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

Let's round this off..

1. For $x$ not a multiple of $\pi$ the problem has a unique solution, contradicting earlier incorrect statements.
2. The Bessel function expansion 1817 converges to the wrong value (and slow) for large $\varepsilon$.
3. The only hope is to write the equation as a least squares interval search (the function is unimodal). And very fast.
4. I also have no hope for Newton Raphson et al either.

example
$\varepsilon = 13.5, x = 0.8$,
$y = -0.064047..$

You can check by plugging this value into the original equation.
.
//
$\varepsilon = 500.0, x = 0.8$ ??
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Jean Piaget

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

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

Finding all the solutions numerically doesn't seem that hard:

1) The overall bounds on the range of y where solutions might be found is a simple function of x and eccentricity.
2) Enumerating the local high and low extrema in x of each oscillation of the sinusoid defines a set of search intervals as well as a pretty decent starting guess for the likely solution in that interval.
3) Then it's a matter of using an iterative finder that stays within the bounds.

There's some housekeeping in starting with a liberal bounds and possibly culling some cycles. There's also a detectable numerical instability where the target value of x is approaching one of the sine's extrema and round-off error in x resolves as large excursions in y.

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

I don't agree. The problem is solved. Besides you have not produced a result to compare against. If you try it you will see what works and what not.

Don't use iterative methods..as mentioned.

See how Lagrange and Bessel tackle this problem.

http://eaton.math.rpi.edu/faculty/Kovac ... Bessel.pdf
Last edited by Cuchulainn on May 19th, 2017, 6:14 pm, edited 1 time in total.
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Jean Piaget

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

defines a set of search intervals as well as a pretty decent starting guess for the likely solution in that interval.

Not exactly. I am using a least squares optimisation for a unimodal function, so 'guess' is irrelevant and not needed.  More precisely, one brackets the solution. Fibonacci., golden mean and evolution algo give the same answer.
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

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

I don't agree. The problem is solved. Besides you have not produced a result to compare against. If you try it you will see what works and what not.

Don't use iterative methods..as mentioned.

See how Lagrange and Bessel tackle this problem.

http://eaton.math.rpi.edu/faculty/Kovac ... Bessel.pdf
LOL! I have absolutely no clue what you consider a valid answer to your question. Is it an animal, mineral, or vegetable?

Methinks this brainteaser is really a trapdoor function.

Paul
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Joined: July 20th, 2001, 3:28 pm

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

There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone...

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

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

Are iterative methods permitted in this fifth dimension? Can one use paper and pencil to raise e to this fifth dimension?

P.S. the middle ground between light and shadow always contains something opaque.

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

Are iterative methods permitted in this fifth dimension?  Can one use paper and pencil to raise e to this fifth dimension?

P.S. the middle ground between light and shadow always contains something opaque.
Look at equation (11) of above link. Look hard and all will reveal..

Plug in $M = n\pi \: for \: n = 1,2,3$. What do you see? Do you see it in a flash? Can you verify it just by looking at it? If not, you may use TI calculator
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: Going off on a tangent (kind of) : find y

Are iterative methods permitted in this fifth dimension?  Can one use paper and pencil to raise e to this fifth dimension?

P.S. the middle ground between light and shadow always contains something opaque.
You still think it is multiple choice!
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

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

eq 11?

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

eq 11?
Indeed; I am trying to impress T4A with Bessel
http://eaton.math.rpi.edu/faculty/Kovac ... Bessel.pdf

Plug in. Boost C++ has this stuff.
Last edited by Cuchulainn on May 19th, 2017, 8:07 pm, edited 1 time in total.
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Cuchulainn
Topic Author
Posts: 61100
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Going off on a tangent (kind of) : find y

There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone...
No, man, like hey, man. Wow. I was watching this object man, li-like the satellite that we saw the other night, right? And, like, it was going right across the sky, man, and then... I mean it just suddenly, uh, it just changed direction and went whizzin right off, man. It flashed...
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Cuchulainn
Topic Author
Posts: 61100
Joined: July 16th, 2004, 7:38 am
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### Re: Going off on a tangent (kind of) : find y

Kepler has played a cameo role in my much visited exponential of five thread. Anyone remember?

viewtopic.php?f=26&t=79451&p=738941&hil ... er#p738941
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

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

eq 11?
Indeed; I am trying to impress T4A with Bessel
http://eaton.math.rpi.edu/faculty/Kovac ... Bessel.pdf

Plug in. Boost C++ has this stuff.
Eq 11 is very clever but how quickly can one compute it (assuming that's even a consideration).

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

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

How quickly does Eq 11 converge?

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