Is the Bisection Technique accurate for finding roots on a closed interval?

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In summary, The conversation was discussing a technique for finding roots or zeros of a function. The technique involves using an algorithm that checks for sign changes on a given interval. However, this technique may not work if the function is not one-to-one or if the interval is not chosen carefully. The article linked in the conversation provides more information on this technique.
  • #1
MathematicalPhysicist
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in this page they are descrbing the forthmentioned technique.
and I am quite puzzled, because if i get this interval:
[1,2]
let's take the equation f(x)=x^2.
now if we follow the algorithm we find that there might not be a root between them, which is ofcourse absurd. (sqrt2 and sqrt3 are ofcourse included inbetween).

anyway, here is the page http://spiff.rit.edu/classes/phys317/lectures/closed_root/closed_root.html
 
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  • #2
'root' means that f(x) = 0

clearly x^2 is not 0 in the interval you mention, so the algorithm is correct

given that everything in the interval [1,2] is the square root of *something*, even your logic there is flawed.
 
  • #3
The roots or zeros of a function are synonymous with the "x-intercepts" of that function.

The article doesn't mention that the function has to be one-to-one or either increasing or decreasing within the interval for the algorithm to work properly. (Consider a parabola with a vertex below the origin between an interval [x1, x2]. If x2-x1 is greater than the distance between the roots, the algorithm doesn't work.)
 
  • #4
The linked algorithm is poorly written. Like signs at the end points of the test interval indicate either no roots or an EVEN NUMBER of roots on the interval. Like wise a sign change on the interval means an odd number of roots on the test interval. The user of the algorithm must have sufficient knowledge of the function to pick a valid starting interval. The best way to get the need information is to plot the function.
 

FAQ: Is the Bisection Technique accurate for finding roots on a closed interval?

What is the bisection technique?

The bisection technique, also known as the bisection method or binary search, is a mathematical algorithm used to find the roots of a continuous function. It works by repeatedly dividing an interval containing a root in half and checking which half the root lies in, ultimately narrowing down the interval to the desired precision.

How does the bisection technique work?

The bisection technique works by first defining an interval [a, b] where the root is known to exist. Then, the midpoint of the interval is calculated as (a+b)/2. The function is evaluated at this midpoint, and depending on whether the function value is positive or negative, the interval is divided into two halves. This process is repeated until the interval becomes small enough to contain the root within the desired precision.

What are the advantages of using the bisection technique?

One advantage of the bisection technique is that it is guaranteed to converge to a root of a continuous function if the function is continuous and changes sign within the interval. It is also relatively simple to implement and does not require any derivative information.

What are the limitations of the bisection technique?

The bisection technique can be slow compared to other root-finding methods, especially when the interval is large. It also requires that the function changes sign within the interval, which may not always be the case. It also does not provide information about multiple roots within the same interval.

How is the bisection technique used in real-world applications?

The bisection technique is commonly used in various fields such as engineering, finance, and computer science to solve problems that involve finding roots of functions. For example, it can be used to find the break-even point in financial analysis or to optimize algorithms that involve searching for a specific value.

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