SERVING THE QUANTITATIVE FINANCE COMMUNITY

 
User avatar
Cuchulainn
Topic Author
Posts: 61589
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Going off on a tangent (kind of) : find y

May 17th, 2017, 11:26 am

[$]x = y - \varepsilon \:  sin y[$] where [$]x = 0.8[$], [$]\varepsilon = 0.2[$]
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
 
User avatar
ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 1:38 pm

maple gives y=0.9643338877

you can solve it by iteration
[$]y_{n+1}=0.8+0.2 \sin y_n[$]
when I tried that, it converges fairly quickly
 
User avatar
ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 2:03 pm

if I write it as [$]1-\epsilon=y-\epsilon \sin y[$]   (why? it feels right, that's y) and expand y as a series in [$]\epsilon[$], the first 4 terms gives 0.9644........
[$]y=y_0+\epsilon y_1 +\epsilon^2 y_2 +\cdots[$]

y0=1
y1= sin 1 -1
etc
 
User avatar
BigAndyD
Posts: 73
Joined: July 10th, 2013, 12:32 pm

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 3:29 pm

To what precision?
 
User avatar
Paul
Posts: 10489
Joined: July 20th, 2001, 3:28 pm

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 3:40 pm

And where does it come from? Possible clues are Cuch's obsessions: Joyce, Russians, numerics, exp(5),...Iceland? But most likely the title of the thread. To get [$]y[$] appearing as a linear and a trig term suggests something involving both triangles and circles...And why have [$]x[$] and [$]\epsilon[$] and not just the numbers? These questions are so much more interesting than 0.964... I'm afraid.
 
User avatar
Traden4Alpha
Posts: 23951
Joined: September 20th, 2002, 8:30 pm

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 4:04 pm

It's a sine he's caught in a vicus of reiteration and the hypotenoose is tightening!
 
User avatar
Cuchulainn
Topic Author
Posts: 61589
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 4:24 pm

And where does it come from? Possible clues are Cuch's obsessions: Joyce, Russians, numerics, exp(5),...Iceland? But most likely the title of the thread. To get [$]y[$] appearing as a linear and a trig term suggests something involving both triangles and circles...And why have [$]x[$] and [$]\epsilon[$] and not just the numbers? These questions are so much more interesting than 0.964... I'm afraid.
Actually ...these are the things to do for relaxation :D

I have deliberately scoped the problem to work with numbers for the moment, to focus the mind.
ppauper gets an A+ for solving the (initial) problem. We can build on his solution (I see about 7 other solutions offhand). One small question is when the eccentricity [$]\varepsilon > 1.[$]
The answer is in the stars :)
As ppauper says: It is developing skill to answer the question that is posed.
Last edited by Cuchulainn on May 17th, 2017, 4:41 pm, edited 4 times in total.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
 
User avatar
Cuchulainn
Topic Author
Posts: 61589
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 4:26 pm

It's a sine he's caught in a vicus of reiteration and the hypotenoose is tightening!
 You are one the right track. Get it working then right then optimised. It is a vicus problem, indeed.
Last edited by Cuchulainn on May 17th, 2017, 4:35 pm, edited 2 times in total.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
 
User avatar
Cuchulainn
Topic Author
Posts: 61589
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Re: Going off on a tangent (kind of) : find y

May 17th, 2017, 4:31 pm

To what precision?
ppauper has two solutions 1) Using Banach fixed point theorem (it is a contraction for [$]\varepsilon < 1[$]) and convergence is linear. It can be improved to second order by Aitken.
2) I don't know the answer in the asymptotic expansion case. But I am sure it is a piece of cake for many Wilmotters.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
 
User avatar
ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

Re: Going off on a tangent (kind of) : find y

May 18th, 2017, 10:43 am

One small question is when the eccentricity [$]\varepsilon > 1.[$]
The answer is in the stars :)
it's straightforward to do other expansions.

if you look at [$]x=y-\epsilon \sin y[$] with [$]x=1-\epsilon[$]

you can expand about [$]\epsilon=1[$], say [$]\epsilon=1+\mu[$]

and for large [$]\epsilon[$], then [$]1/\epsilon[$] is small and expand in that
 
User avatar
Cuchulainn
Topic Author
Posts: 61589
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

Re: Going off on a tangent (kind of) : find y

May 18th, 2017, 11:08 am

I get the same answers as ppauper's findings. Fixed point needs ~ 6 iterations while Aitken does it in  3 for a tolerance of 1.0e-9.

When the eccentricity is 1.5 fixed point needs 36 iteration and Aitken in 6.

The fixed point iteration hangs when [$]\varepsilon = 2[$]. Aitken and Bessel's original solution give 2.29645562918859.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
 
User avatar
ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

Re: Going off on a tangent (kind of) : find y

May 18th, 2017, 2:03 pm

about eps=2.131615 something happens
the curve looks sort of like a cubic and it goes from having 1 rea solution to 3 real solutions
 
User avatar
Paul
Posts: 10489
Joined: July 20th, 2001, 3:28 pm

Re: Going off on a tangent (kind of) : find y

May 18th, 2017, 2:12 pm

Can someone explain why this is of any interest at all?!
 
User avatar
Traden4Alpha
Posts: 23951
Joined: September 20th, 2002, 8:30 pm

Re: Going off on a tangent (kind of) : find y

May 18th, 2017, 2:14 pm

Note that a Russian engineer would say the solution does not exist. The "y" inside the sine must have units of an angle but sin(y) will be a unit-less ratio of distances that cannot be subtracted from an angle (the lead "y").
 
User avatar
Traden4Alpha
Posts: 23951
Joined: September 20th, 2002, 8:30 pm

Re: Going off on a tangent (kind of) : find y

May 18th, 2017, 2:15 pm

Can someone explain why this is of any interest at all?!
Banach has appealch?
ABOUT WILMOTT

PW by JB

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...


Twitter LinkedIn Instagram

JOBS BOARD

JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...


GZIP: On