MaplePrimes - answers and comments on Question, Multiple extrema

Multiple extrema

Robert Israel — Wed, 03 Nov 2010 18:57:47 Z

Is your curve defined by an expression or function? You should be able to take an interval around each maximum, small enough to contain only one critical point, and use fsolve to find the critical point in that interval. Something like:

seq(fsolve(diff(F, x) = 0, x = J), J = [ 687 .. 690, 705 .. 710, 736 .. 740, 758 .. 760 ]);

with(DirectSearch)

hirnyk — Wed, 03 Nov 2010 20:30:38 Z

Try to find all the maximums by the DirectSearch package (see http://www.maplesoft.com/applications/view.aspx?SID=87637 ) . As far as I understand it, this package should work with splines too. Unfortunately, I have not got any such an example, so the following example, where the function has four maximums on 0..25, is attached:
f := x-> (1/100)*x^2+sin(x) :

plot(f, 0 .. 25);

with(DirectSearch):

GlobalSearch(f, {x >= 0, x <= 25}, maximize = true):

op(3, %);

(1, 1) = 6.1176482495205, (1, 2) = (Vector(1, {(1) = 24.9999999997440128})),
(1, 3) = 20, (2, 1) = 5.2563376489113,
(2, 2) = (Vector(1, {(1) = 20.8505054944515998})), (2, 3) = 18, (3, 1) = 3.03967891389065,
(3, 2) = (Vector(1, {(1) = 14.4299297855150002})), (3, 3) = 19, (4, 1) = 1.62946677752837,
(4, 2) = (Vector(1, {(1) = 8.01497572679684112})),
(4, 3) = 32, (5, 1) = 1.02517760626473,
(5, 2) = (Vector(1, {(1) = 1.60285900150238891})), (5, 3) = 24

maximums.mw

Spline

gotamo — Wed, 03 Nov 2010 19:01:33 Z

Hey robert,
the curve is defined by a spline function. i would like to keep the solution as general as possible, since the position of the maxima might shift in later applications.

GlobalSearch

gotamo — Thu, 04 Nov 2010 13:21:44 Z

indeed, the GlobalSearch package works also with spline functions. thanks! i am a bit puzzled with the output: for instance in the example that you created: what does the last command do (op(3,%))? and probably related to that: what is the meaning of the third vector component ((1,1)denotes y, (1,2)=x coordinate but (1,3)=20?)?

Number of steps

hirnyk — Thu, 04 Nov 2010 14:03:15 Z

@gotamo The element (1,3)=20 shows the number of steps, made by the GlobalSearch command to obtain this solution.

Appendix. I could not copy and paste the output of the GlobalSearch command, therefore I extracted its part by op(3,%) and copied and pasted that. See ?op for details.

Example with splines

hirnyk — Sat, 06 Nov 2010 11:55:04 Z

is obtained by the spline approximation of the function f :splines.mw