2
votes
1answer
34 views

Checking tolerance of Newton-Raphson method to calculate square root

Finding the square root of $c$ is finding the solution to: $$x^2 - c = 0.0$$ We can use Newton's method to successively approximate the solution. My question is how to check whether we are within ...
2
votes
1answer
98 views

Solving equation $a^{-x} + \log x/\log a = 0$

Please can you instruct me how should I start writing an algorithm (pseudo-code, to be implemented) for finding all solutions for the following equation: $a^{-x} + \log x/\log a = 0$ where $a$ ($a$ ...
6
votes
1answer
69 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
2
votes
1answer
40 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
3
votes
1answer
50 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
1
vote
2answers
45 views

Fastest way to obtain the parametric value t of a bezier curve, for a given set x coordinates.

The problem is the following: Having a bezier curve B(t) we have coordinate x from the curve, and we need to obtain the y values from it, hence we need to compute the t values. What is the fastest ...
1
vote
1answer
47 views

Laguerre's method and zero division

I'm trying to understand Laguerre's method for root finding and I have hit one road block. Suppose I have a polynomial $p(x) = x^4 + 1$ and an initial guess $x_0 = 0$. This results in division by ...
1
vote
1answer
56 views

Solving a problem using Householder's method

For the following points on a plane: $(-1,1),(0,0),(1,1),(1,-1)$, we look for a polynomial $p(x)=a+bx$ such that: $$ \sum_{i=1}^4{(p(x_i)-y_i)^2} = min $$ How do I formulate this as problem as a ...
2
votes
3answers
92 views

Numerical Solution of $\frac{x}{1-e^{-x}} -5 = 0$

I am working on a problem at the moment which cuts down to the following question: How do I get a numerical solution for: $$\frac{x}{1-e^{-x}} -5 = 0?$$ I've been thinking about using Newton's ...
0
votes
1answer
72 views

Improvement to regula falsi method?

The regula falsi algorithm is based on a linear interpolation between the points $a$ and $b$, which bracket a root we want to find. Would it be any improvement to use a parabolic interpolation ...
3
votes
2answers
132 views

Real roots plot of the modified bessel function

Could anyone point me a program so i can calculate the roots of $$ K_{ia}(2 \pi)=0 $$ here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D My conjecture ...
0
votes
2answers
86 views

How many iterations of the bisection method are needed to achieve full machine precision

Suppose that an equation is known to have a root on the interval $(0,1)$. How many iterations of the bisection method are needed to achieve full machine precision in the approximation to the location ...
1
vote
1answer
42 views

How do you call the following iterative solving method

I have the following implicit equation $$ x= f(x) $$ which I solve by starting with some value for $x$, then setting $x$ to the new value $f(x)$ and so forth until convergence. How is that method ...
1
vote
0answers
37 views

Approximating the smallest positive root of a function

Suppose we have a smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$. Let $S$ denote the set of all positive roots of $f$ and let $x^*$ denote the minimum of $S$ (assuming such a thing exists). What ...
0
votes
1answer
65 views

Estimating the multiplicity of a root (numerically)

I'm working on a modified root finding script that uses the Newton method, but with a modification such that I estimate the order of the root to get faster convergence. The basis of my motivation is ...
0
votes
0answers
31 views

What is a contraction mapping and the use in iterated function?

Like the title said, I really appreciate if anybody can explain the contraction mapping in simple terms with examples for iterated function in numerical analysis. I have looked at the Wikipedia page ...
0
votes
0answers
58 views

What is the order of convergence and multiplicity at each root?

Let $f(x) = x^3 + 3x^2 − 4$. Find two of its roots using Newton’s method. Start with $x_0=2$ and $x_0=−1$ in each case and calculate up to 3 iterations. What is the order of convergence at each root? ...
0
votes
1answer
31 views

Number of needed iterations in finding p'th root of a number with newton method

I need to write a parallel code for finding p'th root of n with newton method. I know how the serial code must be. The only method I found to get rid of the do-while loop in the code is finding a ...
2
votes
2answers
62 views

Numerically finding roots of function - converges?

Well this question was in my homework, I have difficulty to "proof" it (or more correctly: seeing how I would solve it). Consider a floating point system ($s \cdot b^e$ where $1\leq s \leq 10 - 1 ...
0
votes
2answers
80 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
4
votes
2answers
64 views

Newton iteration method

i need some help here. My function is $f(x) =x^{3}$ . I was asked to find the number of iterations that are needed to reach the precission $10^{-5}$ if $x_{0} = 0.9$ I was wondering if there is a ...
3
votes
1answer
88 views

Newton-Raphson's method

Hello MathExchange community ! I am working on some "simple" numerical methods to solve 4th degrees and below equations. To make it easier I am working on the $[0, 1]$ interval and I know for sure ...
1
vote
1answer
72 views

Convergence of order 3 of a Newton's method variant

Let $f\in C^2$ and $x^*$ be a simple root of $f$, i.e. $f(x^*)=0\wedge f'(x^* )\ne 0$. Further, let $U(x^*):=\left\{x : |x-x^* |\le r\right\}$ for some $r>0$ and $\;\;\;\;\;\;\;\;\;\;\displaystyle ...
0
votes
1answer
53 views

Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
2
votes
3answers
118 views

How bad, really, is the bisection method?

We know that the bisection method for root finding is slow (linear convergence), but has the advantage of always working for a continuous function, if we start with a interval which brackets the root. ...
0
votes
1answer
147 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
1
vote
0answers
71 views

Analytical solution(root) for a tenth order polynomial?

is it possible to develop an analytical solution (root) for such a polynomial: $f(x)=\left(x^{10}-c_1^2\right)*\left(c_2-x\right)^2-0.2*\left(x^2-1\right)*c_1^2$ with $c_1$ and $c_2 >0$. Numerical ...
0
votes
0answers
33 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...
0
votes
2answers
99 views

Simplifying equation into Newton Raphson form

Given the equation $\displaystyle{\int_{-x}^x\exp({-t^2})dt}=-\ln(x)$: a. Simplify the integral using Gauss method with 3 points. b. Solve given equation by Newton Raphson iterative ...
3
votes
2answers
118 views

Root of the function $f(x)=xe^x-R$

How can we find the root of the function $f(x)=xe^x - R$ for a general R where $R>=-1/e.$ I don't have any idea as to how to even approach this. Came across this problem during my self-study in ...
2
votes
3answers
110 views

If I know that a polynomial (of order $k \gt 2$) has at most $1$ positive real root - can I find that easily?

[update 2] Urgghh - the time-consumption really stems only from the construction of the h-order polynomial. The time for finding the roots (only 10 to 20 times Newton-iteration because of my nice ...
0
votes
1answer
111 views

Help with finding the roots of a function

I have the following problem: Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy. Any suggestions? :) Thank you for any ...
0
votes
1answer
161 views

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to (a) $0$, if $a+b<2$ (b) $2$, if $a+b>2$ ...
0
votes
1answer
76 views

Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.

Let $f(x)=(x-1)^{10}$. The root of the equation , $p=1$. The approximates of the root, $p_n=1+\frac{1}{n}$ Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires ...
0
votes
1answer
78 views

Difficulty to solve the exercise of Bisection method.

Find an approximation to $ {25}^{\frac{1}{3}}$ correct to within $10^{-4}$ using the Bisection algorithm. How to solve it? Where are the function and interval here?
0
votes
3answers
302 views

I am not understanding what has asked to compute of the following exercise.

let $f(x)=(x+2)(x+1)x(x-1)^3(x-2)$. To which zero of $f$ does the Bisection method converges when applied on the interval $[-3,2.5]$ Have i asked to find the root of $f(x)$ ?
0
votes
1answer
68 views

Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$

i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$ But in answer script, $n=8$. My procedure is $ \frac{(b-a)}{2^n}<\epsilon$ ...
2
votes
1answer
101 views

Correct answer of the following math related to Bisection Method.

Use the Bisection method to find $p_3$ for $$f(x)=\sqrt x-\cos(x)$$ on $[0,1]$ I have got the answer $p_3=0.875$ But in answer script , $p_3=0.625$ Which one is correct? let $[a,b]=[0,1]$ ...
1
vote
1answer
233 views

Do the false position method really need that there exists only one root inside $[a; b]$?

I'm studying the False Position Method for finding zeroes of real functions and in the book I'm reading the author says that it is required that only one root of $f$ is contained inside the initially ...
2
votes
1answer
70 views

Solution to set of three equations

I have the following three equations: $$\cos\theta \left(\cos\psi - k_3\sin\psi\right) = k_1$$ $$\sin\phi\sin\theta\cos\psi - \cos\phi\sin\psi - k_3\left(\cos\phi\cos\psi + ...
2
votes
1answer
233 views

Roots of a finite Fourier series?

In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ...
22
votes
2answers
511 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
2
votes
1answer
925 views

Polynomial root finding

I have an univariate polynomial of some degree - how do I numerically find all of its real roots? I never thought I would ask this question - everyone knows how to find polynomial roots, right..? ...
6
votes
2answers
447 views

Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
3
votes
1answer
301 views

Root Finding Algorithm for Discrete Functions

I was recently working with functions of the form $$N - \sqrt{\frac{N}{x}}\cdot\left\lfloor \frac{N}{\sqrt{N/x}}\right\rfloor + \sqrt{\frac{N}{x}} - \left\lfloor \sqrt{\frac{N}{x}}\right\rfloor$$ ...
1
vote
2answers
3k views

Convergence of Bisection method

I know how to prove the bound on the error after $k$ steps of the Bisection method. I.e. $$|\tau - x_{k}| \leq \left(\frac{1}{2}\right)^{k-1}|b-a|$$ where $a$ and $b$ are the starting points. But ...
0
votes
1answer
79 views

Iterarions count in Newton's method;

How many iterations must I do for getting $n$ signs after floating point in calculating square root by Newton's method P.S Sorry for my bad English. Please mention to me where I've done mistakes. ...
3
votes
1answer
205 views

Understanding accuracy of Newton's Method

In a numerical analysis book I'm reading it says that using the Newton error formula we can find an expression for the number of correct digits in an approximation using Newton's Method. Here's the ...
4
votes
1answer
573 views

How to solve an equation using Newton's method with and without backtracking?

Lets assume I have this equation: $$\log(e^x+e^{-x})=2x+5,\quad x \in (-50,50).$$ As always we have to pick a starting point to solve this by Newton's method, but how can i know for what initial ...
2
votes
1answer
154 views

Fixed-point method in many-dimensions

A well known method of easily solving multi-dimensional non-linear root finding problems, is to bring the equations into the form: $$\bf x = g(x)$$ And then iterating. The problem is, one has to find ...