Questions tagged [rootfinding]
The rootfinding tag has no usage guidance.
29
questions
0
votes
0answers
43 views
Solve non-linear equation in R
I need to solve the following equation for $x$ in [0, 1]. Assume $0<\alpha<1$ and $0<\lambda$.
$$(1 - x)^{\alpha+1} - \lambda (x+1)^{\alpha+1} = -2\lambda (\alpha + 1) x^\alpha$$
Would very ...
1
vote
1answer
66 views
Help understanding Brent's root finding method
Help me understand a part of Brent's root finding algorithm. In a typical iteration we have samples (a,fa), (b,fb), (c,fc) all real with (a<b<c) or (c<b<a) . Also, in the case I am ...
2
votes
1answer
117 views
Rootfinding algorithm that takes advantage of automatic differentiation
Is there any algorithm (or tricks) for rootfinding to take advantages of automatic differentiation (AD)?
Rootfinding algorithms typically solve
$$
\mathbf{f}(\mathbf{y}) = \mathbf{0}
$$
where $\mathbf{...
1
vote
2answers
83 views
Coding up a toy model for gradient-descent — what step size to choose?
I'm coding up a simple model for gradient-descent, and using it to minimize some simple, deterministic functions.
What step size could I choose that's simple enough for me to get started with?
Should ...
0
votes
1answer
72 views
Coding up Newton's method for a mapping from R^2 to R — the Jacobian wouldn't be invertible
I'm trying to code up in Matlab a multivariable Newton's method, for a mapping from R^2 to R, but the Jacobian would be a 2x1 matrix, not square, so it wouldn't be invertible.
Does this mean that ...
0
votes
0answers
32 views
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 ...
1
vote
0answers
65 views
How can I practice multivariable root-finding?
Recently, I've been reading up on various root-finding / optimization algorithms such as the Levenberg-Marquardt method, Gauss-Newton, Conjugate Gradient, trust-region and trust-region-dogleg.
I've ...
0
votes
1answer
71 views
Interpreting multivariable root-finding results from Matlab's fsolve algorithm
Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
1
vote
1answer
44 views
Algorithm to determine if a polynomial has any complex roots
Is there a simple algorithm to determine if a given polynomial (with all real coefficients) has all real roots? I do not need to know what the roots are; I just what to know if a given polynomial has ...
0
votes
1answer
38 views
Fixed-point iteration when image and domain are not the same
I have a function $f(x)$ defined on a domain $D$, but such that the image $f(D)$ may contain extra regions not included in its domain. I am interested in solving the fixed-point equation $x=f(x)$. If ...
5
votes
0answers
203 views
Polynomial rooting - fast root finding
I need to solve a rooting problem of a polynomial (the order of which is 2(N-1), where N=48).
So far I'm using Python Numpy algorithm (that relies on computing the eigenvalues of the companion matrix)...
7
votes
1answer
83 views
Fast computation of the zeros of a trigonometric polynomial
I am wondering what is the fastest way to compute the zeroes of the trigonometric polynomial
$$
T(x) = \sum_{i=1}^La_i\sin\big(2\pi(a_i x) - \phi_i \big), \\
a_i \in \mathbb{N}, \\
\phi_i \in \left[-\...
2
votes
2answers
111 views
What is an efficient way to calculate zeros of Bessel functions?
One approach is the brute force method of evaluating at all points at fixed intervals and when it nears zero write value, this can be combined with adaptive step size. Another approach is ...
2
votes
2answers
72 views
ODE Event detection for calculating multiple roots of continuous sinusoidal equation
Hey everyone I have a paper that has a method for computing rise and set times of a satellite given a closed form solution. It is a complicated sinusoidal function and the paper has a method to ...
1
vote
1answer
78 views
How to avoid gsl root finder evaluate function outside its domain
When I use the newton's method or hybrid solver in the GSL package to deal with 1-D or multidimensional root solving problems, the code frequently crashes when the solver requests function value ...
3
votes
1answer
49 views
Robust ways to find zeros of the Tricomi confluent hypergeometric function as a function of its parameters
I'm solving a quantum mechanical problem, and the quantization condition requires me to solve the equation
$$
U\left(\frac12(\ell+1-E), \ell+1, r^2\right) = 0,
$$
where $U(a,b,z)$ is the confluent ...
1
vote
1answer
61 views
Finding curves where function goes to zero in two dimensions
Suppose $f(x,y)$ is a complex function of two real arguments with roots* that are not discrete points but lie in curves. (Is there are term for this characteristic?) An example is shown below: the ...
6
votes
2answers
127 views
Positive root of $x^q + bx - b$
Is there either a closed-form expression or fast/elegant algorithm for computing the positive root of the polynomial
$$f(x)=x^q + \beta x - \beta,$$
where $\beta>0$ and $q\geq2$? How about the $q\...
2
votes
0answers
30 views
Piecewise-linear Continuations vs Marching Squares/Cubes
It seems that both piecewise-linear continuation and marching squares are methods to produce iso-contours of a scalar function given the function's values on a grid. It seems that piecwise-linear ...
6
votes
1answer
354 views
Defining a condition number and termination criteria for Newton's method
The condition number of function evaluation
$$
\mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right|
$$
is infinite at a root of $f$. Hence it is useless for rescaling a tolerance which defines an ...
1
vote
1answer
114 views
Root finding using Newton's method with quadratic interpolation - Is there a mistake in this textbook?
I've been trying to learn about root finding using Newton's method, which uses a quadratic interpolating polynomial. I found this text, Numerical methods for roots of polynomials, PART 2: McNamee &...
1
vote
1answer
111 views
Muller's method is the same as Newton's method with a quadratic interpolating polynomial?
I'm new to numerical analysis, and have been learning root finding algorithms. I am a bit confused about the difference between Muller's method, and Newton's method using an n-degree interpolating ...
7
votes
3answers
1k views
Finding the first N roots of transcendental equation
I need to find the first $n$ roots of the transcendental equation
\begin{equation}
F(k) = J_m'(kr)Y_m'(k)-J'_m(k)Y'_m(kr)
\end{equation}
for integer values of $m$ and any $r \in [0,1)$ where $J'$ ...
1
vote
0answers
243 views
Levenberg-Marquardt for root-finding: just square the function?
This question might be so obvious and trivial that I'm having a hard time googling it.
I have a multivariate root finding problem that I'm trying to solve in C# and the library that I'm trying to use ...
5
votes
1answer
129 views
Can redundant variables be beneficial for root-finding convergence
Suppose I have $n$ generally nonlinear equations for $n$ variables, like e.g. for $n=2$ the system $F(x,y)=0$
$$
\begin{aligned}
x^2+2y-4&=0\\
\sqrt{8}x+y^2-5&=0
\end{aligned}
$$
By ...
2
votes
2answers
139 views
How does Mathematica compute real and complex solutions to single, non-polynomial equations?
This question on StackOverflow has led me to ask myself what would be involved in solving an equation like this one using tools available to the Python programmer.
$$ \frac{0.125567841}{d^{2.25}} = \...
5
votes
0answers
196 views
Roots of transcendental equation involving bessel functions
I need an efficient way to numerically find the first $n$ positive roots $\lambda_n$ of the transcendental equation
$$ \dfrac{J_0 (\lambda_n r) Y_1 (\lambda_n) - J_1 (\lambda_n) Y_0 (\lambda_n r)}{...
4
votes
3answers
2k views
How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$
I'm interested in finding the roots of the following equation:
$\tan(x) = \frac{2x}{x^2-1}$.
It is easily seen that 0 is a root and the roots are symmetric w.r.t. 0.
I wonder if an analytical ...
1
vote
1answer
646 views
Confusion about determining the jacobian in a rootfinding algorithm
I have written some Python code to determine the numerical roots of the following non-linear equation:
$$f_m=\tan\lambda_m - \frac{\lambda_m}{1+a}$$
where $\lambda_m\gt0$ and $a\geq0$. The code is:
<...
