# Tag Info

### Finding the first N roots of transcendental equation

This is a root-finding problem for an analytic function over a single dimension. One standard technique is to approximate $F(k)$ as a polynomial $F(k) \approx c_0 + c_1 k + c_2 k^2 + \cdots$ using a ...

### Positive root of $x^q + bx - b$

According to Wolfram Alpha, $x^5+3(x-1)=0$ has no closed-form solution, so you can forget about a nice closed-form expression. :) I see nothing wrong with Newton's method; it should be quick and ...
Accepted

### Finding the first N roots of transcendental equation

I will complement @Richard Zhang 's answer (+1) with a python implementation of his suggested approach. The MATLAB package Chebfun has been partially ported in <...

### Fast computation of the zeros of a trigonometric polynomial

It all boils down to building a certain matrix from the polynomial coefficients and computing its eigenvalues. John Boyd did a lot of work in this area, these are some relevant papers: Boyd, John P. "...
Accepted

### Computing powers of diagonal + rank-1 matrix?

This paper shows an algorithm to compute the eigendecomposition of symmetric diagonal-plus-rank-1 matrices in $O(d^2)$.

### What is an efficient way to calculate zeros of Bessel functions?

This is a classical problem in numerical methods research: evaluating the zeros of special functions. Many years of research have gone into devising efficient methods. The canonical starting point for ...
Accepted

### How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$

Here is a simple (Matlab) Newton method as a first attempt to help get started. It finds 1087 roots with error below $10^{-11}$. ...

### How does Mathematica compute real and complex solutions to single, non-polynomial equations?

As the other Answer already touches on the possibility of a symbolic root-solver being applied to this particular equation (by transforming into a polynomial form, albeit of degree $\ge 5$), I'll make ...

### Finding the first N roots of transcendental equation

Complementing the answers, I would like to say that directly discretizing the differential equation might work really well. I used that approach in a previous question. I used Finite Differences and ...

### Secant Method for finding $\sup f^{-1}(0)$

If you need a certified result, you can try interval methods such as the ones in https://github.com/JuliaIntervals/IntervalRootFinding.jl/ . They are at their best for (at least) piecewise ...

### Why is the definition of convergence different for root finding algorithms as compared to sequences?

This is not the definition of convergence. It is the definition of the convergence rate -- that is, how fast the sequence converges to a limit. In other words, the definition you quote can only be ...
Accepted

### Can redundant variables be beneficial for root-finding convergence

In terms of computational effort, it is useless. I mean, you are still having a nonlinear problem with a bigger Jacobian (which is the worst part to be computed quickly). For next thoughts: dense ...
Accepted

### Complexity of recovering all roots of a polynomial

Evaluating a polynomial of degree $<n$ in $n$ points can be done in time $O(n\log n)$. This is called fast multipoint evaluation; see for instance von zur Gathen, Modern Computer Algebra, ch. 10.
Accepted

### ODE Event detection for calculating multiple roots of continuous sinusoidal equation

Assuming it is not too expensive to evaluate the function, I would recommend using the chebfun toolbox (matlab), as shown here: https://www.chebfun.org/docs/guide/guide03.html It builds an ...

### How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$

As this is a transcendent equation, finding all roots is not an option. You cannot (in general) find an expression that gives a closed-form for the roots of the equation. So you need to do some ...

### How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$

You can at least take a quick look at the zeros using gnuplot by plotting contours at zero of the equation $$0 = \frac{2x}{x^2-1} - \tan(x)$$ ...
Any reasonable convergence criterion must be invariant to scaling of the function. A decent stopping criterion is therefore if $\|f(x_k)\|≤ \epsilon\|f(x_0)\|$ where $x_0$ is the starting point of the ...