Re: Solving an equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg51363] Re: [mg51299] Solving an equation*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 15 Oct 2004 02:46:24 -0400 (EDT)*References*: <200410141035.GAA14785@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Scott wrote: > I have an equation, Gamma[a+I b] = some complex number. I need to > solve this equation for a and b. I can write a+I b as z, but it can't > be solved through NSolve. What I have been doing is a double do loop > for a and b and getting some number. Then I compare this to the > number I have. Then I narrow down my possibilities for a and b, and > go through the process again. Does anyone know of a better way to do > this problem? Is there a way to have Mathematica compare each result > of the do loop to a given value, and given certain conditions spit out > an answer for a and b? > > Hope that makes sense. > I'm not sure what might be the best way to do this, but one workable approach is to treat it as an optimization. Say your complex number is 3+5*I. Belwo we find roots near 1-2*I and 5+4*I respectively. In[9]:= FindMinimum[Abs[Gamma[a+I*b]-(3+5*I)], {a,1}, {b,-2}] FindMinimum::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances. -8 Out[9]= {2.39838 10 , {a -> 0.0910323, b -> -0.133628}} In[10]:= FindMinimum[Abs[Gamma[a+I*b]-(3+5*I)], {a,5}, {b,4}] FindMinimum::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances. -7 Out[10]= {1.10535 10 , {a -> 5.24561, b -> 4.37032}} If you like you can use a root finder instead, or in conjunction with this. I tend to prefer the second approach, using it as a polishing step, because it does not always perform well without a good initial point. Also, as the functions are not analytic, we should either provide a gradient or use a secant method by specifying two initial values per argument. eqns = Thread[{Re[Gamma[a+I*b]],Im[Gamma[a+I*b]]}=={3,5}]; In[16]:= FindRoot[eqns, {a,5.24,5.25}, {b,4.3,4.4}, WorkingPrecision->20, AccuracyGoal->20, PrecisionGoal->20] Out[16]= {a -> 5.2456145798271348496, b -> 4.3703225075594339402} We could instead have requested higher precision/accuracy in FindMinimum but that is slower and also, to judge from residuals, not quite as accurate. Daniel Lichtblau Wolfram Research

**References**:**Solving an equation***From:*jujio77@yahoo.com (Scott)

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