Find Roots: Explaining the Need for Numerical Methods

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In summary, finding the zeros of a function is a fundamental task in engineering, used in designing transfer functions and evaluating stability. It is also used in optimization, where the maximal value is found by taking derivatives and finding the points where they are zero. This process can be applied to any situation represented by a rational or polynomial equation to find the solution for the unknown variable.
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vigintitres
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I am in a numerical methods class, which uses MATLAB and c to do methods like regular falsi and Newton raphson. I should know this, but why do we bother finding the value of x that makes our function evaluate to zero? Is it so that we have some basis as to where to start or stop a certain process or something? I guess I'm really asking why do we need to find roots when you have an equation for some thing that you are analyzing. Thank you
 
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vigintitres said:
I am in a numerical methods class, which uses MATLAB and c to do methods like regular falsi and Newton raphson. I should know this, but why do we bother finding the value of x that makes our function evaluate to zero? Is it so that we have some basis as to where to start or stop a certain process or something? I guess I'm really asking why do we need to find roots when you have an equation for some thing that you are analyzing. Thank you

Good question. We find roots of the numerator and denominator of a transfer function in order to find where the zeros and poles of the tranfer funtion are located in frequency. This is used in the design of the transfer function, to shape it as a LP, BP, HP, or other function. It also helps us evaluate the stability of the transfer function for various input signals, etc.

http://en.wikipedia.org/wiki/Pole–zero_plot

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  • #3
Optimization is one of the most important and common tasks in engineering. Optimization means finding the maximal value of something with respect to some variables. To find the maximal value, you take derivatives with respect to your variables, and then find the places where those derivatives are zero.

Thus, finding the zeros of a function is one of the most fundamental mathematical tasks in all engineering.

- Warren
 
  • #4
chroot said,
finding the zeros of a function is one of the most fundamental mathematical tasks in all engineering.
You can say that again!

Any time a situation can be represented with a rational or polynomial (or who knows what other type of) equation, you may transform that equation into a form of an expression equal to zero. If this is an equation in one variable, then certainly, you are essentially looking for one or more roots as the solution to the unknown variable. Note that the roots may not always be rational; they may not always be Real; but you would still be interested in finding their values. A person does not need to be an engineer in order to accept that. (this topic is in the General Engineering board)
 

1. What are numerical methods for finding roots?

Numerical methods are computational techniques used to approximate the roots of a mathematical equation. This is necessary when the roots cannot be found analytically or when the equation is too complex to solve using traditional algebraic methods.

2. Why is it important to find roots accurately?

Roots are essential in many scientific and engineering applications, such as optimization, modeling, and simulation. Accurate root finding is crucial for obtaining reliable and precise results in these fields.

3. What are some common numerical methods for finding roots?

Some commonly used numerical methods for finding roots include the bisection method, the Newton-Raphson method, and the secant method. These methods differ in their approach and complexity, but all aim to approximate the roots of an equation.

4. How do numerical methods differ from analytical methods for finding roots?

Analytical methods involve solving equations symbolically using algebraic techniques, while numerical methods use numerical approximations to find the roots. Numerical methods are generally more efficient for complex equations, while analytical methods are limited to simpler equations that can be solved algebraically.

5. Can numerical methods always guarantee accurate root finding?

No, numerical methods are subject to round-off errors and other sources of numerical instability, which can lead to inaccuracies in the computed roots. It is important to choose the appropriate method and consider the limitations of numerical methods when using them for root finding.

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