difficult inequality to solve
显示 更早的评论
abs((cos(x)+1/2*x^2)-1)*x^4)<=1/24, does anybody have any idea how to solve this on matlab or write down some possible codes?
采纳的回答
since your function is even it is symmetric and it is monotone for x>0 or x<0
use fzero
>> f=@(x)((cos(x)+1./2.*x.^2)-1).*x.^4 - 1/24 % subtract the value to transform it into an issue of roots
>> solution=abs(fzero(f,0))
solution =
1.0042
>> range=[-solution solution]
range =
-1.0042 1.0042
19 个评论
bk97
2017-1-25
Niels
2017-1-25
you are missing a * before x^4
Niels
2017-1-25
and the abs does not change anything
bk97
2017-1-25
Niels
2017-1-25
the answer is still the same
Niels
2017-1-25
maybe you meant
(cos(x)+1./ ->(2.*x.^2)<-)-1
and not
(cos(x)+ (1./2) .*x.^2)-1 ??
bk97
2017-1-25
Niels
2017-1-25
nope
it does not:
>> f=@(x)abs(((cos(x)+1./2.*x.^2)-1).*x.^4) - 1/24 % subtract the value to transform it into an issue of roots
solution=abs(fzero(f,0))
range=[-solution solution]
f =
function_handle with value:
@(x)abs(((cos(x)+1./2.*x.^2)-1).*x.^4)-1/24
solution =
1.0042
range =
-1.0042 1.0042
Niels
2017-1-25
since your function is >0 for all x the abs does not change anything anyway, so it is pretty unnecessary
bk97
2017-1-25
Walter Roberson
2017-1-25
The abs() makes a difference if you are expecting complex numbers for input.
bk97
2017-1-25
Walter Roberson
2017-1-26
Are you looking for real valued solutions or complex valued solutions?
bk97
2017-1-26
Walter Roberson
2017-1-26
There are an infinity of complex solutions, symmetric in positive and negative real components, and symmetric in positive and negative imaginary components. The boundaries on the imaginary components are +/- -.9958714409867068 approximately and the boundaries on the real components are +/- -1.004205445912837 approximately.
The area is not circular in real and imaginary components, but it is approximately circular.
For any given real component inside the given range, there are two imaginary components that lead to solution; likewise for any given imaginary component within the range, there are two real components that lead to solution.
When I talk about infinity of solutions, I am referring just to the boundary; because if the inequality, everything within the boundary is included too.
The boundary is the solutions for
(1/2)*sqrt((2*cosh(2*b)+2*cos(2*a)+(4*a^2-4*b^2-8)*cos(a)*cosh(b)-8*a*b*sin(a)*sinh(b)+a^4+(2*b^2-4)*a^2+b^4+4*b^2+4)*(a^2+b^2)^4)-1/24
where a is the real component and b is the imaginary component.
bk97
2017-1-26
Walter Roberson
2017-1-26
The +/- -1.004205445912837 I show occurs when the imaginary component is 0. It is a higher precision version of the value that Niels posted. The code Niels posted is fine.
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