Computer Science/Numerical Methods/ Bisection method example

Computer Science/Numerical Methods/ Bisection method example

Bisection Method – Numerical Methods (Computer Science)

The Bisection Method is a root-finding technique that applies to continuous functions where the root lies between two given points. It is a simple and reliable method based on the Intermediate Value Theorem.

 Algorithm of Bisection Method

 Select two points a and b such that $f(a)$ and $f(b)$ have opposite signs (i.e., $f(a)\times f(b)<0$), ensuring a root exists in between.
 Compute the midpoint $c$:

c=a+b2c = \frac{a + b}{2}c=2a+b​

 Check $f(c)$:

• If $f(c)=0$, then $c$ is the root.

• If $f(a)\times f(c)<0$, then the root lies in [a, c], set $b=c$.

• Else, the root lies in [c, b], set $a=c$.
   Repeat the process until the error tolerance is met:

$$|b-a|<\text{tolerance}$$

 Example: Find the root of f(x)=x3−x−2f(x) = x^3 – x – 2f(x)=x3−x−2 in [1, 2]

Step 1: Check function values

f(1)=13−1−2=−2f(1) = 1^3 – 1 – 2 = -2f(1)=13−1−2=−2
f(2)=23−2−2=4f(2) = 2^3 – 2 – 2 = 4f(2)=23−2−2=4
Since $f(1)\times f(2)<0$, a root exists between 1 and 2.

Step 2: Iterations

The root is approximately 1.5938 with a small error.

Python Code for Bisection Method

 Applications of Bisection Method

Solving non-linear equations in engineering & science
Finding zero crossings in signal processing
Used in numerical simulations & modeling

 Conclusion:

The Bisection Method is a simple, robust but slow root-finding method. It is best suited for cases where guaranteed convergence is needed.

Computer Science/Numerical Methods/ Bisection method example

Numerical Methods – Australian Mathematical Sciences Institute

Solutions of Equations in One Variable The Bisection Method

Solution of Algebraic and Transcendental Equations

Here’s a clear explanation of the Bisection Method with a complete example – ideal for students in Computer Science/Numerical Methods.

Bisection Method – Numerical Methods in Computer Science

What is Bisection Method?

The Bisection Method is a numerical technique used to find roots of a continuous function f(x) within a given interval [a, b], where:

• f(a) and f(b) have opposite signs (i.e., f(a) * f(b) < 0)

• This implies a root exists between a and b (by Intermediate Value Theorem)

Step-by-Step Algorithm:

1. Check that f(a) and f(b) have opposite signs.

2. Calculate midpoint c = (a + b)/2.

3. Check f(c):

   • If f(c) = 0, then c is the root.

   • Else, replace a or b with c based on sign change:

     • If f(a) * f(c) < 0, set b = c

     • Else, set a = c

4. Repeat steps until desired accuracy is achieved (|f(c)| < ε, small error tolerance).

Example: Find a root of f(x) = x³ – x – 2 using Bisection Method

Step 1: Choose initial interval [1, 2]

Check signs:

• f(1) = 1³ – 1 – 2 = -2

• f(2) = 8 – 2 – 2 = 4
  Opposite signs → root lies in [1, 2]

Step 2: Start Iteration

Repeat until |f(c)| < 0.001 or desired precision.

Final Answer:

The root lies approximately near x ≈ 1.52 (can be refined with more iterations).

Use Cases in Computer Science:

• Solving nonlinear equations in simulations

• Numerical solvers in compilers and symbolic computation

• Used in control systems, AI pathfinding (as a logic component)

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• A step-by-step worksheet for manual solving?

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Computer Science/Numerical Methods/ Bisection method example

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