Minimization
------------

The `a N' (`calc-find-minimum') [`minimize'] command finds a minimum
value for a formula.  It is very similar in operation to `a R'
(`calc-find-root'): You give the formula and an initial guess on the
stack, and are prompted for the name of a variable.  The guess may be
either a number near the desired minimum, or an interval enclosing the
desired minimum.  The function returns a vector containing the value
of the the variable which minimizes the formula's value, along with
the minimum value itself.

Note that this command looks for a *local* minimum.  Many functions
have more than one minimum; some, like `x sin(x)', have infinitely
many.  In fact, there is no easy way to define the "global" minimum of
`x sin(x)' but Calc can still locate any particular local minimum for
you.  Calc basically goes downhill from the initial guess until it
finds a point at which the function's value is greater both to the
left and to the right.  Calc does not use derivatives when minimizing
a function.

If your initial guess is an interval and it looks like the minimum
occurs at one or the other endpoint of the interval, Calc will return
that endpoint only if that endpoint is closed; thus, minimizing `17 x'
over `[2..3]' will return `[2, 38]', but minimizing over `(2..3]'
would report no minimum found.  In general, you should use closed
intervals to find literally the minimum value in that range of `x', or
open intervals to find the local minimum, if any, that happens to lie
in that range.

Most functions are smooth and flat near their minimum values.  Because
of this flatness, if the current precision is, say, 12 digits, the
variable can only be determined meaningfully to about six digits.
Thus you should set the precision to twice as many digits as you need
in your answer.

The `H a N' [`wminimize'] command, analogously to `H a R', expands the
guess interval to enclose a minimum rather than requiring that the
minimum lie inside the interval you supply.

The `a X' (`calc-find-maximum') [`maximize'] and `H a X' [`wmaximize']
commands effectively minimize the negative of the formula you supply.

The formula must evaluate to a real number at all points inside the
interval (or near the initial guess if the guess is a number).  If the
initial guess is a complex number the variable will be minimized over
the complex numbers; if it is real or an interval it will be minimized
over the reals.