Numerical Methods | Module 1
Study Guide: Bisection Method | Prepared by REINNIEL JUNE AQUINO
NUMERICAL METHODS
Course Study Guide & Bisection Method
Prepared by:
REINNIEL JUNE AQUINO
Computer Engineering
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Guidelines and tips
Prepare for your exams
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Numerical Methods: Bisection Method Complete Guide + Excel Tutorial, Study Guides, Projects, Research of Numerical Methods in Engineering
Course: Numerical Methods / Computational Engineering Topic: Roots of Equations - The Bisection Method What’s inside: • Clear, plain-English definitions of key terms (Bracketing, Iteration, Tolerance). • Step-by-step breakdown of the formal algorithm. • A custom high-quality graph visually explaining the root of f(x) = x - 2 + ln(x). • A fully worked-out example problem featuring a clean, 22-iteration table formatted to 6-decimal precision. • BONUS: A complete step-by-step Excel Automation guide. Learn how to use IF statements to automate the sign-change logic and generate your own 20+ row iteration tables in under 60 seconds. Perfect for 2nd/3rd-year engineering students who need to actually understand the math, not just memorize it.
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NUMERICAL METHODS
Course Study Guide & Bisection Method
Prepared by:
REINNIEL JUNE AQUINO
Computer Engineering
Course Coverage
- Roots of Equations
- Systems of Linear Algebraic Equations
- Curve Fitting
- Differentiation
- Integration
- Ordinary Differential Equations
Note: This module focuses specifically on the first topic—Roots of Equations—using foundational bracketing algorithms.
Definition of Terms
- Numerical Methods: Techniques by which complex mathematical problems are formulated so that they can be solved
with arithmetic operations.
- Root: A value that makes the equation equal to zero. Visually, it is the exact coordinate where a plotted function crosses the
x-axis.
- Iteration: A repetitive process where the output of one sequence becomes the input of the next, gradually closing in on the
final answer.
- Bracketing: Identifying two initial guess values (a and b) that "trap" the root between them.
- Tolerance (ε): The acceptable margin of error. Since computers can calculate indefinitely, we set a tolerance (e.g., |f(x)| <
10 ⁻⁶) to tell the algorithm when to stop.
💡 Engineering Context: Analytical math (like factoring algebra) is great for paper, but computers only know
how to add, subtract, multiply, and divide. When the Ryzen 7 processor in your Laptop calculates complex game
physics, it can't use algebra. Instead, it uses numerical methods—brute-forcing arithmetic millions of times a
second to zero in on the exact answer.
Visualizing the Root:
Example Problem:
Use the bisection method to find the root of: f(x) = x - 2 + ln(x)
Solution:
Step 1: Let f(x) = 0 and set the tolerance to ε = 10 ⁻ ⁶ (0.000001).
Step 2: Find two initial values (a and b) that bracket the root.
Assume: a = 1 and b = 2
Evaluating f(x):
- f(1) = - 1
- f(2) = 0.
Because f(a) is negative and f(b) is positive, the sign-change condition is met.
Step 3 to 6: Iteration Table (Stopping Criterion: |f(x_m)| < 10 ⁻ ⁶)
💡 Scientific Notation: If you see a number like - 2.74E-05 in the table below, that is just computer-speak for
scientific notation. It means - 2.74 × 10 ⁻ ⁵, or - 0.0000274. It just means the number is getting incredibly close to
zero!
Iter # a x_m b f(a) f(x_m) f(b)
💡 Optional Bonus: How to Automate this in Excel
Nobody calculates 22 iterations by hand. Here is the exact logic to generate the table above in Excel in under 60 seconds:
A B C D E F
1 a x_m b f(a) f(x_m) f(b) 2 1 =(A2+C2)/2 2 =A2-2+LN(A2) =B2-2+LN(B2) =C2-2+LN(C2) 3 =IF(D2*E2<0, A2, B2)
=(A3+C3)/2 =IF(D2*E2<0, B2,
C2)
=A3-2+LN(A3) =B3-2+LN(B3) =C3-2+LN(C3)
- Row 2 (Initialization): Plug in your starting boundaries (A2, C2). The rest of the row uses formulas to find the midpoint and
evaluate the functions.
- Row 3 (The 'Sign-Change' Logic): This is the magic part. We use an IF statement to automatically decide which boundary
to keep based on the signs from the row above:
- Cell A3 checks if f(a) and f(x_m) have opposite signs (D2*E2 < 0). If True, the root is on the left, so we keep 'a' (A2). If False,
we swap 'a' for the midpoint (B2).
- Cell C3 checks the exact same thing (D2*E2 < 0). If True, the root is on the left, so 'b' doesn't change and we keep C2. If
False, the root is on the right, so we swap 'b' for the midpoint (B2).
- Automate: Highlight all of Row 3, grab the little green square in the bottom right corner, and drag it down to Row 25. Excel
will automatically step through the iterations.
Conclusion