![]() Since f(x) = 0 is a quadratic equation, it must have two roots, and those two points are in fact ±(0.877659 - 0.142424 i), as can be verified by direct calculation. It follows that the intersections of those curves are the points where Re(f(x)) = Im(f(x)) = 0, which means they are the roots of the equation f(x) = 0. ![]() The cross-sections of the two surfaces by the horizontal plane c = 0 are pairs of curves where each function is zero, respectively. For example, f(x) has the following graphs for the real and imaginary parts. Such functions can be plotted in 3D by representing x = a + i b as a point in the (a, b) plane, and the value of the function along the third dimension, say c. So my question is: is it possible to identify the roots of such an equation by simply looking at the real and imaginary parts of the plot?į(x) = x^2 - 1 + i (x^2 - 0.5) is a complex function of a complex variable, which maps a complex variable x = a + i b to the complex value f(x) = Re(f(x)) + i Im(f(x)).Įach of Re(f(x)) and Im(f(x)) is a real function of a complex variable. However, this equation has no real roots, so the crossing points are different. (3) The roots are all within 1-2 orders of magnitude in size. I would expect numeric methods to have trouble. (2) The polynomial is a perturbation of one with substantial multiplicity. ![]() Here FindRoot will use 100 digits in x in each step of the iteration towards the root, and when it has found the root to 15 digit precision, it stops, and return that x, which still has a precision of 100. (1) The ones that find all polynomial roots have the most difficult job, in that they may need to deal with deflated polynomials. If they both crossed the horizontal axis at the same point(s), that would mean the equation has real root(s), since both real and imaginary parts would be zero for some real value of x. 1,506 9 14 2 I think the precision goal is how close you need to get to the true root. Those are graphs of the real and imaginary parts plotted for real values of x. We have a real part which crosses zero, and an imaginary part which also crosses zero but at a different x. ![]()
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