# Equation root finding

1. What is an equation?

An equation is a mathematical statement indicating that two algebraic expressions are equal. Single variable equations have the form: f(x)=0.

2. What is root finding?

Determining a solution of an equation is essentially the same as finding a root of a function, that is, a zero of the function.

3. What is the purpose of root finding?

Many engineering problems involve single variable equations. Some applications for root finding are: systems of equilibrium, elliptical orbits, the van der Waals equation, and natural frequencies of spring systems.

4. What algorithms does DeadLine use to solve equations?

Roots of equations are used to solve many engineering problems. Some nonlinear single variable equations are hard to solve analytically, and therefore numerical methods are used to solve these equations. The software solves not just algebraic equations, but also exponential equations and trigonometric equations. While there is no flawless technique to solve equations, DeadLine combines the most successful methods in order to deliver you the right answer.

5. What does root bracketing mean?

Bracketing algorithms begin with a bounded region known to contain a root. The size of this bounded region is reduced, until it encloses the root to a desired tolerance. Algorithms that proceed by bracketing a root are guaranteed to converge.

6. How can I use the graph of the function?

The freeware plots the graph of the function and automatically displays all the real roots of the equation in a scrollable list. If you want to locate a root on the graph, select it from the list and analyze the graph. You can zoom in, zoom out or move the view area to the left, to the right, upwards or downwards.

7. Can I save the roots?

If you want to include the roots in your documents, you have the option to export them in a html file located in a special folder. You can save equation roots and the extrema. Save the graph in one of the many file formats, including GIF, PNG, BMP, TIFF, or copy the graph to clipboard, in order to paste it in other applications.

8. What types of functions does DeadLine support?

The most important math functions:

 Category Function name Short name trigonometric functions sine sin cosine cos tangent tan inverse trigonometric functions inverse sine asin inverse cosine acos inverse tangent atan hyperbolic functions hyperbolic sine sinh hyperbolic cosine cosh hyperbolic tangent tanh inverse hyperbolic functions inverse hyperbolic sine asinh inverse hyperbolic cosine acosh inverse hyperbolic tangent atanh exponential functions and other functions natural logarithm ln logarithm base 10 log exponential (e raised to the power of a number) exp square root sqrt absolute value (the absolute value of a number is the number without its sign) abs sign sgn

The constant e, the base of the natural logarithm, equals approximately 2.71828182845904.

9. I want to know if a future upgrade will include second derivative and inflexion points.

DeadLine 2 evaluates and displays the second derivative. To find the inflexion points, you should solve the equation f ′′(x) =0. If x0 is a root of the equation, test if f ′′ changes the sign in x0. If that's the case, x0 is an inflexion point.

10. How does DeadLine evaluate the functions? It is very fast.

DeadLine converts the function to a parse tree representation. To evaluate the function, the software traverses the tree in pre-order. That's faster than parsing the expression at every evaluation. Check this VB6 code to see how to evaluate functions differently. The algorithm is a VB module from DeadLine 1.0 source code that also includes root-finding functions. The code for symbolic derivation solves an ACM problem.

11. For a given function I would rather have a tool to edit the parameters and then see the influence on the graph.
From DeadLine 2.25, you can solve parametric equations. The equation can include any number of parameters that should be given initial values. Then you can modify their values and have a look at the new graph and the corresponding roots. Root-finding just got better.

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