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CME NOTE.pdfhbbbbjjjbhjhgjhgkhghghjfhcfkck

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  Bisection Method 1What isbisection method?/statement Bisection method is based on the intermediate value theorem, which states that if a continuous function f(x) changes sign between two points a and b, then there exists a point c between them such that f(c) is zero. The method works by dividing the interval [a,b] into two equal subintervals, and checking which one contains the root. Then, it repeats the process with the chosen subinterval, until the desired accuracy is reached. . Bisection Method Formula is: x2 = (x0 + x1) / 2 Bisection Method Procedure To solve bisection method problems, given below is the step-by-step explanation of the working of the bisection method algorithm for a given function f(x):
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  Step 1: Choosetwo values, a and b such that f(a) > 0 and f(b) < 0 . Step 2: Calculate a midpoint c as the arithmetic mean between a and b such that c = (a + b) / 2. This is called interval halving. Step 3: Evaluate the function f for the value of c. Step 4: The root of the function is found only if the value of f(c) = 0. Step 5: If (c) ≠ 0, then we need to check the sign: a. we replace a with c if f(c) has the same sign as f(a) and we keep the same value for b b. we replace b with c if f(c) has the same sign as f(b), and we keep the same value for a To get the right value with the new value of a or b, we go back to step 2 And recalculate c. Advantages of Bisection Method 1. Convergence is guarenteed: Bisection method is bracketing method and it is always convergent. 2. Error can be controlled: In Bisection method, increasing number of iteration always yields more accurate root. 3. Does not involve complex calculations: Bisection method does not require any complex calculations. To perform Bisection method, all we need is to calculate average of two numbers. 4. Guaranteed error bound: In this method, there is a guaranteed error bound, and it decreases with each successive iteration. The error bound decreases by ½ with each iteration. 5. Bisection method is very simple and easy to program in computer. 6. Bisection method is fast in case of multiple roots. Disadvantages of Bisection Method 1. Slow Rate of Convergence: Although convergence of Bisection method is guaranteed, it is generally slow.
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  2. Choosing oneguess close to root has no advantage: Choosing one guess close to the root may result in requiring many iterations to converge. 3. Can not find root of some equations. For example: f(x) = x2 as there are no bracketing values. 4. It has linear rate of convergence. 5. It fails to determine complex roots. 6. It can not be applied if there are discontinuities in the guess interval. 7. It can not be applied over an interval where the function takes values of the same sign. Q: How to compare bisection method with other numerical methods? When searching for the root of a function, there are many numerical methods to choose from, such as Newton's method, secant method, regula falsi method, and fixed point iteration method. The best choice depends on the nature and behavior of the function, the accuracy of the initial guess and derivative, if needed, the desired precision and speed of convergence, and the computational cost and complexity of the method. To compare bisection method with other numerical methods, criteria like the number of iterations and evaluations required to reach a certain accuracy or tolerance, the rate of convergence and order of convergence, and the error analysis and bounds can be used. These can help determine how close the approximate solution is to the exact solution and how much it can vary due to errors.
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  Comparison between BisectionMethod and Newton Raphson Method Sr. No. Bisection Method Newton Raphson Method 1. In the Bisection Method, the rate of convergence is linear thus it is slow. In the Newton Raphson method, the rate of convergence is second-order or quadratic. 2. In Bisection Method we used following formula x2= (x0 + x1) / 2 In Newton Raphson method we used following formula x1 = x0 – f(x0)/f'(x0) 3. In this method, we take two initial approximations of the root in which the root is expected to lie. In this method, we take one initial approximation of the root. 4. The computation of function per iteration is 1. The computation of function per iteration is 2. 5. The initial approximation is less sensitive. The initial approximation is very sensitive. 6. In the Bisection Method, there is no need to find derivatives. In the Newton Raphson method, there is a need to find derivatives. 7. This method is not applicable for finding complex, multiple, and nearly equal two roots. This method is applicable for finding complex, multiple, and nearly equal two roots. 8. Accuracy is less than Newton raphson Accuracy is more than bisection Advantages of Newton Raphson Method o The method has one of the fastest convergences to the root. o It is easy to program as it has a simple formula
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  o It isused to further improve a root found by other methods. o It requires only one guess to find the root. o Derivation of the method is more intuitive, so it is easier to understand its behaviour, like when it is to converge and when it is to diverge. Differences between Bisection Method and Regula False Method Basis Bisection Method Regula Falsi Method Definition In mathematics, the bisection method is a root-finding method that applies to continuous function for which knows two values with opposite signs. In mathematics, the false position method is a very old method for solving equations with one unknown this method is modified form is still in use. Simplicity it is simple to use and easy to implement. Simple to use as compared to Bisection Method Computational Efforts Less as compared to Regula Falsi Method More as compared to Bisection Method Iteration required In the bisection method, if one of the initial guesses is closer to the root, it will take a large number of iterations to reach the root. Less as compared to Bisection Method. This method can be less precise than bisection – no strict precision is guaranteed. Convergence The order of convergence of the bisection method is slow and linear. This method faster order of convergence than the bisection method.
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  Basis Bisection MethodRegula Falsi Method General Iterative Formula Formula is : X3 =( X1 + X2)/2 Formula is : X3 = X1(fx2) – x2(fx1)/ f(x2) -f(x1) Other Names It is also known as the Bolzano method, Binary chopping method, half Interval method. It is also known as the False Position method. Advantages of Secant Method:  The speed of convergence of secant method is faster than that of Bisection and Regula falsi method.  It uses the two most recent approximations of root to find new approximations, instead of using only those approximations which bound the interval to enclose root Disadvantages of Secant Method:  The Convergence in secant method is not always assured.  If at any stage of iteration this method fails.  Since convergence is not guaranteed, therefore we should put limit on maximum number of iterations while implementing this method on computer.