Description:DESCRIPTION:
Field of the invention:
The present disclosure generally relates to the technical field of computational devices and, in specific, relates to a system and method that require minimum number of iterations for efficiently resolving non-linear dynamics.
Background of the invention:
Non-linear dynamics are common in a variety of scientific and technical fields. Finding the roots (solutions) of non-linear dynamics is critical for activities such as physical phenomenon modelling, process optimization, and data fitting and analysis. Modelling physical phenomena requires accurately capturing real-world systems, which frequently feature nonlinear interactions. Solving these dynamics allows us to predict and analyze behavior. Optimization problems can be expressed as nonlinear dynamics. Finding the roots allows you to determine the best settings for your system. Curve fitting approaches usually require solving non-linear dynamics to determine the best-fit model for a given dataset.

There is a high demand for computational devices as technology has become more complicated. Furthermore, computational approaches allow computational devices to solve a wide range of mathematical problems, making them invaluable instruments in scientific study, engineering, and data processing. To be explicit, computational approaches such as the Bisection method, Secant method, Newton Raphson method, and others are widely utilized in various software programs to determine the roots of a real function. Many programs presently use the bisection approach to compute the initial approximation of the needed root. For instance: The MATLAB function 'fzero ()' employs the bisection method to locate the root of a real valued function. OCTAVE also uses the function 'fzero ()' to find the root. The module 'SciPy. Optimize' in Python with SciPy has the 'bisection' function for solving dynamics using the bisection method.

In existing technology, GNUOCTAVE provides a built-in function called 'fzero ()'. Mathematica's built-in function 'find root' employs the bisection approach, as does Sagemath's 'find root' function. Many software applications currently employ the Bisection approach to calculate an initial approximation of the needed root. The disadvantage of the Bisection approach is that it takes many iterations to discover the root with the requisite accuracy. The SS weighted averaging approach can replace the bisection method because it requires fewer iterations. Online calculators like codesansar.com, planetcalc.com, and AtoZmath.com include built-in capabilities for calculating roots using bisection method, false position method and many other methods. Thus, the online calculators can create routines to calculate the SS weighted averaging approach.

In numerous industries, the users frequently confront complex challenges that are difficult to solve. The majority of these problems are solved using mathematical tools such as differentiation, integration, curve fitting, and least squares methods, which entail modelling the problem. The solution of these models frequently yields non-linear dynamics, which can be solved using the SS weighted averaging method. Because manual computation takes more time and effort, computer programming of the approach is required to execute it on a digital electronic computer for calculation. Thus, the SS weighted averaging method can be used as a computational tool in computational devices.

The SS weighted averaging and Bisection methods are bracketing methods that use the intermediate value theorem, are relatively simple to comprehend and apply, and will undoubtedly lead to the solution. The SS weighted averaging method finds the solution using the weighted averaging methodology, and weights are assigned in each iteration based on the functional values of the selected interval's end points. In contrast, the Bisection approach gets the solution by averaging the specified interval's end points. This causes the SS weighted averaging approach to converge faster than the Bisection method.

The SS weighted averaging method has all of the same advantages as the bisection method. In addition, the SS weighted averaging method is more efficient than the bisection method. As a result, this can be employed in practically any situation where the bisection approach is used to reduce the number of iterations. The SS weighted averaging approach has applications in several sectors of engineering, including electrical engineering, mechanical engineering, chemical engineering, and computer science. Particularly in domains such as current flow, circuit analysis, mechanical oscillations, beam forces, optimization, and data analysis.

In existing technology, for example, a patent US20010051936A1 discloses a form of a method and system for returning an optimum (or near-optimum) solution to a nonlinear programming problem. By specifying a precision coefficient, the user can influence the flexibility of the returned solution. A population of possible solutions is initialized based on input parameters defining the problem. The input parameters may include a minimum progress and a maximum number of iterations having minimum progress. The solutions are mapped into a search space that converts a constrained problem into an unconstrained problem. Through multiple iterations, a subset of solutions is selected from the population of solutions, and variation operators are applied to the subset of solutions so that a new population of solutions is initialized and then mapped. However, the method and system require maximum number of iterations for returning an optimum (or near-optimum) solution to the nonlinear programming problem.

By addressing all the above mentioned facts, there is a need for a system that requires a minimum number of iterations to efficiently resolve non-linear dynamics. There is also need for a system that utilizes weighted averaging techniques to find the solution to a nonlinear equation. There is also need for a system that obtains the solution within 10 to 20 percentage points lesser number of iterations for a permissible error greater than 0.0001, when compared to the bisection method, depending upon the complexity of the function. There is also need for a system that assigns weights in each iteration through functional values.

There is also need for a system that may decrease 5 to 10 percent of the calculation time by investigating the weights when compared to the bisection method based on the complexity of the function. There is also need for a system that applies in many fields of engineering, such as electrical engineering, mechanical engineering, chemical engineering, computer science. Particularly in domains like flow of current, analysis of circuits, mechanical oscillations, forces on beams, optimization, data analysis etc.
Objectives of the invention:
The primary objective of the present invention is to provide a system that requires a minimum number of iterations to efficiently resolve non-linear dynamics.

Another objective of the present invention is to provide a system that utilizes weighted averaging techniques to find the solution to a nonlinear equation.

The other objective of the present invention is to provide a system that obtains the solution within 10 to 20 percentage points lesser number of iterations for a permissible error greater than 0.0001, when compared to the bisection method, depending upon the complexity of the function.

The other objective of the present invention is to provide a system that applies in many fields of engineering, such as electrical engineering, mechanical engineering, chemical engineering, computer science. Particularly in domains like flow of current, analysis of circuits, mechanical oscillations, forces on beams, optimization, data analysis etc.

Yet another objective of the present invention is to provide a system that may decrease 5 to 10 percent of the calculation time by investigating the weights when compared to the bisection method based on the complexity of the function.

Further objective of the present invention is to provide a system that assigns weights in each iteration through functional values.
Summary of the invention:
The present disclosure proposes a system for resolving non-linear dynamics with lesser iterations in numerous sectors and method thereof. The following presents a simplified summary in order to provide a basic understanding of some aspects of the claimed subject matter. This summary is not an extensive overview. It is not intended to identify key/critical elements or to delineate the scope of the claimed subject matter. Its sole purpose is to present some concepts in a simplified form as a prelude to the more detailed description that is presented later.

In order to overcome the above deficiencies of the prior art, the present disclosure is to solve the technical problem to provide a system and method that require minimum number of iterations for efficiently resolving non-linear dynamics.

According to one aspect, the invention provides a system for resolving non-linear dynamics with lesser iterations. The system comprises a computing device, a network and an application server. The computing device having a processor and a memory for storing and executing one or more instructions. The computing device is in communication with the application server via the network. The processor is configured to execute plurality of modules for performing operations. The plurality of modules comprises a communication module, a display module, an input module, a processing module and an output module.

In one embodiment, the communication module is configured to enable communication between the computing device and the application server via the network. The display module is configured to display one or more attributes for users, thereby enabling to select at least one attribute based on requirements. The input module is configured to enable the users to enter input data upon selecting the at least one attribute. The input data includes a function f(x), the interval (p, q) in which the solution to be obtained, permitted error (?) and weights (m), (n). The input data includes a non-linear function, interval, permitted error and weights.

In one embodiment, the processing module is configured to analyses the input data using a weighted average method, thereby providing solutions for the function f(x). The weighted average method for efficiently resolving non-linear dynamics. At one step, the input data such as the function f(x), the interval in which the solution is obtained (p, q), permitted error (?) and the positive real numbers (m), (n) such that (m) is greater than (n) are collected. At one step, the absolute values |f(p)| and |f(q)| are compared, thereby resulting that the value of |f(p)| is greater than the value of |f(q)|, then the iteration value of (x) is (xi), which is equal to the weighted average of the (p) and (q) by assigning larger weight (m) to (q) and smaller weight (n) to (p).

At one step, the value of |f(p)| is less than the value of |f(q)|, then the iteration value of (x) is (xi), which is equal to the weighted average of (p) and (q) by assigning a larger weight (m) to (p) and a smaller weight (n) to (q). At one step, the f(xi) is calculated, thereby resulting f(xi) is equal to zero, and then the (xi) is required solution and terminate the method.

At one step, the resultant value of f(xi) is not equal to zero, thereby assigning p=p, q=xi if f(p) and f(xi) are opposite in signs or assigning p=xi, q=q if f(q) and f(xi) are opposite in signs. At one step, the method is terminated if absolute value of p-q is less than the permitted error (?), then xi is the required solution, and otherwise repeat the method.

In one embodiment, the system is configured to obtain solution to the non-linear equation within a specified interval using a weighted averaging technique. The functional values f(p) and f(q) are used to assign weights (m), (n) in each iteration where (m), (n) are real numbers greater than zero and m>n.

In one embodiment, the system utilizes the computational devices to build functions such as fzero, find root and bisection. The weighted average method of the system is compared with to bisection method through conducting tests to detect optimum time to obtain the required solution. The solution of the non-linear dynamics is obtained within 10 to 20 percentage lesser number of iterations for a permissible error greater than 0.0001, when compared to the bisection method, depending upon the complexity of the function. The investigation of weights may decrease 5 to 10 percentage of the calculation time, when compared to the bisection method based on the complexity of the function.

According to another aspect, the invention provides a method for resolving non-linear dynamics with lesser iterations through the system. At one step, the communication module enables communication between the computing device and the application server via the network. At one step, the display module displays one or more attributes for users, thereby enabling to select at least one attribute based on requirements. At one step, the input module enables the users to enter the input data upon selecting the at least one attribute.

At one step, the processing module analyzes the input data using the weighted average method, thereby providing solutions for the function f(x). At one step, the output module displays the solution for the function f(x) as output data.

Further, objects and advantages of the present invention will be apparent from a study of the following portion of the specification, the claims, and the attached drawings.
Detailed description of drawings:
The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate an embodiment of the invention, and, together with the description, explain the principles of the invention.

FIG. 1 illustrates a block diagram of a system for resolving non-linear dynamics with lesser iterations, in accordance to an exemplary embodiment of the invention.

FIG. 2 illustrates a flowchart of a weighted average method of the system for resolving non-linear dynamics with lesser iterations, in accordance to an exemplary embodiment of the invention.

FIG. 3 illustrates a flowchart of a method for resolving non-linear dynamics with lesser iterations through the system, in accordance to an exemplary embodiment of the invention.
Detailed invention disclosure:
Various embodiments of the present invention will be described in reference to the accompanying drawings. Wherever possible, same or similar reference numerals are used in the drawings and the description to refer to the same or like parts or steps.

The present disclosure has been made with a view towards solving the problem with the prior art described above, and it is an object of the present invention to provide a system and method that require minimum number of iterations for efficiently resolving non-linear dynamics.

According to one exemplary embodiment of the invention, FIG. 1 refers to a block diagram of a system 100 for resolving non-linear dynamics with lesser iterations. The system 100 applies in many fields of engineering, such as electrical engineering, mechanical engineering, chemical engineering, computer science. Particularly in domains like flow of current, analysis of circuits, mechanical oscillations, forces on beams, optimization, data analysis etc. The system 100 assigns weights to each iteration through functional values. The system 100 comprises a computing device 102, a network 118 and an application server 120.

The computing device 102 having a processor 104 and a memory 106 for storing and executing one or more instructions. The computing device 102 is in communication with the application server 120 via the network 118. The processor 104 is configured to execute plurality of modules 108 for performing operations. The plurality of modules 108 comprises a communication module 110, a display module 111, an input module 112, a processing module 114 and an output module 116.

In one embodiment, the communication module 110 is configured to enable communication between the computing device 102 and the application server 120 via the network 118. The computing devices 102 includes a computer, a tab, a laptop and a smart phone. The network 118 includes wireless network, Wi-Fi and Bluetooth. The application server 120 enables the user to install the required application based on the processor 104 of the system 100. The display module 111 is configured to display one or more attributes for users, thereby enabling to select at least one attribute based on requirements. The one or more attributes include an application icon and calls for the inbuilt function of the application. The input module 112 is configured to enable the users to enter input data upon selecting the at least one attribute. The input data includes a function f(x), the interval (p, q) in which the solution obtained, a permitted error (?) and weights (m), (n). The input data includes a non-linear function, interval, the permitted error (?) and weights.

In one embodiment, the processing module 114 is configured to analyses the input data using weighted average method, thereby providing solutions for the function f(x). The weighted average method for efficiently resolving non-linear dynamics. At one step, the input data such as the function f(x), the interval in which the solution lies (p, q), permitted error (?) and the positive real numbers (m), (n) such that (m) is greater than (n) are collected. At one step, the absolute values |f(p)| and |f(q)| are compared, thereby resulting that the value of |f(p)| is greater than the value of |f(q)|, then the iteration value of (x) is (xi), which is equal to the weighted average of the (p) and (q) by assigning larger weight (m) to (q) and smaller weight (n) to (p).

At one step, the value of |f(p)| is less than the value of |f(q)|, then the iteration value of (x) is (xi), which is equal to the weighted average of (p) and (q) by assigning a larger weight (m) to (p) and a smaller weight (n) to (q). At one step, the f(xi) is calculated, thereby resulting f(xi) is equal to zero, and then the (xi) is required solution and terminate the method.

At one step, the resultant value of f(xi) is not equal to zero, thereby assigning p=p, q=xi if f(p) and f(xi) are opposite in signs or assigning p=xi, q=q if f(q) and f(xi) are opposite in signs. At one step, the method is terminated if absolute value of p-q is less than the permitted error (?), then xi is the required solution, and otherwise repeat the method.

In one embodiment herein, the output module 116 is configured to display the solutions for the function f(x) as output data. The output module 116 is a user interface. The system 100 is configured to obtain solution to the non-linear equation within a specified interval using a weighted averaging technique. The functional values of (p) and (q) are used to assign weights (m), (n) in each iteration where (m), (n) are real numbers greater than zero and m>n.

In one embodiment herein, the system 100 utilizes the computational devices to build functions such as fzero, find root and bisection. The weighted average method of the system 100 is compared with to bisection method through conducting tests to detect optimum time to obtain the required solution. The solution of the non-linear dynamics is obtained within 10 to 20 percentage lesser number of iterations for a permissible error greater than 0.0001, when compared to the bisection method, depending upon the complexity of the function. The investigation of weights decreases 5 to 10 percentage of the calculation time, when compared to the bisection method based on the complexity of the function.

According to one exemplary embodiment of the invention, FIG. 2 refers to a flowchart of the weighted average method of the system for resolving non-linear dynamics with lesser iterations. At step 202, the function f(x), the interval in which the root lies say, (p, q) and the permitted error be ? are determined, thereby choosing two weights say (m), (n) such that (m), (n) are real numbers greater than zero and m>n. At step 204, the absolute values of f(p) and f(q) are compared and represented as |f(p)|, |f(q)| respectively. At step 206, If |f(p)| is greater than |f(q)| then iteration value of x, say xi equals weighted average of p and q where larger weight m is assigned to q and smaller weight n is assigned to p or else iteration value of x, say xi equals weighted average of p, q by assigning larger weight m to p and smaller weight n to q, at step 208. At step 210, determining the f(xi).

At step 212, if f(xi) equals zero then the xi is the required solution and then we can terminate the process or Else, If f(p) and f(xi) are of opposite signs is observe at step 214, then p=p, q = xi is changed at step 216. At step 218, if f(q) and f(xi) are of opposite signs then p = xi, q=q is changed. At step 220, if absolute value of (p-q) is less than ? then terminate the process and then xi will be the required root or otherwise repeat the process from step 204.

In one embodiment herein, a first test procedure is conducted for finding optimum time by using the SS weighted averaging method and the Bisection method, thereby comparing the test results of the SS weighted averaging method with the Bisection method to determine the efficient method for resolving the non-linear equation within less iterations.

In many industries, the temperature maintenance of devices, machines, and substances is crucial to avoid damage. If the temperature exceeds the machine’s tolerance, then the machine stops working, and resulting in significant damage to the industry. Therefore, it is necessary to prevent machine overheating. Hence, it is essential to determine the optimum time required for the machine to cool down, ensuring that the temperature remains below the cutoff point. The machine can operate continuously without interruption. The estimation of time taken for cooling is modeled by using newton’s law of cooling. The newton’s law of cooling states that the rate of change of temperature of a body is proportional to the difference between the temperature of the body and surrounding temperature. Let u0 be the surrounding temperature and u be temperature of the body at any time t, then the rate of change of temperature du/dt? (u-u_0) which implies du/dt=-k (u-u_0) where k is a constant. This differential equation can be easily simplified by means of integration to obtain the solution
u=u_0+ce^(-kt) (1)
Where c is constant. This equation can be used to estimate time t for any given u, c and k. Now rewriting the equation (1) as u_0-u+ce^(-kt)=0 which is a nonlinear equation of the form f(t)=0 when u, c and k are known values. Therefore, t value will be solution of this nonlinear equation which can be solved by SS weighted averaging method.

In an embodiment herein, in an industry, the machine can tolerate a temperature of 49.5oC. It is observed that, initially the machine is at 80oC and cools down to 60oC in 15 minutes when the surrounding temperature is 40oC. What is the time required for the machine to cool down to its tolerance temperature. If u is the temperature of the machine at any time t, then
du/dt=-k (u-40)
Where k is a constant
On integrating, u=40+ce^(-kt) (2)
Substituting initial conditions in equation (2) we obtain c=40 and k= 0.0346573.

Now the time taken by the machine to cool down to its tolerance temperature implies finding the solution of the nonlinear equation 49.5=40+40e^(-0.0346573t) ------(3)which is solved by SS weighted averaging method. At first rewriting the equation (3) we get 40e^(-0.0346573t)-9.5=0.
Assume
f(t) = 40e^(-0.0346573t)-9.5

The first test result for finding optimum time by using SS weighted averaging method. Let
f(t) = 40e^(-0.0346573t)-9.5

The device obtain the optimum time (t) lies between the interval (41, 42) by using intermediate value theorem. So assume p=41 q=42 i = iteration number. Taking m=10, n=5 and permitted error ?=0.0001.

Table 1:
i P q ti f(p) f(q) f(ti)
1 41.000000 42.000000 41.333333 0.159387 -0.169647 0.048439
2 41.333333 42.000000 41.555556 0.048439 -0.169647 -0.024817
3 41.333333 41.555556 41.481481 0.048439 -0.024817 -0.000461
4 41.333333 41.481481 41.432099 0.048439 -0.000461 0.015811
5 41.432099 41.481481 41.465021 0.015811 -0.000461 0.004960
6 41.465021 41.481481 41.475995 0.004960 -0.000461 0.001346
7 41.475995 41.481481 41.479652 0.001346 -0.000461 0.000142
8 41.479652 41.481481 41.480262 0.000142 -0.000461 -0.000059
9 41.479652 41.480262 41.480059 0.000142 -0.000059 0.000008
10 41.480059 41.480262 41.480127 0.000008 -0.000059 -0.000015

Iteration value of optimum time t, t_i={¦((mp+nq)/(m+n) for |f(p)|<|f(q)| @( mq+np)/(m+n) for |f(p)|>|f(q)|)¦

After 10 iterations p =41.480059 q =41.480127 Error = |p-q| = 0.000068< ? .Hence the time taken by the machine to cool down to its tolerance temperature is equal to 41.480 minutes.

The first test result for finding optimum time by using bisection method.

Let f(t) = 40e^(-0.0346573t)-9.5. By intermediate value theorem the required optimum time, t lies in the interval (41, 42). So assume p=41 q=42 i = iteration number.
Iteration value of time t, t_i=(p+q)/2

Table 3:
i P q ti f(p) f(q) f(ti)
1 41.000000 42.000000 41.500000 0.159387 -0.169647 -0.006556
2 41.000000 41.500000 41.250000 0.159387 -0.006556 0.076056
3 41.250000 41.500000 41.375000 0.076056 -0.006556 0.034661
4 41.375000 41.500000 41.437500 0.034661 -0.006556 0.014030
5 41.437500 41.500000 41.468750 0.014030 -0.006556 0.003732
6 41.468750 41.500000 41.484375 0.003732 -0.006556 -0.001413
7 41.468750 41.484375 41.476562 0.003732 -0.001413 0.001159
8 41.476562 41.484375 41.480469 0.001159 -0.001413 -0.000127
9 41.476562 41.480469 41.478516 0.001159 -0.000127 0.000516
10 41.478516 41.480469 41.479492 0.000516 -0.000127 0.000194
11 41.479492 41.480469 41.479980 0.000194 -0.000127 0.000034
12 41.479980 41.480469 41.480225 0.000034 -0.000127 -0.000047
13 41.479980 41.480225 41.480103 0.000034 -0.000047 -0.000007
14 41.479980 41.480103 41.480042 0.000034 -0.000007 0.000013

After 14 iterations p=41.480042 q=41.480103, Error = |p-q|= 0.000061< ?. Hence the time taken by the machine to cool down to its tolerance temperature, t = 41.480 minutes. From the observation of Table 1 and Table 2, it is to be noted that in this case SS weighted averaging method obtains the solution in 10 iterations whereas Bisection method requires 14 iterations to obtain the solution.

In one embodiment herein, a second test procedure is conducted for estimating volume from the given set of data by using the SS weighted averaging method and the bisection method, thereby comparing the test results of the SS weighted averaging method with the bisection method to determine the efficient method for resolving the non-linear equation within less iterations.

Data analysis is crucial in many industries it is required to form an empirical law to properly express the given data, obtained from observations. To obtain empirical laws from available data it is necessary to fit a curve. This can be done by the principle of least squares. These empirical laws are generally in the form of non-linear dynamics which should be solved to analyze the data. The working rule of fitting an equation of the form pv^?=k to the given data where ? and k are constants is explained below. Convert the equation pv^?=k into linear equation of the form Y=A+BX by taking log on both sides. Here Y=log v, A=? ??^(-1) log k, B= ?- ??^(-1), X=log p. Then obtain the normal equation ?¦Y=nA+B?¦X, ?¦XY=A?¦X+B?¦X^2 using the given set of p and v values, here n is number of p values or number of v values in the data. Now solve these two normal equation to find values of A and B. Calculate the values of ? and k using A and B. Substituting the values of ? and k in equation pv^?=k gives required equation that fit the given data. Now for any given value of p we can compute v by using SS weighted averaging method.

For example, in an experiment conducted on a gas, the data of pressure p and volume v of a gas are collected. Now it required to relate pressure and volume of the gas by the relation pv^?=k and estimate the volume when pressure equals 2.69 kg/cm2.

Table 3:
p(kg/cm2) 0.5 1.0 1.5 2.0 2.5 3.0
v(liters) 1.62 1.00 0.75 0.62 0.52 0.46

The set of observation are we get the equation of the best fit for given data as
pv^1.276=1.039 (3)

By using method of least square. now to find the v when p=2.6 rewrite the equation (2) as
v^1.276-0.3996=0 (4)

The solution of equation (4) is solved by SS weighted averaging method to get required volume v.

The second test results for estimating volume from the given set of data by using the SS weighted averaging method. Let f(v) = v^1.276-0.3996 p=0 q=1 i = iteration number. Taking m=10, n=5 and permitted error ?=0.001.
Iteration value of volume v, v_i={¦((mp+nq)/(m+n) for |f(p)|<|f(q)| @( mq+np)/(m+n) for |f(p)|>|f(q)|)¦

Table 4:
i P q Vi f(p) f(q) f(vi)
1 0.000000 1.000000 0.333333 -0.399600 0.600400 -0.153454
2 0.333333 1.000000 0.555556 -0.153454 0.600400 0.072759
3 0.333333 0.555556 0.481481 -0.153454 0.072759 -0.006076
4 0.481481 0.555556 0.506173 -0.006076 0.072759 0.019855
5 0.481481 0.506173 0.489712 -0.006076 0.019855 0.002528
6 0.481481 0.489712 0.486968 -0.006076 0.002528 -0.000344
7 0.486968 0.489712 0.487883 -0.000344 0.002528 0.000613

After 7 approximations p = 0.486968 q =0.487883 and Error = |p-q| = 0.000915. Hence the required volume v= 0.48 liters.

The second test results for estimating volume from the given set of data by using the bisection method. Let f(v) = v^1.276-0.3996, p=0, q=1, i = iteration number and permitted error ?=0.001.
Iteration value of volume v, v_i=(p+q)/2
Table 5:
i P q vi f(p) f(q) f(vi)
1 0.000000 1.000000 0.500000 -0.399600 0.600400 0.013339
2 0.000000 0.500000 0.250000 -0.399600 0.013339 -0.229082
3 0.250000 0.500000 0.375000 -0.229082 0.013339 -0.113536
4 0.375000 0.500000 0.437500 -0.113536 0.013339 -0.051353
5 0.437500 0.500000 0.468750 -0.051353 0.013339 -0.019305
6 0.468750 0.500000 0.484375 -0.019305 0.013339 -0.003056
7 0.484375 0.500000 0.492188 -0.003056 0.013339 0.005124
8 0.484375 0.492188 0.488281 -0.003056 0.005124 0.001030
9 0.484375 0.488281 0.486328 -0.003056 0.001030 -0.001014
10 0.486328 0.488281 0.487305 -0.001014 0.001030 0.000007

After 10 iterations, p=0.486328, q=0.487305, Error=|p-q|=0.000977. Hence the required volume v= 0.48 liters. From the observation of Table 4 and Table 5, it is to be noted that in this case SS weighted averaging method obtains the solution in 7 iterations, whereas bisection method requires 10 iterations to obtain the solution.

In one embodiment herein, a third test procedure is conducted for resolving non-linear dynamics to get solution by using the SS weighted averaging method and the bisection method, thereby comparing the test results of the SS weighted averaging method with the bisection method to determine the efficient method for resolving the non-linear equation within less iterations.

Now finding the solution of the nonlinear equation cos x – x ex =0 with in the interval (0.5, 1) and permissible error is 0.001. The third test results for resolving non-linear dynamics to get solution by using the SS weighted averaging method. Let f(x) = cos x – x ex, p=0.5, q=1, i = iteration number. Taking m=10 and n=5.
Iteration value of x, x_i={¦((mp+nq)/(m+n) for |f(p)|<|f(q)| @( mq+np)/(m+n) for |f(p)|>|f(q)|)¦

Table 6:
i P q xi f(p) f(q) f(xi)
1 0.500000 1.000000 0.666667 0.053222 -2.177980 -0.512602
2 0.500000 0.666667 0.555556 0.053222 -0.512602 -0.118675
3 0.500000 0.555556 0.518519 0.053222 -0.118675 -0.002317
4 0.500000 0.518519 0.512346 0.053222 -0.002317 0.016389
5 0.512346 0.518519 0.516461 0.016389 -0.002317 0.003940
6 0.516461 0.518519 0.517833 0.003940 -0.002317 -0.000229
7 0.516461 0.517833 0.517375 0.003940 -0.000229 0.001162

After 7 iterations p = 0.517375 q= 0.517833 Error = |p-q|=0.000458. The solution of the given nonlinear equation with in the interval (0.5,1) with permissible error 0.001, x= 0.517.

The third test results for resolving non-linear dynamics to get solution by using the bisection method. Let f(x) = cos x – x ex, p=0.5, q=1, i = iteration number.
Iteration value of x, x_i=(p+q)/2
Table 7:
i P q xi f(p) f(q) f(xi)
1 0.500000 1.000000 0.750000 0.053222 -2.177980 -0.856061
2 0.500000 0.750000 0.625000 0.053222 -0.856061 0.356691
3 0.500000 0.625000 0.562500 0.053222 -0.356691 -0.141294
4 0.500000 0.562500 0.531250 0.053222 -0.141294 -0.041512
5 0.500000 0.531250 0.515625 0.053222 -0.041512 0.006475
6 0.515625 0.531250 0.523438 0.006475 -0.041512 -0.017362
7 0.515625 0.523438 0.519531 0.006475 -0.017362 -0.005404
8 0.515625 0.519531 0.517578 0.006475 -0.005404 0.000545
9 0.517578 0.519531 0.518555 0.000545 -0.005404 -0.002427
10 0.517578 0.518555 0.518066 0.000545 -0.002427 -0.000940

After 10 approximations p=0.517578, q=0.518066, Error = |p-q|=0.000488. The solution of the given nonlinear equation with in the interval (0.5, 1) with permissible error 0.001, x= 0.517. From the observation of Table 4 and Table 5, it is to be noted that in this case SS weighted averaging method obtains the solution in 7 iterations, whereas bisection method requires 10 iterations to obtain the solution bisection method.

According to another exemplary embodiment of the invention, FIG. 3 refers to a flowchart 300 of a method for resolving non-linear dynamics with lesser iterations through the system 100. At step 302, the communication module 110 communicates with the application server 120 via the network 118, thereby enabling users to install the required computational application. At step 304, the display module 111 displays one or more attributes for users, thereby enabling to select at least one attribute based on requirements. At step 306, the input module 112 enables the users to enter input data upon calling for inbuilt function in the application.

At step 308, the processing module 114 analyzes the input data using the weighted average method, thereby providing solution for the function f(x). At step 310, the output module 116 displays the solution for the function f(x) as output data.

Numerous advantages of the present disclosure may be apparent from the discussion above. In accordance with the present disclosure a system 100 for resolving non-linear dynamics with lesser iterations in numerous sectors and method thereof is disclosed. The proposed invention provides the system 100 that requires a minimum number of iterations to efficiently resolve non-linear dynamics. The system 100 utilizes weighted averaging techniques to find the solution to a nonlinear equation. The system 100 obtains the solution within 10 to 20 percentage points lesser number of iterations for a permissible error greater than 0.0001, when compared to the bisection method, depending upon the complexity of the function.

The proposed invention provides the system 100 applies in many fields of engineering, such as electrical engineering, mechanical engineering, chemical engineering, computer science. Particularly in domains like flow of current, analysis of circuits, mechanical oscillations, forces on beams, optimization, data analysis etc. The system 100 may decrease 5 to 10 percent of the calculation time by investigating the weights when compared to the bisection method based on the complexity of the function. The system 100 assigns weights to each iteration through functional values.

It will readily be apparent that numerous modifications and alterations can be made to the processes described in the foregoing examples without departing from the principles underlying the invention, and all such modifications and alterations are intended to be embraced by this application.
, Claims:CLAIMS:
I/We Claim:
1. A system (100) for resolving non-linear dynamics with lesser iterations, comprising:
a computing device (102) having a processor (104) and a memory (106) for storing and executing one or more instructions, wherein the computing device (102) is in communication with an application server (120) via a network (118),
wherein the processor (104) is configured to execute plurality of modules (108) for performing operations,
wherein said plurality of modules (108) comprises:
a communication module (110) configured to enable communication between the computing device (102) and the application server (120) via the network (118);
a display module (111) configured to display one or more attributes for users, thereby enabling to select at least one attribute based on requirements;
an input module (112) configured to enable the users to enter input data upon selecting the at least one attribute, wherein the input data includes a function f(x), an interval (p, q) in which a solution lies, a permitted error (?) and weights m, n;
a processing module (114) configured to analyze the input data using a weighted average method, thereby providing solutions for the function f(x); and
an output module (116) configured to display the solutions for the function f(x) as output data,
whereby the system (100) provides solutions for the non-linear dynamics with lesser iterations using the weighted average method.
2. The system (100) as claimed in claim 1, wherein the weighted average method comprises:
collecting input data that is the function f(x), the interval (p, q) in which the solution lies, permitted error (?) and
choosing the positive real numbers (m), (n) as weights such that (m) is greater than (n);
comparing the absolute values of |f(p)| and |f(q)|, thereby resulting that the value of |f(p)| is greater than the value of |f(q)|, then the iteration value of (x) is (xi), which is equal to the weighted average of the (p) and (q) by assigning larger weight (m) to (q) and smaller weight (n) to (p);
observing the value of |f(p)| is less than the value of |f(q)|, then the iteration value of (x) is (xi), which is equal to the weighted average of (p) and (q) by assigning a larger weight (m) to (p) and a smaller weight (n) to (q);
calculating f(xi), thereby observing the resultant value of f(xi) is equal to zero, then the (xi) is required solution and terminate the method;
observing the resultant value of f(xi) is not equal to zero, thereby assigning p=p, q=xi if f(p) and f(xi) are opposite in signs or assigning p=xi, q=q if f(q) and f(xi) are opposite in signs; and
terminating the method If absolute value of p-q is less than ?, then xi is the required solution, and otherwise repeat the weighted average method.
3. The system (100) as claimed in claim 1, wherein the input data includes a non-linear function, time intervals, the permitted error (?) and weights m, n;
4. The system (100) as claimed in claim 1, wherein the system (100) is configured to obtain solutions to the non-linear equations within a specified interval using the weighted averaging technique.
5. The system (100) as claimed in claim 1, wherein the functional values f(p) and f(q) are used to assign weights (m), (n) in each iteration such that (m), (n) are real numbers greater than zero and m>n.
6. The system (100) as claimed in claim 1, wherein the system (100) is utilized the computational device (102) to build functions such as fzero, find root and bisection.
7. The system (100) as claimed in claim 1, wherein the weighted average method of the system (100) is compared with a bisection method through conducting tests to detect optimum time to obtain the required solution.
8. The system (100) as claimed in claim 1, wherein the solution of the non-linear dynamics is obtained within 10 to 20 percentage points lesser number of iterations for a permissible error greater than 0.0001 when compared to the bisection method depending upon the complexity of the function.
9. The system (100) as claimed in claim 1, wherein the investigation of weights decreases 5 to 10 percentage of the calculation time when compared to the bisection method based on the complexity of the function.
10. A method for resolving non-linear dynamics with lesser iterations through a system (100), comprising:
enabling, by a communication module (110), communication between a computing device (102) and an application server (120) via a network (118);
displaying, by a display module (111), one or more attributes for users, thereby enabling to select at least one attribute based on requirements;
enabling, by an input module (112), the users to enter input data upon selecting the at least one attribute;
analyzing, by a processing module (114), the input data using a weighted average method, thereby providing solutions for the function f(x); and
displaying, by an output module (116), the solutions for the function f(x) as output data.