Which converges faster

Fixed Point Theory Appl , Download citation. Received : 20 November Revised : 06 February Published : 07 June Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article. Also, the Ishikawa iteration method is defined by. The Noor iteration method is defined by. In , Agarwal et al. In , Abbas et al. In , Thakur et al. Also, the Picard S-iteration was defined by.

Now, we are ready to provide our main results for contractive maps. Consider the first case for Mann iteration. In this case, we have. This completes the proof. One can use similar conditions instead of the conditions which we will use in our results.

As we know, we can consider four cases for writing the Ishikawa iteration method. In the next result, we indicate each case by different enumeration. Similar to the last result, we want to compare the Ishikawa iteration method with itself in the four possible cases. Consider the following cases of the Ishikawa iteration method :. Then we have.

Thus, we obtain. By using a similar condition, one can show that the iteration 3. Now consider eight cases for writing the Noor iteration method.

We enumerate the cases of the Noor iteration method during the proof of our next result. Consider the case 2. First, we compare the case 2. Now, we compare the case 2. This implies that. Then we get. By using similar proofs, one can show that the case 2. By using similar conditions, one can show that the case 3.

One can easily show that the case 3. We record it as the next lemma. Also by using a similar condition, one can show that the case 3. Similar to Theorem 3. Also, one can show that for contractive maps the case 2. We record these results as follows. Consider the following case in the Abbas iteration method :. In this section, we compare the rate of convergence of some different iteration methods for contractive maps.

Our goal is to show that the rate of convergence relates to the coefficients. Also , the case 2. Thus, the case 3. Now for the case 2. By using a similar proof, we can compare the Thakur-Thakur-Postolache and the Agarwal iteration methods as follows.

Also by using similar proofs, we can compare some another iteration methods. We record those as follows. It is notable that there are some cases which the coefficients have no effective roles to play in the rate of convergence.

By using similar proofs, one can check the next result. One can obtain some similar cases. This shows us that researchers should stress more the probability of the efficiency of coefficients in the rate of convergence for iteration methods.

Then the case 2. It is easy to see that T is a contraction. In Tables 1 - 3 , we first compare two cases of the Mann iteration method and also four cases of the Ishikawa and Agarwal iteration methods separately. From a mathematical point of view, one can see that the Mann iteration 3. We first add our CPU time in Tables 1 - 3 for each iteration method. Also, we provide Figure 1 by using at least 30 times calculating of CPU times for our faster cases in the methods. From a computer-calculation point of view, we get a different answer.

As one can see in the CPU time table, we found that the Agarwal iteration 3. This note emphasizes the difference of the mathematical results and computer-calculation results which have appeared many times in the literature. The next example illustrates Lemma 3. Table 4 shows us that the Abbas iteration 3. One can get similar results about difference of the mathematical and computer-calculating points of views for this example. The Runge-Kutta method finds approximate value of y for a given x.

Only first order ordinary differential equations can be solved by using the Runge Kutta 4th order method. The value of n are 0, 1, 2, 3, …. Some implicit methods have such good stability properties that they can solve stiff initial value problems with step sizes that are appropriate to the behavior of the solution if they are evaluated in a suitable way. The backward Euler method and the trapezoidal rule are examples. The rate of change is how fast the output changes relative to the input, or, on a graph, how fast y changes relative to x.

You can use initial value and rate of change to figure out all kinds of information about functions. Compare them to the exact values of the solution at these points.

For a given differential equation with initial condition. In the Euler method, the tangent is drawn at a point and slope is calculated for a given step size …. This property depends on the mesh and initial condition and differential equations you have considered.

If the exact solution to the differential equation is a polynomial of order n, it will be solved exactly by an n-th Runge-Kutta method.

Because convergence rate of RK4 method is more than Euler. In numerical analysis, predictor—corrector methods belong to a class of algorithms designed to integrate ordinary differential equations — to find an unknown function that satisfies a given differential equation.

Runge-Kutta method is better since higher order derivatives of y are not required. Taylor series method involves use of higher order derivatives which may be difficult in case of complicated algebraic equations. Begin typing your search term above and press enter to search.

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