Statistical Study of Bisection Method for Cubic Equations with Random Coefficients

: The bisection method is an iterative approach used in numerical analysis to find nonlinear equations' solutions. It is an easily understandable and simple method that assured convergence. The main purpose of this paper is to study how the parameters of a probability distribution which covers both discrete and continuous distribution that characterizes the coefficient of a cubic polynomial influence the convergence of the bisection method. Distributions covered among discrete and continuous are discrete uniform distribution, continuous uniform distribution and normal distribution. In case of both discrete and continuous uniform distribution inputs, the parameter r indicates the distribution interval [-r, r]. In case of normal distribution input, the parameters m and sigma implies mean and standard deviation respectively. The parameter values are gradually changed. In discrete as well as continuous uniform distribution inputs, and it is found that a second degree polynomial equation can be used to predict the average iteration for a given parameter value, i.e., second-degree polynomial is the best fit. On comparison between discrete and continuous uniform distributions, even the coefficients of both the second-degree polynomial are almost same. For normal distribution input, the average iteration does not depend upon the standard deviation when the mean is fixed and the standard deviation is varying whereas second degree polynomial is again the best fit when the standard deviation is fixed and the mean is varying. This means the average iteration depends upon mean of the normal distribution. Overall, our paper concludes that: I


Introduction 1.1 Introduction to bisection method
It is sometimes difficult to locate the solution of equations f(x)=0 in scientific and technical inquiries. Either f(x) is a quadratic or cubic or biquadratic equation, there are algebraic approach for locating the roots in terms of the coefficients. However, if f(x) is a higher-degree polynomial or an equation with transcendental functions, algebraic approaches are not available. For instance, when 0 , g, u, 0 , and are supplied, the equation is a nonlinear equation for t. This kind of equation is used in rocket research. It is difficult to locate the roots of such nonlinear equations which is frequently encountered in engineering. As a result, several numerical methods have been developed with the goal of offering effective ways to discover numerical solutions to such issues, among which the bisection method was one of the first numerical methods to be developed. The next section will include the explanation of the bisection method.

Bisection method
To "bisect" something implies to slice it down the middle. The bisection method reduces the search area by half at each stage while looking for a solution. The bisection method is thought to possess a linear rate of convergence and provides acceptable accuracy.

Literature review
The fundamental approach to finding a root is the bisection method. Every loop implies a halving of the interval. Since f(a) and f(b) should have different signs and "f" is a continuous function in the interval [a, b], then method will surely converge to a root of "f". The role of bisection method by C Solanki, P Thapliyal, and K Tomar [1] helped us to learn more about the area of the bisection method. In addition to focusing on the bisection method's importance in computer science research, this work also offered a novel method that combines bisection with other methods, such as the Newton-Raphson method which keeps the root bracketed while enabling us to take advantage of the Newton-Raphson method's speed. To further address the issue of nonlinear unconstrained minimization, E.Y. Morozova [2] suggested an adaptation of a new multidimensional bisection method for minimizing function over the simplex. This method does not need the function to be differentiable and is guaranteed to converge to the minimizer for the class of strictly unimodal functions.
Another interesting research study on the bisection method for triangles was conducted by Andrew Adler [3]. According to this study, the longest edge of a triangle was picked and bisected to give birth to two daughter triangles, and continued the bisection procedure indefinitely. He established precise estimates for the longest j th generation edge and demonstrated that the infinite family of triangles so formed falls into a finite number of similarity classes. A geometrical approach to the bisection method by Cl audio Gutierrez, Flavio Gutierrez and Maria-Cecilia Rivara [4] helped us to know that how the behavior of the bisection method depends on the classification of triangles which is to be bisected. Since, the bisection method is the consecutive bisection of a tri angle by the median of the longest side. Therefore, partition of the triangles into classes reflects this behavior by taking into account some fundamental geometrical properties. Its main finding is an asymptotic upper constraint on the total number of triangle similarity classes that may be established on an iterative bisection-created mesh, that was previously unknown. It shows that there are finite number of directions on the plane that may be given by the sides of the resultant triangles. In addition to these research articles, we made use of other books and works that are cited in the reference section ([5]-[10]).
For further literature on bisection method, the reader is referred to [11]- [21].

Motivation and problem statement
Sometimes we can solve problems in a good, easy, or accurate way. For instance, equations like quadratic and linear equations may be solved precisely. However, certain equations might be considerably trickier to solve than others precisely. In rare cases, it may even be difficult to pen down an accurate expression for a solution.
Exercises in mathematics textbooks for schools are frequently purposefully made to provide exact solutions. However, there is no reason to anticipate a particularly good outcome when solving many mathematical equations derived from real-life situations. Most of the time, what we can expect is an approximate solution with the required level of precision. We can use approximate numerical methods to obtain a solution when equations are challenging to solve. Sometimes getting an approximate solution is more effective.
This paper primarily focuses on the bisection method, one of the widely used numerical methods for locating roots. It determines how the parameter of a probability distribution which characterizes the coefficient of a cubic polynomial influence the convergence of the bisection method. Therefore, this study would help us in taking a decision whether to recommend or not to recommend bisection method for a solution if we have prior knowledge that the coefficient of the equations to be solved coming from a particular probability distribution. Further if we have knowledge of parameters of the underlined distributions we may be able to predict the average iterations as the function of the distribution parameter. Research in this direction might be important in tackling many issues emerging in diverse fields of higher mathematics. Such a statistical analysis of the bisection method speeds up the process of solving a problem.

Paper alignment
The alignment of this paper is as follows: Section 1: This section presents the background and basis for the paper. It will provide an overview of bisection method, it's methodology and research problem. Section 2: This section examines the performance of bisection method for a cubic equation with Discrete Uniform coefficients through a statistical approach.
Section 3: This section examines the performance of bisection method for a cubic equation with Continuous Uniform coefficients through a statistical approach.
Section 4: This section examines the performance of bisection method for a cubic equation, with coefficients normally distributed, through a statistical approach.
Section 5: This section provides a summary of the results and makes recommendations for further study.

Study on Cubic Equation for Discrete Uniform Coefficients 2.1 Overview
The term "uniform distribution" in statistics refers to a kind of probability distribution wherein each potential outcome has an equal chance of occurring, i.e., the probability is constant while every variable has an equal number of chances of being the outcome.
For illustration, each person who passes by has an equal chance of receiving the 100-rupee note, if you were to start randomly handing out the note while standing on a street corner. The probability as a percentage is equals to 1/ entire number of possibilities (the number of onlookers). On the other hand, the chances of short persons or women receiving the 100-rupee note are higher than those of other onlookers if you favour them. But this is not what is meant by uniform probability.
Based on the types of probable outcomes, uniform distribution can be divided into two categories: Discrete and Continuous. This section will cover the discrete uniform distribution in detail.

Discrete uniform distribution
The discrete uniform distribution is a statistical distribution in probability theory and statistics where the probability of outcomes is equal and has finite values. For instance, the possible results of throwing a 6-sided die. 1, 2, 3, 4, 5 and 6 are the possible values. Each of the six numbers has a similar chance of appearing in this scenario. Consequently, each time the 6-sided die is thrown, each side gets a chance of 1/6.
There is a finite number of values. When rolling a fair die, it is impossible to obtain a value of 1.3, 4.2, or 5.7. The distribution, however, is no longer uniform if a second die is added, and they are both thrown, as the likelihood of the sums is not the same. The likelihood of tossing a coin is another straightforward illustration. There can only be two outcomes in such a situation. Consequently, 2 is the finite value.
To be more specific, take x to be a discrete random variable with ɱ values in interval [a,b]. Let X has a discrete uniform distribution if its probability mass function (pmf) can be expressed as follows:

Expected value and variance
Two statistics which are often obtained are the expected value and the variance. The discrete uniform random variable's expected value is given by: which, for discrete uniform variate X is, The expression for variance is given by: Where, ℰ( 2 ) is given by: which, in our case, gives Where, is standard deviation.

Methodology
This section will provide the method that has been used to fulfil the objective of this section. Here, coefficients of cubic equation have been generated using the function dis_unirand() which includes inbuilt function rand and rng.

dis_unirand()
u=dis_unirand(r) generates random numbers from the discrete uniform distribution specified by r which implies the range upto which random number is generated i.e.
[-r, r]. The function dis_unirand() includes following functions: rand: X=rand() generates a random scalar in the range (0,1) that is chosen at random from the uniform distribution.

Rng:
In random number generator rng is used to change the seed in MATLAB to prevent using the same random numbers repeatedly. By producing a seed based on the current time, rng provides a simple method for doing that. 'Shuffle' reseeds the generator with a different seed each time you use it. rng can be called without any inputs to reveal the exact seed that was used. Example:

INPUT ARGUMENTS
In MATLAB, the `rand()` function is used to obtain random numbers from a uniform distribution. The function can take one or two input arguments. When called with a single argument, ` ŋ`, `rand(ŋ)` obtain a ` ŋ`-by-` ŋ` matrix of random numbers uniformly distributed between 0 and 1.
Alternatively, when called with two arguments, ` ŋ1` and ` ŋ2`, `rand(ŋ1, ŋ2)` generates a ` ŋ1`-by-` ŋ2` matrix of random numbers with same distribution. In both cases, the resulting matrix of random numbers is returned as the output of the `rand()` function.
MATLAB CODE -REFER APPENDIX A

Steps to follow in matlab code
1.Enter the range of the random number as r and the value of tolerance is assumed to be 0.001. 2.Using the function dis_unirand(), the cubic equation's coefficients are generated at random from a discrete uniform distribution.

Results
Following the above instructions in the MATLAB code stated in APPENDIX A, the mean and standard deviation of iteration (k) for a certain range [-r, r] of parameters over 100 trials were obtained, as shown in Table 1.  From fig. 1, it is clear that resultant second degree polynomial equation can be used to predict the average iteration for a given parameter.

Study on Cubic Equation for Continuous Uniform Coefficients 3.1. Overview
Instead of being discrete, some uniform distributions are continuous. The most commonly used distribution among the two is a continuous uniform distribution. Every outcome in this distribution has an equal chance of appearing though the number of outcomes is infinite. In the previous section, discrete uniform distribution has been discussed broadly. Now, this section contains a detailed analysis of continuous uniform distribution.

Continuous uniform distribution
The random variable in a continuous uniform distribution, X, can have values between γ and δ (lower and upper bounds). Both γ and δ are referred to as the continuous uniform distribution parameters in the discipline of statistics. We cannot have a result that is either bigger than δ or smaller than γ. Every variable has an identical chance of appearing in a continuous uniform distribution, also known as a rectangle distribution, where the density function is constant or flat. It has infinite number of outcomes in a given range.
For example: The time it takes a student to finish a mathematics test ranges evenly between 30 and 60 minutes, despite the fact that there are an unlimited number of points between 30 and 60. To be more specific, take x as a continuous random variable within the interval [γ, δ], X ~ U(γ, δ), if its probability distribution function (pdf) can be expressed as follows:

Expected value and variance
The continuous uniform random variable's expected value is given by: Hence, the expected value is The expression for continuous uniform distribution's variance is given by: Where, ℰ( 2 ) is given by: Hence, the variance is Where, is standard deviation.

Methodology
This section will provide the method that has been used to fulfil the objective of this section. Here, coefficients of the cubic equation have been generated using the function con_unirand() which includes inbuilt function rand and rng.

con_unirand()
u=con_unirand(r) generates random numbers from the continuous uniform distribution specified by r which implies the range upto which random number is generated i.e.

Rng:
In random number generator rng is used to change the seed in MATLAB to prevent using the same random numbers repeatedly. By producing a seed based on the current time, rng provides a simple method for doing that. 'Shuffle' reseeds the generator with a different seed each time you use it. rng can be called without any inputs to reveal the exact seed that was used.

INPUT ARGUMENTS
In MATLAB, the `rand()` function is used to obtain random numbers from a uniform distribution. The function can take one or two input arguments. When called with a single argument, ` ŋ`, `rand(ŋ)` obtain a ` ŋ`-by-` ŋ` matrix of random numbers uniformly distributed between 0 and 1.
Alternatively, when called with two arguments, ` ŋ1` and ` ŋ2`, `rand(ŋ1, ŋ2)` generates a ` ŋ1`-by-` ŋ2` matrix of random numbers with same distribution. In both cases, the resulting matrix of random numbers is returned as the output of the `rand()` function.
MATLAB CODE -REFER APPENDIX B

Steps to follow in matlab code
1.Enter the range of the random number as r and the value of tolerance is assumed to be 0.001. 2.Using the function con_unirand(), the cubic equation's coefficients are generated at random from a continuous uniform distribution.

Results
Following the above instructions in the MATLAB code stated in APPENDIX B, the mean and standard deviation of iteration (k) for a certain range [-r, r] of parameters over 100 trials were obtained, as shown in Table 2.  From fig. 2, it is clear that resultant second degree polynomial equation can be used to predict the average iteration for a given parameter.

Comparison between discrete and continuous uniform distribution results
In discrete as well as continuous uniform distribution, second degree polynomial equation can be used to predict the average iterations for a given parameter .i.e., second degree polynomial is the best fit. On comparison, even the coefficients of both the y = -5E-08x 2  Mean iterations Parameter r of continuous uniform distribution second-degree polynomial are almost same which implies that the average iteration does not depend on whether the distribution is discrete or continuous but rather depend on the range of the distribution which is a parameter. Hence, in uniform distribution second degree polynomial is the best fit.

Study on Cubic Equation for Coefficients Normally Distributed 4.1. Overview
The normal distribution, sometimes known as the Gaussian distribution, is the most important probability distribution in statistics for independent, random variables. Its well-known bell-shaped curve is readily noticed in statistics reports.
Most of the observations tend to cluster around the central peak of a normal distribution, which is a type of continuous probability distribution that is symmetrically distributed around its mean. The probability of obtaining values that are further away from the mean decreases at an equal rate in both directions, and extreme values in the tails of the distribution are also infrequent. It's worth noting that while the normal distribution is symmetrical, not all symmetrical distributions are normal.
The normal distribution is a probability distribution that describes how a variable's values are distributed, similar to other probability distributions. Due to its ability to accurately represent the distribution of values for many natural phenomena, it is considered the most important probability distribution in statistics. Normal distributions are commonly used to describe characteristics that are the result of multiple independent processes. For example, the normal distribution is commonly observed in traits such as height, blood pressure, measurement error, and IQ scores.

Parameters of normal distribution
The normal distribution's parameters ultimately determine its structure and probabilities, just like with any other probabili ty distribution. The mean and standard deviation are the two variables that jointly form the normal distribution. There is not just one variant of the Gaussian distribution. Instead, the form alters according to the values of the parameter.

Mean:
The mean of the normal distribution represents its center of tendency and identifies the location of the peak of the bell curve. The majority of the data is clustered around the mean. When the mean is changed on a graph, the entire curve shifts to the left or right on the X-axis, as illustrated in

Standard deviation:
The standard deviation is a measure of variability that determines the spread of the normal distribution. It quantifies how much the data typically deviates from the mean and represents the typical distance between observations and the mean. It is used to display the normal separation between the average and the observations, with a larger standard deviation indicating that the data is more spread out, while a smaller standard deviation indicates that the data is more tightly clustered around the mean.
As shown in fig 4, altering the standard deviation causes the width of the distribution along the X-axis to either tighten or spread out. Wider distributions are produced by higher standard deviations. When distributions are narrow, there is a higher probability that values will not deviate significantly from the mean. Conversely, as the dispersion of the bell curve widens, there is a greater likelihood that observations will deviate further away from the mean. In other words, the risk of significant deviations from the mean increases as the normal distribution becomes more spread out.

Normal distribution probability density function
Normal distribution is obtained as a limiting case of Binomial distribution. Let X be a Binomial variate with parameters n and p. Let where, np=θ and √ =σ, which means Naturally Z is a standardized form of Binomial variate. When the two limiting conditions: i. n→∞ ii. p is neither small nor large are applied on Z, then it can be shown that Z becomes a standard normal variate with probability density function (pdf) given by: is probability density function of Z; φ(z)dz is called probability differential of Z. X becomes a normal variate with probability density function given by: f(x)is probability density function of X; ( ) is called probability differential of X. X is a normal variate with mean θ and variance σ 2 . In symbols, X~N(θ, σ 2 ). Z is a standard normal variate. In symbols, Z~N(0,1). Normal distribution is so called because there was an attempt to project this distribution as the benchmark for all continuous probability distribution. The attempt failed as many continuous distributions turned out to be very different from normal. However, the nomenclature "normal" stayed.

Methodology
This section will provide the method that has been used to fulfil the objective of this section. Here, coefficients of the cubic equation have been generated using the function normal() which includes inbuilt function rand and rng.

normal()
u=normal(m, sigma) generates random numbers from the normal distribution specified by m and sigma which implies the mean and standard deviation respectively. The function normal() includes following functions: rand: X=rand() generates a random scalar in the range (0,1) that is chosen at random from the uniform distribution.  (ŋ1, ŋ2..., ŋn) generates an array of random numbers of size ŋ1- by-ŋ2-by-....-by-ŋn, where ŋ1, ŋ2..., ŋn indicate the size of each dimension.

Rng:
In random number generator rng is used to change the seed in MATLAB to prevent using the same random numbers repeatedly. By producing a seed based on the current time, rng provides a simple method for doing that. 'Shuffle' reseeds the generator with a different seed each time you use it. rng can be called without any inputs to reveal the exact seed that was used.
Example: rng shuffle r=rand() r=0.1929 INPUT ARGUMENTS In MATLAB, the `rand()` function is used to obtain random numbers from a uniform distribution. The function can take one or two input arguments. When called with a single argument, ` ŋ`, `rand(ŋ)` obtain a ` ŋ`-by-` ŋ` matrix of random numbers uniformly distributed between 0 and 1.
Alternatively, when called with two arguments, ` ŋ1` and ` ŋ2`, `rand(ŋ1, ŋ2)` generates a ` ŋ1`-by-` ŋ2` matrix of random numbers with same distribution. In both cases, the resulting matrix of random numbers is returned as the output of the `rand()` function.
MATLAB CODE -REFER APPENDIX C

Results
Following the above instructions in the MATLAB code stated in APPENDIX C, the mean and standard deviation of iteration (k), once for a fixed parameter-mean(m) and varied parameter-standard deviation(sigma) of normal distribution, as shown in Table 3(a) and Table 3(b) and another for a fixed parameter-standard deviation(sigma) and varied parameter-mean(m) of normal distribution, as shown in Table 4(a) and Table 4(b) over 100 trials were obtained.

Conclusion and Future Scope 5.1 Conclusion
Having knowledge of statistics allow us to make informed decisions about the most appropriate data collection techniques, apply suitable statistical analyses, and effectively communicate the resulting findings. Making decisions based on data, predicting future outcomes, and making scientific discoveries all rely on statistical knowledge and skills. In other words, statistics is a crucial tool that allows us to draw meaningful conclusions from data and make evidence-based decisions, therefore applying statistical analysis on bisection method to analyse its performance in a cubic equation when coefficients are coming from particular distribution.
The convergence of the bisection method for solving a cubic equation will depend on the coefficients of the equation. Now, if the coefficients are coming from some probability distribution, then it is logical that the parameters of that probability distribution will influence the convergence. Hence, our study is to investigate in what way the parameter influences the convergence and considered both uniform and non-uniform distribution which includes discrete uniform distribution, continuous uniform distribution and normal distribution. In order to generate random number from these distribution various functions have been created like dis_unirand(), con_unirand(), normal() with the help of inbuilt functions rand and rng in MATLAB.
After analysis, in all the three distributions second degree polynomial equation can be used to predict the average iteration for a given parameter. On the other hand, coefficients of the polynomial are almost same for discrete and continuous uniform distribution whereas in the normal distribution the coefficients are different, which conclude the following: i. In case of uniform distribution input, the average iteration does not depend on whether the distribution is discrete or continuous but rather depend on the range of the distribution which is a parameter.
ii. In case of non-uniform distribution input, the average iteration depends on the mean of the distribution but not on the standard deviation. Thus, it depends on the location parameter but not on the scale parameter.
Our statistical study of bisection method for cubic equations is intended to inspire other scholars to conduct related research.

Future scope of this work
Other distributions can also be used to study bisection method like Bernoulli distribution, exponential distribution, etc. Since, bisection method is guaranteed to converge though it is slow therefore, a new improvised method can be generated to improve the rate of convergence of bisection method by combining it with regula falsi method or Newton-Raphson method. Even the statistical study of new improvised method can also be done along with comparisons to the traditional method but new methods add new limitations to the data set, for instance, in case of Newton-Raphson method 1 st derivative of polynomial should not be equal to 0, therefore demands new data set for resultant output and comparison. Hence, it provides new scope to this work in future with more improvised algorithm and appropriate data set and make it more reliable to find the root of any equation in real life for wider applications.