MaplePrimes - answers and comments on Question, How many solutions?

4950

Kitonum — Wed, 12 Dec 2012 03:16:59 Z

It is easy to prove that in any interval RealRange(n, n+1) , where n is positive integer, function x -> frac(x*floor(x)) - 1/2 has exactly n roots. Thereforу in RealRange(1,100) there are add(n, n=1..99) = 4950 roots.

It is not so simple

Markiyan Hirnyk — Wed, 12 Dec 2012 11:34:49 Z

Thank you for your interest to the question and a partial answer to it. However,
if we consider RealRange(1,100+frac(evalf,Pi,20))) or/and frac((x+sin(10*x))*
floor(x+cos(10*sqrt(2)*x))), then your arguments do not work.
A Maple procedure proc(f::function, a::real, b::real,N::posint) is required which returns
the number of the solutions of f(x)=0 belonging to RealRange(a,b) if that number is
less than or equal to N and "The number of the solutions is greater than N" otherwise.

Another question

Kitonum — Wed, 12 Dec 2012 21:24:25 Z

@Markiyan Hirnyk

I already answered your original question (What is the number of all the solutions of the equation frac(x*floor(x)) = 1/2 belonging to RealRange(1,100)? ) . Gotten the exact result and specified the idea of solution. Why do you think my answer is partial?

Your new question a lot more difficult. If the function f(x)=frac(x*floor(x))-1/2 is the same, then the problem can be solved. For an arbitrary fuction f question is too general. In any case, create a new topic, in which accurately define a new question!

Putting the finishing touches

Markiyan Hirnyk — Wed, 12 Dec 2012 21:50:09 Z

@Kitonum You wrote: "It is easy to prove that in any interval RealRange(n, n+1) , where n is positive integer, function x -> frac(x*floor(x)) - 1/2 has exactly n roots". This most important part of your answer is not made with Maple.

PS. > assume(n::integer): simplify(floor(x)) assuming x >= n, x < n+1;

floor(x)
> is(floor(x)-n = 0) assuming x >= n, x < n+1;

FAIL

PPS. > fsolve(frac(x*floor(x)) = 1/2, x = 3 .. 4);

3.500000000

NULL

Man and machine

Kitonum — Wed, 12 Dec 2012 22:47:40 Z

@Markiyan Hirnyk

You see that Maple can not solve a much simpler example! It is therefore important to combine human and machine capabilities.

How about this?

Markiyan Hirnyk — Wed, 12 Dec 2012 22:55:17 Z

@Kitonum > sol := DirectSearch:-SolveEquations(frac(x*floor(x)) = 1/2, {x >= 3, x <=4}, AllSolutions);

PS.

The output - 95 solutions.