Bisection Method
The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a sub-interval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods.
This method is also called the Interval Halving Method, the Binary Search Method, the Dichotomy Method, Robust Root Finding Method and Bolzano’s Method. It’s strategy is to begin with two values of x, say xL and xU, that brackets a root of f(x) = 0. It determines that the initial values x = xL and x = xU. The method treats the interval distance between the initial values as line segment then successively divides the interval in half and replaces one endpoint with the midpoint so that again the root is bracketed.
Note : xL for lower limit
xU for upper limit
xR for average of upper and lower limit ()
At each step, the method divides the interval in two by computing the midpoint xR = (xL + xU) / 2 of the interval and the value of the function f(xR) at that point. Unless xR is itself a root (which is very unlikely, but possible) there are now two possibilities: either f(xL) and f(xR) have opposite signs and bracket a root, or f(xR) and f(xU) have opposite signs and bracket a root. The method selects the subinterval that is a bracket as a new interval to be used in the next step. In this way the interval that contains a zero of f is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.
Explicitly, if f(xL) and f(xR) are opposite signs, then the method sets xR as the new value for xU, and if f(xL) and f(xR) are same signs then the method sets xR as the new xL. (If f(xR)=0 then xR may be taken as the solution and the process stops.)In both cases, the new f(xL) and f(xU) have opposite signs, so the method is applicable to this smaller interval.
Steps / Procedures for Bisection Method:
1. First, choose lower limit/guess (xL) and the upper limit (xU) for the root such that the function changes sign over the interval. This can be checked by ensuring that f(xL)*f(xU) < 0.
2. Solve for xR. Use the equation ().
3. To get f(xL), substitute the value of xL to the given function. Likewise to get f(xR), substitute the value of xR to the given function. The f(xU) can be don’t get since it will not be use on the next steps.
4. Multiply the f(xL) and f(xR).
a. If f(xL) * f(xR) < 0, the root lies in the upper subinterval or limit. Therefore set xU = xR. Then return to step 2 and repeat the steps.
b. If f(xL) * f(xR) > 0, the root lies in the lower subinterval or limit. Therefore set xL = xR. Then return to step 2 and repeat the steps.
c. If f(xL) * f(xR) = 0, the root equals xR. Terminate now the computation.
Note: the value of the label not being changed is still the same to its recent value. For example if xL = 0 and xU = 1, then the value of xU was changed to the value of xR which is 2, the new values of the for the next iteration will be: xL = 0 and xU = 2.
5. Lastly, after repeating the steps from step 2 to step 4, if you the answers to xR became the same then STOP. It only means that the same value occurring to xR is the root being solved.
This method is also called the Interval Halving Method, the Binary Search Method, the Dichotomy Method, Robust Root Finding Method and Bolzano’s Method. It’s strategy is to begin with two values of x, say xL and xU, that brackets a root of f(x) = 0. It determines that the initial values x = xL and x = xU. The method treats the interval distance between the initial values as line segment then successively divides the interval in half and replaces one endpoint with the midpoint so that again the root is bracketed.
Note : xL for lower limit
xU for upper limit
xR for average of upper and lower limit ()
At each step, the method divides the interval in two by computing the midpoint xR = (xL + xU) / 2 of the interval and the value of the function f(xR) at that point. Unless xR is itself a root (which is very unlikely, but possible) there are now two possibilities: either f(xL) and f(xR) have opposite signs and bracket a root, or f(xR) and f(xU) have opposite signs and bracket a root. The method selects the subinterval that is a bracket as a new interval to be used in the next step. In this way the interval that contains a zero of f is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.
Explicitly, if f(xL) and f(xR) are opposite signs, then the method sets xR as the new value for xU, and if f(xL) and f(xR) are same signs then the method sets xR as the new xL. (If f(xR)=0 then xR may be taken as the solution and the process stops.)In both cases, the new f(xL) and f(xU) have opposite signs, so the method is applicable to this smaller interval.
Steps / Procedures for Bisection Method:
1. First, choose lower limit/guess (xL) and the upper limit (xU) for the root such that the function changes sign over the interval. This can be checked by ensuring that f(xL)*f(xU) < 0.
2. Solve for xR. Use the equation ().
3. To get f(xL), substitute the value of xL to the given function. Likewise to get f(xR), substitute the value of xR to the given function. The f(xU) can be don’t get since it will not be use on the next steps.
4. Multiply the f(xL) and f(xR).
a. If f(xL) * f(xR) < 0, the root lies in the upper subinterval or limit. Therefore set xU = xR. Then return to step 2 and repeat the steps.
b. If f(xL) * f(xR) > 0, the root lies in the lower subinterval or limit. Therefore set xL = xR. Then return to step 2 and repeat the steps.
c. If f(xL) * f(xR) = 0, the root equals xR. Terminate now the computation.
Note: the value of the label not being changed is still the same to its recent value. For example if xL = 0 and xU = 1, then the value of xU was changed to the value of xR which is 2, the new values of the for the next iteration will be: xL = 0 and xU = 2.
5. Lastly, after repeating the steps from step 2 to step 4, if you the answers to xR became the same then STOP. It only means that the same value occurring to xR is the root being solved.
Therefore, one root is 1.397748.