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No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Solution of non-linear equations 1. Urroz, September In this document I present methods for the solution of single non-linear equations as well as for systems of such equations.

SSoolluuttiioonn ooff aa ssiinnggllee nnoonn--lliinneeaarr eeqquuaattiioonn Equations that can be cast in the form of a polynomial are referred to as algebraic equations.

Equations involving more complicated terms, such as trigonometric, hyperbolic, exponential, or logarithmic functions are referred to as transcendental equations.

The methods presented in this section are numerical methods that can be applied to the solution of such equations, to which we will refer, in general, as non-linear equations.

We will present the Newton-Raphson algorithm, and the secant method. In the secant method we need to provide two initial values of x to get the algorithm started.

In the Newton-Raphson methods only one initial value is required. The Newton-Raphson method consists in obtaining improved values of the approximate root through the recurrent application of equation above.

The figure below illustrates the way in which the solution is found by using the Newton- Raphson method. This line intersects the x-axis i.

Upcoming SlideShare. Like this presentation? Why not share! Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode.

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Prepared by T. Sathiyabama M. Sahaya Pretha K.

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Solution of non-linear equations 1. Urroz, September In this document I present methods for the solution of single non-linear equations as well as for systems of such equations.

SSoolluuttiioonn ooff aa ssiinnggllee nnoonn--lliinneeaarr eeqquuaattiioonn Equations that can be cast in the form of a polynomial are referred to as algebraic equations.

Equations involving more complicated terms, such as trigonometric, hyperbolic, exponential, or logarithmic functions are referred to as transcendental equations.

The methods presented in this section are numerical methods that can be applied to the solution of such equations, to which we will refer, in general, as non-linear equations.

We will present the Newton-Raphson algorithm, and the secant method. In the secant method we need to provide two initial values of x to get the algorithm started.

In the Newton-Raphson methods only one initial value is required. The Newton-Raphson method consists in obtaining improved values of the approximate root through the recurrent application of equation above.

The figure below illustrates the way in which the solution is found by using the Newton- Raphson method.

This line intersects the x-axis i. Proceeding with the iteration we can see that the intersection of consecutive tangent lines with the x-axis approaches the actual root relatively fast.

The main disadvantage is that you need to know not only the function f x , but also its derivative, f' x , in order to achieve a solution.

The secant method, discussed in the following section, utilizes an approximation to the derivative, thus obviating such requirement.

The programming algorithm of any of these methods must include the option of stopping the program if the number of iterations grows too large. How large is large?

That will depend of the particular problem solved. However, any Newton-Raphson, or secant method solution that takes more than iterations to converge is either ill-posed or contains a logical error.

Debugging of the program will be called for at this point by changing the initial values provided to the program, or by checking the program's logic.

It uses as arguments an initial value and expressions for f x and f' x. The solutions found are stored in variable xs while the required number of iterations is stored in variable is.

Most of the time, about 50 iterations are required. The following example shows a case in which the solution actually diverges: 6.

It requires two initial values and an expression for the function, f x. The following commands call the function secant.

Notice also that, the closer the initial guess is to zero, the less number of iterations to convergence are required. NOTE: If the function of interest is defined by an m-file, the reference to the function name in the call to newton.

We can also use as convergence criteria the difference between consecutive values of the solution, i.

As the number of equations and unknowns, n, increases, so does the number of elements in the Jacobian, n2. Solution using function newtonm Next, we use function newtonm to solve the problem postulated earlier.

This approach is a generalization of the secant method for a single non-linear equation. To calculate the Jacobian we proceed by columns, i.

Therefore, we try one more approximation to the solution, i. For the solution of systems of non-linear equations Matlab provides function fsolve as part of the Optimization package.

Since this is an add-on package, function fsolve is not available in the student version of Matlab. You just clipped your first slide!

Clipping is a handy way to collect important slides you want to go back to later. Show related SlideShares at end. WordPress Shortcode.

Published in: Engineering. Full Name Comment goes here. Are you sure you want to Yes No. They can help you with any type of writing - from personal statement to research paper.

Due to this service you'll save your time and get an essay without plagiarism. Tracy Duncan Hello! I can recommend a site that has helped me. Wanda Whitley Snaptits big-breasted girls snaps only.

View pics Show More. Vinjamuri Vijayalakshki , No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds.

No notes for slide. Sharpening spatial filters 1. Prepared by T. Sathiyabama M. Sahaya Pretha K. Shunmuga Priya R. Algorithm 1. Using Laplacian filter to original image 2.

And then add the image result from step 1 and the original image 3. Its result only horizontal and vertical.

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