 Cuchulainn
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### Find inverse function

Babyonians and the square root of 2
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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Jean Piaget list1
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### Find inverse function

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnFor 1, see thisFor my example 2, it is the fixed point form for sqrt(2) known to the ancient Babyloninansin this thei no words similar to contractor. two curves intersected in one point and how does this graph is related to contractor notion?My question was how 1-2 are related to contractors?First, it is called contraction. Second, you need to find it, it is easy. It is high-school piddling calculus.hintfor x = g(x) we need to have |g'(x)| < 1.now let g(x) = cos(x). BTW this is a contractorIn your reference "contractions" we can read that "In mathematics, a contraction mapping, or contraction or contractor, ..." and then they use 'f' and 'd'. In your example for x = cos (x) you probably think that f ( x ) = cos x. Then for example $[\,cos x\,]\,^{-\,1}\,\,=\,\,-\,sin\,x$. Therefore $|\,-\, sin\, (\frac{\pi}{2}\, +k \pi \,) \,|\,\, =\,\,1$. Thus these points should be excluded from domain of the'contraction'. Therefor Then f is a contraction or as Wiki admits a contractor. But it seems that this fact as well as a solution of the eq x = cos x by no means directly relates to a construction of inverse function which commonly denote as x = arccos y.
Last edited by list1 on December 14th, 2015, 11:00 pm, edited 1 time in total. Cuchulainn
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### Find inverse function

QuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnFor 1, see thisFor my example 2, it is the fixed point form for sqrt(2) known to the ancient Babyloninansin this thei no words similar to contractor. two curves intersected in one point and how does this graph is related to contractor notion?My question was how 1-2 are related to contractors?First, it is called contraction. Second, you need to find it, it is easy. It is high-school piddling calculus.hintfor x = g(x) we need to have |g'(x)| < 1.now let g(x) = cos(x). BTW this is a contractorIn your reference "contractions" we can read that "In mathematics, a contraction mapping, or contraction or contractor, ..." and then they use 'f' and 'd'. In your example for x = cos (x) you probably think that f ( x ) = cos x. Then f is a contraction or as Wiki admits a contractor. But it seems that this fact as well as a solution of the eq x = cos x by no means directly relates to a construction of inverse function which commonly denote as x = arccos y.I leave it as an exercise for you to generalize it.'contactor' is slang iMO.
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Jean Piaget list1
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### Find inverse function

One should also note that $[\,cos x\,]\,^{\prime}\,\,=\,\,-\,sin\,x$ and $|\,-\, sin\, (\frac{\pi}{2}\, +k \pi \,) \,|\,\, =\,\,1$. Thus these points should be excluded from domain of the 'contraction'.
Last edited by list1 on December 14th, 2015, 11:00 pm, edited 1 time in total. Cuchulainn
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### Find inverse function

QuoteOriginally posted by: list1One should also note that $[\,cos x\,]\,^{-\,1}\,\,=\,\,-\,sin\,x$ It's been a while since I did cos, but are you sure on this one?
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Jean Piaget list1
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### Find inverse function

It should be prime which signifies derivative. I incorrectly used '- 1' and I already corrected it.
Last edited by list1 on December 14th, 2015, 11:00 pm, edited 1 time in total. Cuchulainn
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### Find inverse function

QuoteOriginally posted by: list1It should be prime which signifies derivative. I incorrectly used '- 1' and I already corrected it.You had me real scared for a moment.
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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Jean Piaget Cuchulainn
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### Find inverse function

QuoteOriginally posted by: list1One should also note that $[\,cos x\,]\,^{\prime}\,\,=\,\,-\,sin\,x$ and $|\,-\, sin\, (\frac{\pi}{2}\, +k \pi \,) \,|\,\, =\,\,1$. Thus these points should be excluded from domain of the 'contraction'.The derivative having absolute value < 1 is probably sufficient but not necessary.I have provided code here You need to write your own x =g(x) stuff, of course.
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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Jean Piaget Cuchulainn
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### Find inverse function

Inverse normal CDF list,Do you (still) think a closed formula is possible?
Last edited by Cuchulainn on December 15th, 2015, 11:00 pm, edited 1 time in total.
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### Find inverse function

QuoteOriginally posted by: CuchulainnInverse normal CDF list,Do you (still) think a closed formula is possible?In case of cdf of normal distribution the latter is given by the integral. In math it is called closed form. In many cases we can expect that inverse function will be given in a similar form. Sometimes inverse function can be given by a ode. Actually it is not a big difference if a function is given by an integral or when the function is cos x. Actually my question was originated by the change time $t \,\rightarrow\, \tau_t$ in sde where $\tau_t$ is defined by equality$t\, \, = \,\, \int_0^{ \tau_t} g\,(\,\xi\,(\,s\,)\,)\,ds$ Here $\xi\,(t\,)$ is a solution of the sde. It seems I got the details of the transformation. Cuchulainn
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### Find inverse function

What the use of a closed form? We can look at it but it is barren. The problem in 2 variables; can we find a closed form?I formulate it as a Goursat PDE and solve using FDM. The right side of the PDE is
Last edited by Cuchulainn on December 15th, 2015, 11:00 pm, edited 1 time in total.
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### Find inverse function

There is indeed a difference between closed form formula and its program for calculation. To see essense of such problem it is easier to consider more common problem as it might relate more people than N (0,1) distribution. For example whether one think that the formula $S \,\, =\,\,2\pi\,R$ is barren compare with the program which calculate this formula taking $\pi$ and R with 100 decimals.
Last edited by list1 on December 15th, 2015, 11:00 pm, edited 1 time in total. Cuchulainn
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### Find inverse function

QuoteOriginally posted by: list1There is indeed a difference between closed form formula and its program for calculation. To see essense of such problem it is easier to consider more common problem as it might relate more people than N (0,1) distribution. For example whether one think that the formula $S \,\, =\,\,2\pi\,R$ is barren compare with the program which calculate this formula taking $\pi$ and R with 100 decimals.I agree. This formula can be computed by hand using addition + and multiply *. It is closed, linear(yikes!) and computable. it is a direct problem of the form y = F(x) where x is given to compute y. The inverse problem is also trivial.But it is not in the same league as the inverse cumulative normal distribution because the latter is a nonlinear problem. It is a nonlinear problem of the formy = F(x) where y is given and we find x.So, F(x) is an integral from 0 to x. So in general F(a+b) != F() + F(b). Example; take f(t) = t ==> F(a) = a^2/2 etc.. a^2/2 + b^2/2 != (a+b)^2/2."It is by logic that we prove, but by intuition that we discover."H. Poincare
Last edited by Cuchulainn on December 15th, 2015, 11:00 pm, edited 1 time in total.
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Jean Piaget list1
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### Find inverse function

it was not a sarcastic joke, sorry
Last edited by list1 on December 16th, 2015, 11:00 pm, edited 1 time in total. Paul
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### Find inverse function

Why, oh why???P  