On Fri, Nov 21, 2014 at 9:37 AM, Bill Page <bill.p...@newsynthesis.org> wrote: > On 20 November 2014 22:08, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: >> On Thu, Nov 20, 2014 at 7:53 PM, Bill Page <bill.p...@newsynthesis.org> >> wrote: >> ... >>> This problem can be reduced to finding an algorithm to determine >>> if f(x) is everywhere non-negative. Richardson proves that no such >>> algorithm exists. >> >> I see. But what does this have to do with the derivative of |f(x)| that >> we are trying to figure out? >> > > This has to do with 'conjugate' in general, not just derivatives of > expressions containing 'conjugate'. The problem is that 'conjugate' > is transcendental but it cannot be written in terms of log and exp. > >> As you pointed out, the challenge is that if you include conjugate(x), >> then you might be out of luck. But aren't you out of luck already if >> you have abs(x) in the expression in the first place? I.e. taking a >> derivative is not going to change anything, you are still out of luck. >> > > You are right about the derivative. But my limited understanding is > that the strategy is not to avoid 'abs(x)' but rather to avoid 'sin'. > We cannot similarly avoid 'conjugate' and in general the effect of > including 'conjugate' is apparently unknown. But one effect of > including 'conjugate' is that we can have expressions like > > x+conjugate(x) > > which is necessarily real-valued, rather like 'abs(x)' for x > real-valued is non-negative. So it would be nice to know, for example > for any expression composed of x, integers, +, *, sin, and conjugate, > if there is an algorithm to determine if this expression is everywhere > real-valued.
I am still confused about one thing: is this issue is already present in FriCAS before your changes? Because you can already use conjugate, sin, +, *, ..., even without defining the derivative for abs(x). I fail to see how defining the abs(x).diff(x) in the way you did it can introduce issues that weren't present in the first place. ----- I have finished the writeup, it starts here (you might want to refresh your browser to see the latest changes): http://www.theoretical-physics.net/dev/math/complex.html#complex-conjugate and it was implemented with these two PRs: https://github.com/certik/theoretical-physics/pull/39 https://github.com/certik/theoretical-physics/pull/40 I must say one thing that I like about the "theta" is that it tells you immediately if the function is analytic or not (if theta is present it is not, if it is not present, then the expression does not depend on theta, and thus is analytic). For example, for log(z), the theta cancels, and so the result 1/z is analytic. I found a bug in these results from FriCAS: > (4) -> D(abs(f(x)),x) > > , _ _ , > f(x)f (x) + f(x)f (x) > > (4) --------------------- > 2abs(f(x)) > Type: > Expression(Integer) > (5) -> D(abs(log(x)),x) > > _ _ > xlog(x) + x log(x) > (5) ------------------ > _ > 2xxabs(log(x)) > Type: > Expression(Integer) The bar must be over the whole f(x) as well as log(x), because conjugate(log(x)) is only equal log(conjugate(x)) if x is not negative real number. See the example here: http://www.theoretical-physics.net/dev/math/complex.html#id1 where I have it explicitly worked out. You can also check that easily in Python: In [1]: from cmath import log In [2]: x = -1+1j In [3]: log(x).conjugate() Out[3]: (0.34657359027997264-2.356194490192345j) In [4]: log(x.conjugate()) Out[4]: (0.34657359027997264-2.356194490192345j) In [5]: x = -1 In [6]: log(x).conjugate() Out[6]: -3.141592653589793j In [7]: log(x.conjugate()) Out[7]: 3.141592653589793j In [8]: log(x.conjugate()) - 2*pi*1j Out[8]: -3.141592653589793j Where [3] and [4] are equal, but [6] and [7] are not (you need to subtract 2*pi*i from [7], as in [8], in order to recover [6], consistent with the formula in the writeup). Apart from this issue, all other examples that you posted seem to be correct and agree with my hand based step by step calculation in the writeup. Ondrej -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
