{3}^{2} - 9 = 0
4w_{1}^{2}w_{3}^{2} + 2w_{2}^{3}w_{3} - 37 = 0
These equations can be described by F(W) = (0, 0, 0). The elements of unrestricted F(Z) are
F_{1} = 7z_{1} z_{2}^{2} - 3z_{2} z_{3}^{4} + 7
F_{2} = z_{1}^{3}z_{2} + z_{1}^{4}z_{3}^{2} - 9
F_{3} = 4z_{1}^{2}z_{3}^{2} + 2z_{2}^{3}z_{3} - 37
(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
DZ = Z - W
DF = F(Z) - F(W)
Since F(W) = (0, 0, 0), the increment DF is reduced to 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 using the differential equations of F(Z). The differentials of F(Z) are defined by the partial derivatives of F(Z), and the differentials of Z:
dF_{2} = [dF_{2}/dz_{1}]dz_{1} + [dF_{2}/dz_{2}]dz_{2} + [dF_{2}/dz_{3}]dz_{3}
dF_{3} = [dF_{3}/dx_{1}]dz_{1} + [dF_{3}/dz_{2}]dz_{2} + [dF_{3}/dz_{3}]dz_{3}
This system of differential equations can be written as a matrix product:
dF_{1} dF_{2} dF_{3} | = |
dF_{1}/dz_{1} dF_{2}/dz_{1} dF_{3}/dz_{1} |
dF_{1}/dz_{2} dF_{2}/dz_{2} dF_{3}/dz_{2} |
dF_{1}/dz_{3} dF_{2}/dz_{3} dF_{3}/dz_{3} | * |
dz_{1} dz_{2} dz_{3} |
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:
Restricted Equations | Unrestricted Functions |
7w_{1} w_{2}^{2} - 3w_{2} w_{3}^{4} + 7 = 0 | F_{1} = 7z_{1} z_{2}^{2} - 3z_{2} z_{3}^{4} + 7 |
w_{1}^{3}w_{2} + w_{1}^{4}w_{3}^{2} - 9 = 0 | F_{2} = z_{1}^{3}z_{2} + z_{1}^{4}z_{3}^{2} - 9 |
4w_{1}^{2}w_{3}^{2} + 2w_{2}^{3}w_{3} - 37 = 0 | F_{3} = 4z_{1}^{2}z_{3}^{2} + 2z_{2}^{3}z_{3} - 37 |
The differentials of the functions are
dF_{1} = [7z_{2}^{2}] dz_{1} + [14z_{1} z_{2} - 3z_{3}^{4}] dz_{2} + [-12z_{2} z_{3}^{3}] dz_{3}
dF_{2} = [3z_{1}^{2} z_{2} + 4z_{1}^{3} z_{3}^{2}] dz_{1} + [z_{1}^{3}] dz_{2} + [2z_{1}^{4} z_{3}] dz_{3}
dF_{3} = [8z_{1} z_{3}^{2}] dz_{1} + [6z_{2}^{2} z_{3}] dz_{2} + [8z_{1}^{2} z_{3} + 2z_{2}^{3}] dz_{3}
We can write the differentials as a matrix expression, dF = J(Z)*dZ:
dF_{1} dF_{2} dF_{3} | = |
7 z_{2}^{2} 3 z_{1}^{2} z_{2} + 4z_{1}^{3} z_{3}^{2} 8 z_{1} z_{3}^{2} |
14 z_{1} z_{2} - 3 z_{3}^{4} z_{1}^{3} 6 z_{2}^{2} z_{3} |
-12 z_{2} z_{3}^{3} 2 z_{1}^{4} z_{3} 8z_{1}^{2} z_{3} + 2 z_{2}^{3} | * |
dz_{1} dz_{2} dz_{3} |
(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
where F(Z) is a column-vector of functions based on the set of equations, J(Z) is the Jacobian matrix, and DZ is a column-vector of increments between estimate Z and solution W.
We will now derive complex approximation functions by allowing W, Z, DZ, F(Z) and J(Z) to have complex values:
Z = X + iY = (x_{1}, x_{2}, x_{3}) + i(y_{1}, y_{2}, y_{3})
F(Z) = Fr + iFi = (Fr_{1}, Fr_{2}, Fr_{3}) + i(Fi_{1}, Fi_{2}, Fi_{3})
J(Z) = Jr + iJi
DZ = DX + iDY = (Dx_{1}, Dx_{2}, Dx_{3}) + i(Dy_{1}, Dy_{2}, Dy_{3})
Since DZ = Z - W, we can also write the relations
DY = Y - V
Example 3. Referring to the functions in Example 1, we will resolve the first element of F(Z) into its real and imaginary parts. Note that R and q represent the absolute value and imaginary argument of a variable:
Fr_{1} = 7R_{1} R_{2}^{2}Cos(q_{1} + 2q_{2}) - 3R_{2} R_{3}^{4}Cos(q_{2} + 4q_{3}) + 7
Fi_{1} = 7R_{1} R_{2}^{2}Sin(q_{1} + 2q_{2}) - 3R_{2} R_{3}^{4}Sin(q_{2} + 4q_{3})
The other elements of F(Z) and the elements of J(Z) are resolved in similar manner.
(End of Example 3)
Now we will return to the approximation F(Z) ~ J(Z)*DZ, and substitute the complex quantities:
Multiplying the terms on the right-hand-side yields
The real and imaginary terms can be separated into two approximations:
Fi ~ Ji*DX + Jr*DY
We will use these approximations to find expressions for DX and DY. To solve for DX, we left-multiply the approximation of Fr by 1/Ji, and left-multiply the approximation of Fi by 1/Jr:
(1/Jr)*Fi ~ (1/Jr)*Ji*DX + (1/Jr)*Jr*DY
This gives us
(1/Jr)*Fi ~ (1/Jr)*Ji*DX + DY
Adding these approximations will eliminate DY:
The term [(1/Ji)*Fr + (1/Jr)*Fi] on the left-hand-side is a column vector, and the term
Now we'll return to the approximations of Fr and Fi, and find an expression for DY. We left-multiply the approximation of Fr by 1/Jr, and left-multiply the approximation of Fi by 1/Ji:
(1/Ji)*Fi ~ (1/Ji)*Ji*DX + (1/Ji)*Jr*DY
This gives us
(1/Ji)*Fi ~ DX + (1/Ji)*Jr*DY
Subtracting the top approximation from the bottom will eliminate DX:
The term [(1/Ji)*Fi - (1/Jr)*Fr] on the left-hand-side is a column vector, and the term
Earlier we defined DX as DX = X - U, and DY as
V = Y - DY
Substituting the approximations of DX and DY into the above gives us approximations of U and V:
V ~ Y - 1/[(1/Ji)*Jr + (1/Jr)*Ji] * [(1/Ji)*Fi - (1/Jr)*Fr]
These are the approximation functions for computing complex solutions of several equations. 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.
Comment: In the procedure for finding complex solutions of several equations, and in the Sample Excel Sheet accompanying this procedure, the vectors and matrices were assigned the following nicknames:
matrix 1/Marie = 1/[(1/Ji)*Jr + (1/Jr)*Ji]
vector Vince = [(1/Ji)*Fr + (1/Jr)*Fi]
vector Vito = [(1/Ji)*Fi - (1/Jr)*Fr]
URL http://members.aceweb.com/patrussell/approximations/Theory2.htm
Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.