The utility functions
and
show search data for
FindMinimum and
FindRoot for one- and two-dimensional functions. They work with essentially the same arguments as
FindMinimum and
FindRoot except that they additionally take options, which affect the graphics functions they call to provide the plots, and they do not have the
HoldAll attribute as do
FindMinimum and
FindRoot.
Note that to simplify processing and reduce possible confusion about the function
f,
does not accept equations; it finds a root
.
Steps and evaluation points are color coded for easy detection as follows:
- plot is the graphics object shown.
This shows in two dimensions the steps and evaluations used by
FindMinimum to find a local minimum of the function
starting at the point
. Options are given to
ContourPlot so that no contour lines are shown and the function value is indicated by grayscale. Since
FindMinimum by default uses the
"quasi-Newton" method, there are only evaluations of the function and gradient that occur at the same points, indicated by the red circles with green centers.
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This shows in two dimensions the steps and evaluations used by
FindMinimum to find a local minimum of the function
starting at the point
. Since the problem is a sum of squares,
FindMinimum by default uses the "
Gauss-Newton"/Levenberg-Marquardt method that derives a residual function and only evaluates it and its Jacobian. Points at which the residual function is evaluated are shown with yellow dots. The yellow dots surrounded by a large purple circle are points at which the Jacobian was evaluated as well.
| Out[3]= |  |
This shows in two dimensions the steps and evaluations used by
FindMinimum to find a local minimum of the function
starting at the point
using
"Newton's" method. Points at which the function, gradient, and Hessian were all evaluated are shown by concentric green, red, and cyan circles. Note that in this example, all the Newton steps satisfied the Wolfe conditions, so there were no points where the function and gradient were evaluated separately from the Hessian, which is not always the case. Note also that Newton's method finds a different local minimum than the default method.
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This shows the steps and evaluations used by
FindMinimum to find a local minimum of the function
with two starting values superimposed on the plot of the function. Options are given to
Plot so that the curve representing the function is thick and purple. With two starting values,
FindMinimum uses the derivative-free principal axis method, so there are only function evaluations, indicated by the green dots.
| Out[5]= |  |
This shows in two dimensions the steps and evaluations used by
FindRoot to find a root of the function
starting at the point
. As described earlier, the function is a residual, and the default method in
FindRoot evaluates the residual and its Jacobian as shown by the yellow dots and purple circles. Note that this plot is nearly the same as the one produced by
with the default method for the function
since the residual is the same.
also shows the zero contour of each component of the residual function in red and green.
| Out[6]= |  |
, superimposed on a plot of f versus x; opts may include options from both FindMinimum and Plot
, superimposed on a contour plot of f as a function of x and y; opts may include options from both FindMinimum and ContourPlot
, superimposed on a plot of f versus x; opts may include options from both FindRoot and Plot
, superimposed on a contour plot of the merit function f as a function of x and y; opts may include options from both FindRoot and ContourPlot
