Questions tagged [rootfinding]
For questions about the theory and process of finding the roots of a function (values where the function returns zero).
38
questions
3
votes
0answers
39 views
Sufficient condition for real roots of a polynomial of order $n>5$ with arbitrary real coefficients
I ask for help in solving the problem. I am developing an optimization program that selects the coefficients of a polynomial of order $n> 5$ so that all its zeros are just real numbers. And I ...
2
votes
0answers
89 views
Solve Rational Equation for Root Music in MATLAB
I'm trying to estimate DOA in the Hybrid architecture using root music so I need to solve the attached equation to find the roots for the Root_Music equation in Matlab. Does anyone have an idea for ...
1
vote
0answers
79 views
Solving general initial value problem $\mathbf{f}(\mathbf{x}, \mathbf{x'}, t)=\mathbf{0}$
The common form of initial value problem that can be solved using ODE integrator is
$$
\mathbf{x'}=\mathbf{g}(\mathbf{x}, t)
$$
where $\mathbf{x'}=\partial\mathbf{x}/\partial t$. The initial ...
4
votes
0answers
43 views
Pass forward intermediate results during iterative optimization
To investigate a counter-current flow heat exchanger while considering temperature dependent physical properties (such as specific heat $c_\textrm{p,i}$, heat conductivity $\lambda_\textrm{i}$, ...
3
votes
1answer
94 views
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 ...
1
vote
1answer
282 views
What is the use of arrayfun if a for loop is faster?
This may not be so much of a scientific computing question but more of a MATLAB question, if that is the case, please feel free to close or migrate the question.
Root-finding problems are commonly ...
2
votes
0answers
102 views
Providing Jacobian as LinearOperator in scipy.optimize.root
I asked this question a few days ago on stackoverflow, but I figure scicomp.stackexchange is probably a better place. Sorry for the double post.
I want to solve a system of nonlinear equations using ...
3
votes
1answer
119 views
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, ...
0
votes
1answer
415 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
77 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
166 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
101 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
108 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
33 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
89 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
132 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
48 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
54 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
272 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
92 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
$$
T(x) = \sum_{i=1}^La_i\sin\big(2\pi(a_i x) - \phi_i \big), \\
a_i \in \mathbb{N}, \\
\phi_i \in \left[-\...
2
votes
2answers
327 views
What is an efficient way to calculate zeros of Bessel functions?
One approach is the brute force method of evaluating at all points at fixed intervals and when it nears zero write value, this can be combined with adaptive step size. Another approach is ...
2
votes
2answers
104 views
ODE Event detection for calculating multiple roots of continuous sinusoidal equation
I found a paper [1] that has a method for computing rise and set times of a satellite given a closed form solution. It is a complicated sinusoidal function and the paper has a method to calculate a ...
2
votes
1answer
178 views
How to avoid gsl root finder evaluate function outside its domain
When I use the newton's method or hybrid solver in the GSL package to deal with 1-D or multidimensional root solving problems, the code frequently crashes when the solver requests function value ...
3
votes
1answer
76 views
Robust ways to find zeros of the Tricomi confluent hypergeometric function as a function of its parameters
I'm solving a quantum mechanical problem, and the quantization condition requires me to solve the equation
$$
U\left(\frac12(\ell+1-E), \ell+1, r^2\right) = 0,
$$
where $U(a,b,z)$ is the confluent ...
1
vote
1answer
127 views
Finding curves where function goes to zero in two dimensions
Suppose $f(x,y)$ is a complex function of two real arguments with roots* that are not discrete points but lie in curves. (Is there are term for this characteristic?) An example is shown below: the ...
6
votes
2answers
137 views
Positive root of $x^q + bx - b$
Is there either a closed-form expression or fast/elegant algorithm for computing the positive root of the polynomial
$$f(x)=x^q + \beta x - \beta,$$
where $\beta>0$ and $q\geq2$? How about the $q\...
2
votes
0answers
34 views
Piecewise-linear Continuations vs Marching Squares/Cubes
It seems that both piecewise-linear continuation and marching squares are methods to produce iso-contours of a scalar function given the function's values on a grid. It seems that piecwise-linear ...
6
votes
1answer
557 views
Defining a condition number and termination criteria for Newton's method
The condition number of function evaluation
$$
\mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right|
$$
is infinite at a root of $f$. Hence it is useless for rescaling a tolerance which defines an ...
1
vote
1answer
252 views
Root finding using Newton's method with quadratic interpolation - Is there a mistake in this textbook?
I've been trying to learn about root finding using Newton's method, which uses a quadratic interpolating polynomial. I found this text, Numerical methods for roots of polynomials, PART 2: McNamee &...
1
vote
1answer
214 views
Muller's method is the same as Newton's method with a quadratic interpolating polynomial?
I'm new to numerical analysis, and have been learning root finding algorithms. I am a bit confused about the difference between Muller's method, and Newton's method using an n-degree interpolating ...
7
votes
3answers
1k views
Finding the first N roots of transcendental equation
I need to find the first $n$ roots of the transcendental equation
\begin{equation}
F(k) = J_m'(kr)Y_m'(k)-J'_m(k)Y'_m(kr)
\end{equation}
for integer values of $m$ and any $r \in [0,1)$ where $J'$ ...
1
vote
0answers
313 views
Levenberg-Marquardt for root-finding: just square the function?
This question might be so obvious and trivial that I'm having a hard time googling it.
I have a multivariate root finding problem that I'm trying to solve in C# and the library that I'm trying to use ...
5
votes
1answer
141 views
Can redundant variables be beneficial for root-finding convergence
Suppose I have $n$ generally nonlinear equations for $n$ variables, like e.g. for $n=2$ the system $F(x,y)=0$
$$
\begin{aligned}
x^2+2y-4&=0\\
\sqrt{8}x+y^2-5&=0
\end{aligned}
$$
By ...
2
votes
2answers
158 views
How does Mathematica compute real and complex solutions to single, non-polynomial equations?
This question on StackOverflow has led me to ask myself what would be involved in solving an equation like this one using tools available to the Python programmer.
$$ \frac{0.125567841}{d^{2.25}} = \...
5
votes
0answers
246 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
$$ \dfrac{J_0 (\lambda_n r) Y_1 (\lambda_n) - J_1 (\lambda_n) Y_0 (\lambda_n r)}{...
4
votes
3answers
3k views
How to solve the transcendental equation: $\tan(x) = \frac{2x}{x^2-1}$
I'm interested in finding the roots of the following equation:
$\tan(x) = \frac{2x}{x^2-1}$.
It is easily seen that 0 is a root and the roots are symmetric w.r.t. 0.
I wonder if an analytical ...
1
vote
1answer
726 views
Confusion about determining the jacobian in a rootfinding algorithm
I have written some Python code to determine the numerical roots of the following non-linear equation:
$$f_m=\tan\lambda_m - \frac{\lambda_m}{1+a}$$
where $\lambda_m\gt0$ and $a\geq0$. The code is:
<...
4
votes
3answers
851 views
Best method to find the zero of a decreasing function numerically
I need to find the zero of a function $f(\lambda)$ which is of the form $\sum \frac{c_i^2}{(1+\lambda d_i)^2} -1 $. I tried using Newton's method, and it works sometimes, but it is highly dependent on ...
