# Tagged Questions

1answer
37 views

### Numerical root finding for 5th degree polyomial

I have the equation $y^5 -ay -b=0$. I need to get a solution whether numerical or analytical. I heard $5$th order polynomials are not solvable analytically, so how can I get the root numerically. ...
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35 views

### What is the (currently) optimal root finding algorithm for multivariate functions? [closed]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
1answer
81 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 ...
1answer
112 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 $a$ ($a$ ...
1answer
77 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 ...
1answer
46 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: ...
1answer
64 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 ...
2answers
59 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 ...
1answer
53 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 ...
1answer
60 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 ...
3answers
94 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 ...
1answer
80 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 ...
2answers
139 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 ...
2answers
154 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 ...
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 ...
0answers
44 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 ...
1answer
71 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 ...
0answers
33 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 ...
0answers
59 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? ...
1answer
77 views

### Rate of convergence of an iterative root finding method similar to Newton-Raphson

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. ...
1answer
36 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 ...
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### 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 ...
3answers
133 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. ...
2answers
181 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 ...
0answers
77 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 ...
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 ...
2answers
104 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 ...
2answers
121 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 ...
3answers
114 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 ...
1answer
113 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 ...
1answer
181 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 $a+b>2$ ...
1answer
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### 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 ...
1answer
89 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?
3answers
337 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 $f(x)$ ?
1answer
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### 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 $\frac{(b-a)}{2^n}<\epsilon$ ...
1answer
105 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 $[a,b]=[0,1]$ ...
1answer
240 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 ...
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### 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. ...