How to solve equations using continued fractions?

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In summary, continued fractions can be used to solve polynomial equations, including those with low degrees, and can also be employed to find rapid converging sequences to the roots of equations. They can also be used to solve differential equations. However, there are other algorithms available for solving equations with degrees greater than 5.
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Arian.D
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Is it possible to solve any equation if we use continued fractions? I've heard that polynomial equations could be solved using continued fractions, and I used to obtain one of the several roots of a polynomial equation of low degrees using continued fractions in high school, but I read somewhere that continued fractions could be employed to find rapid converging sequences to the roots of an equation and I guess I even read somewhere that they could be used to solve differential equations!

Could someone inform me about the use of continued fractions to solve equations please? Thanks in advance.
 
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I've found some applications of continued fractions ( http://www.rnta.eu/SecondRNTA/Waldschmidt-Sanna.pdf ) and in section 5 exponential Diophantine equations. Also Pell's theorem plays a role, but I haven't seen applications on general roots. I guess there are better algorithms, and from degree 5 onwards we only have numerical methods.
 

1. What is a continued fraction?

A continued fraction is a mathematical expression that represents a number as a sequence of nested fractions. It is written in the form [a0; a1, a2, a3...], with the a0 term representing the integer part and the remaining terms representing the fractional part.

2. How are continued fractions used to solve equations?

Continued fractions can be used to solve equations by converting the equation into an infinite continued fraction and then approximating the solution by truncating the fraction at a certain point.

3. What is the algorithm for solving equations using continued fractions?

The algorithm for solving equations using continued fractions involves converting the equation into a continued fraction, finding a convergent of the fraction that is close to the actual solution, and then using this convergent to obtain a better approximation of the solution. This process is repeated until the desired level of accuracy is achieved.

4. Can continued fractions be used to solve all types of equations?

No, continued fractions can only be used to solve certain types of equations, such as linear equations with one variable. They are not suitable for solving equations with multiple variables or equations involving trigonometric functions.

5. What are the advantages of using continued fractions to solve equations?

One advantage of using continued fractions to solve equations is that they can provide a more accurate approximation of the solution compared to other methods. They also have a more systematic approach and can be applied to a wide range of equations.

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