Summary notes - Finding roots
1. Why We Find Roots
A root of an equation is a value of 𝑥such that
𝑓(𝑥) = 0.
Many engineering problems reduce to this form (equilibrium, energy balances, flow
equations, etc.), but most real functions can’t be solved analytically—so we estimate the
root numerically.
Graphical Methods
Idea:
Plot the function 𝑓(𝑥) and visually see where it crosses the x-axis.
Why Useful:
•
Quick sense of where roots may exist.
•
Good for detecting multiple roots or wild oscillations.
Caution I Emphasized:
•
Graphical methods do NOT give accurate roots. They are just for “locating
neighborhoods.”
•
A misleading plot range or poor resolution can make you miss a root completely.
Bracketing Methods
These methods require you to start with an interval [𝑎, 𝑏]where
𝑓(𝑎) ⋅ 𝑓(𝑏) < 0.
That means the function crosses zero somewhere inside.
2. Incremental Search (a crude bracketing tool)
Idea:
Walk along the x-axis in steps (e.g., ∆x = 0.1) until you notice a sign change.
Warnings:
Step size matters a LOT
•
If ∆x is too large, you might step over a root and never see the sign change.
•
If ∆x is too small, it wastes computation time.
Tangent roots
•
If the root touches the x-axis but doesn’t cross it (even multiplicity), then
𝑓(𝑎) ⋅ 𝑓(𝑏) > 0
—so incremental search will miss it!
Discontinuous functions
•
A jump from +10 to –10 does not mean there's a root—just a discontinuity.
Be careful.
3. Bisection Method
Idea:
•
Start with a bracket [𝑎, 𝑏] where the sign changes.
•
Cut the interval in half repeatedly.
Why It’s Nice:
•
Bisection always converges as long as the sign change exists.
•
Very robust.
Precision Estimate (What I emphasized!)
One of the coolest features:
After 𝑛 bisections, the interval width is
𝑏−𝑎
.
2𝑛
If you want a maximum error of 𝜖:
𝑏−𝑎
).
𝑛 ≥ log 2 (
𝜖
You can estimate in advance how many steps you need to reach the desired accuracy.
Students really like this because it feels predictable.
4. False Position
Idea:
Similar to bisection, but instead of cutting the interval in half, you draw a straight line
between
(𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏))
and use the intercept with the x-axis as the next estimate.
Caution I Emphasized:
Slow-slope issue:
If one side has a very shallow slope, the false-position method keeps choosing points near
the other side.
This means the bracket shrinks very slowly, and the method converges painfully slowly.
That’s its main drawback.
Open Methods
These do not require a bracketing interval.
They are typically faster but less stable.
5. Fixed-Point Iteration
Idea:
Rewrite
𝑓(𝑥) = 0
as
𝑥 = 𝑔(𝑥),
then iterate
𝑥𝑛+1 = 𝑔(𝑥𝑛 ).
Key Concept:
You interpret the root as the intersection of the curves:
•
𝑦 = 𝑔(𝑥)
•
𝑦=𝑥
Cautions:
The method only converges if ∣ 𝑔′ (𝑥) ∣< 1 at the root.
Students often forget this.
Divergence problem:
If the slope is steep or poorly behaved, the next guesses can spiral away from the line 𝑦 =
𝑥.
Graphically, you bounce farther and farther from the intersection (see Figure 6.3 in
textbook).
6. Wegstein Method
An acceleration technique applied to fixed-point iterations.
Idea in simple terms:
•
It watches how the sequence of iterates is moving. Remember we use the
intersection of the line passing through two initial guesses and the y=x line rather
than projecting just one initial guess onto y=x.
•
It estimates a relaxation factor (amount of move towards the next guess) that can
“speed up” convergence. When you seem to know where you are going, instead of
taking the normal step, jump farther!
When useful:
•
Works well when fixed-point iteration is slowly converging but still stable.
Caution:
•
If applied blindly, an over-aggressive relaxation factor can actually destabilize the
iteration.
7. Newton–Raphson Method
Idea:
Use the tangent line (derivative) at the current point.
𝑥𝑛+1 = 𝑥𝑛 −
𝑓(𝑥𝑛 )
.
𝑓 ′ (𝑥𝑛 )
Why Engineers Love It:
•
Fast (quadratic) convergence near the root.
•
Usually the go-to open method.
Warnings I Emphasized:
Needs the derivative, which isn’t always easy.
If the slope is very small (flat function),
𝑓 ′ (𝑥𝑛 ) ≈ 0
→ the update becomes enormous → it jumps far away.
This can cause divergence or meaningless estimates.
If starting far from the root, Newton can “run wild.”
8. Secant Method
Idea:
A Newton-like method that replaces the derivative with a numerical approximation using
two past points (the finite divided difference which we'll discuss in full in Chapter 21):
𝑓 ′ (𝑥) ≈
𝑓(𝑥𝑛 ) − 𝑓(𝑥𝑛−1 )
.
𝑥𝑛 − 𝑥𝑛−1
Then:
𝑥𝑛+1 = 𝑥𝑛 − 𝑓(𝑥𝑛 ) ⋅
𝑥𝑛 − 𝑥𝑛−1
.
𝑓(𝑥𝑛 ) − 𝑓(𝑥𝑛−1 )
Why It’s Useful:
•
Doesn’t require symbolic derivatives.
•
Faster than bisection.
•
Often nearly as fast as Newton.
Cautions:
Using two points can confuse students—what is the “delta”?
The update isn’t simply “add a small step.”
Instead, the computed correction can be multiplied and added to the current guess,
making it look unintuitive.
Perturbation variation (“finite-difference Newton”)
Sometimes we compute
𝑓 ′ (𝑥) ≈
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ
with a tiny ℎ.
Why?
•
Because using a very small perturbation gives a more accurate derivative
estimate.
•
This can improve stability when the secant method is unstable.
In Summary — What I Emphasized the Most
✔ Graphical methods → just for locating candidate intervals.
✔ Bracketing methods → safe, reliable, but sometimes slow (false-position slope issue!).
✔ Bisection → predictable number of steps!
✔ Incremental search → easily misses roots based on step size.
✔ Fixed-point iteration → only converges when ∣ 𝑔′ ∣< 1; watch for divergence.
✔ Wegstein → speeds up fixed-point but can destabilize.
✔ Newton–Raphson → fast but sensitive to flat slopes.
✔ Secant and perturbation variation → derivative-free Newton; careful about interpreting
updates.