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Cuchulainn
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Find inverse function

December 15th, 2015, 3:47 pm

Babyonians and the square root of 2
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list1
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Find inverse function

December 15th, 2015, 3:48 pm

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.
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Find inverse function

December 15th, 2015, 3:55 pm

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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Find inverse function

December 15th, 2015, 4:09 pm

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'.
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December 15th, 2015, 4:15 pm

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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December 15th, 2015, 4:21 pm

It should be prime which signifies derivative. I incorrectly used '- 1' and I already corrected it.
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December 15th, 2015, 4:36 pm

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.
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December 15th, 2015, 6:33 pm

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.
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December 16th, 2015, 6:08 am

Inverse normal CDF list,Do you (still) think a closed formula is possible?
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December 16th, 2015, 10:02 am

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.
 
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December 16th, 2015, 11:07 am

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
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list1
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Find inverse function

December 16th, 2015, 5:27 pm

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.
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December 16th, 2015, 6:04 pm

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
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Find inverse function

December 16th, 2015, 10:05 pm

it was not a sarcastic joke, sorry
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Paul
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Find inverse function

December 16th, 2015, 10:23 pm

Why, oh why???P
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