Re: Re: equation question

*To*: mathgroup at smc.vnet.net*Subject*: [mg69997] Re: [mg69973] Re: [mg69921] equation question*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 30 Sep 2006 05:12:17 -0400 (EDT)*References*: <200609271005.GAA00230@smc.vnet.net> <200609281016.GAA25761@smc.vnet.net>

On 28 Sep 2006, at 19:16, Daniel Lichtblau wrote: > dimmechan at yahoo.com wrote: >> Hello. >> >> Consider the following simple examples of FindRoot application. >> >> FindRoot[Sin[x] == 2, {x, I}] >> {x -> 1.5707963267948966 + 1.3169578969248168*I} >> >> FindRoot[Sin[x^2] == 2, {x, I + 1}] >> {x -> 1.3454777060580754 + 0.4894016047219337*I} >> >> FindRoot[Sin[x^2] == 2, {x, 3*I + 2}] >> {x -> 0.3004695589886017 + 2.1914997002654357*I} >> >> Is it possible for FindRoot (or in general in another way) to search >> for solutions >> in the complex plane in an particular domain e.g. searching in the >> domain that >> is made by the lines Re[x]=a1, Re[x]=a2 and Im[x]=b1, Im[b]=b2 ? >> >> I really appreciate any assistance. >> >> Regards >> Dimitris > > You can set it up as a minimization of a square (for multiple > equations, > a sum of squares) and use NMinimize, which handles such constraints. > > Example: > > In[1]:= NMinimize[{Abs[(Sin[(a+I*b)^2]-2)^2], > {5<=a<=9, 0<=b<=2}}, {a,b}] > -17 > Out[1]= {3.89097 10 , {a -> 5.16912, b -> 0.127387}} > > > Daniel Lichtblau > Wolfram Research > > A slightly different approach is: NMinimize[{1, {5 <= a <= 9, 0 <= b <= 2, Abs[Sin[(a + I*b)^2] - 2] == 0}}, {a, b}] Out[1]= {1., {a -> 5.1691164655883295, b -> 0.12738713670869523}} In fact, in this approach, it makes no difference, if one uses NMinimize or NMaximize: In[2]:= NMaximize[{1, {5 <= a <= 9, 0 <= b <= 2, Abs[Sin[(a + I*b)^2] - 2] == 0}}, {a, b}] Out[2]= {1., {a -> 5.1691164655883295, b -> 0.12738713670869523}} On the other hand, one advantage of having Abs[Sin[(a + I*b)^2] - 2] as the objective function is that we automatically get an idea of the size of the residual. Andrzej Kozlowski

**References**:**equation question***From:*dimmechan@yahoo.com

**Re: equation question***From:*Daniel Lichtblau <danl@wolfram.com>