# Questions tagged [rootfinding]

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### Finding the first N roots of transcendental equation

I need to find the first $n$ roots of the transcendental equation \begin{equation} F(k) = J_m'(kr)Y_m'(k)-J'_m(k)Y'_m(kr) \end{equation} for integer values of $m$ and any $r \in [0,1)$ where $J'$ ...
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### How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$

I'm interested in finding the roots of the following equation: $\tan(x) = \frac{2x}{x^2-1}$. It is easily seen that 0 is a root and the roots are symmetric w.r.t. 0. I wonder if an analytical ...
1answer
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### Calculating the Jacobian for a function containing a derivative

I have the equation $F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$ and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$. I am testing out a new algorithm to ...
1answer
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### Nonlinear root solving libraries which accept a Jacobian in band-storage

I'm in search for a library for solving large systems of non-linear equations, similar to MINPACK, but unlike MINPACK, can accept a Jacobian in band-storage. My Jacobian is sometimes not invertible, ...
1answer
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### Robust ways to find zeros of the Tricomi confluent hypergeometric function as a function of its parameters

I'm solving a quantum mechanical problem, and the quantization condition requires me to solve the equation $$U\left(\frac12(\ell+1-E), \ell+1, r^2\right) = 0,$$ where $U(a,b,z)$ is the confluent ...
2answers
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0answers
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### How to describe function convergence and function tolerance for numerical root-finding?

I'm currently doing some practice problems on root-finding and am writing up some notes / comments on my code. In my solver code, if my function value is below the tolerance that I've set, should I ...