Questions tagged [rootfinding]

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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 ...
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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
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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{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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[-\...
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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
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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 ...
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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
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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 ...
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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
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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\...
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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
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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 ...
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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 &...
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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 ...
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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 $J'$ ...
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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
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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 ...
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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}} = \...
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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)}{...
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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 ...
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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: <...