EXPLANATION FILE OF PROGRAM MUELLER2
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Two-Dimensional Form of Mueller's Method
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In this section, Mueller's method for one dimension is combined with the
concept of successive substitution to produce an algorithm that can be called
on to find a root of the function µ (x,y). We will construct this algorithm
by considering each dimension separately.
The starting point is the initial guess and two bounds: B1 for the X direc-
tion and B2 for the Y direction. The first of these bounds is used to genera-
te three evauation points:
X3 = X0
X1 = X0 - B1
X2 = X0 + B1
Y1, Y2, Y3 = Y0
Mueller's method for one dimension is then applied to find a better estimate
for the X coordinate of the root, Xnew. B1 is replaced with |Xnew - X3|.
For the iteration in the Y direction, the three evaluation points are
Y3 = Y0
Y1 = Y0 - B1
Y2 = Y0 + B1
X1, X2, X3 = Xnew
The one-dimensional form of Mueller's method is again used to find a new es-
timate for the Y coordinate of the root, Ynew. As with the X iteration, B2 is
replaced with |Ynew - Y3|. If the change in X and Y satisfies the prechosen
error criterion,
|B1| + |B2| < E
then the iteration is ended. Otherwise, X0 --> Xnew, Yo --> Ynew, and the
procedure is repeated.
A program for performing these calculations is shown in MUELLER2. It is ba-
sed primarily on the routine given in the previous section (see Mueller.txt).
It requires as inputs an initial guess for the location of the root, (X0, Y0);
two measures of the bound on the range of the starting two-dimensional parabo-
la (B1 for the X direction and B2 for the Y direction); a convergence crite-
rion, E; and a bound on the maximum number of iterations to be performed, N.
The outputs of the subroutine are the estimate of the location of the root
(X, Y), and the actual number of iterations performed, K.
To demonstrate the basic operation of MUELLER2, the following function was
examined:
5 5
µ (x,y) = (x + 1) (y - 1)
As expected from the example in the previous section, convergence occurs,
but it is not rapid. The reason for this is that the curvature of the Function
is high (quintic) because of the multiple roots.
Another function one might inadvertently attempt to solve is
µ (x,y) = x² - y² + 1
This is the real part of the complex function f(z) = Z² + 1. MUELLER2 comple-
tely fails in this exercise by diverging toward infinity, and eventually rea-
ching a numeric overflow. The reason for the failure is that the function has
no unique root! Such errors in the key inputs are not guarded against by this
subroutine.
The precautions regarding the use of MUELLER2 center on the two-dimensional
shape of the function. If the calculation is started in the wrong place, the
iteration can diverge. Also, the algorithm is particularly vulnerable to the
divergence associated with roots that occur at or near saddle points or local
minima, although recovery is possible because of the error checks within the
subroutine.
From [BIBLI 01].
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End of file Mueller2.txt