Loops

Loops#

Every Numerical Method Needs This#

Numerical methods are fundamentally about repetition. Whether it’s:

  • Euler’s method: repeat “compute next value” 1000 times

  • Newton-Raphson: repeat “improve guess” until error is small enough

  • Simulations: repeat “update state” for each time step

You can’t do computational engineering without loops. They’re the engine that drives every numerical algorithm.

For Loop#

Use for when you know how many times you want to repeat something.

A Simple Counting Example#

Let’s start with something basic. Say you want to print “Hello” five times:

for i in range(5):
    print(f"Hello! This is repetition number {i + 1}")

Output:

Hello! This is repetition number 1
Hello! This is repetition number 2
Hello! This is repetition number 3
Hello! This is repetition number 4
Hello! This is repetition number 5

The range(5) generates the numbers 0, 1, 2, 3, 4 (five numbers, starting from 0). For each number, Python runs the indented code.

Summing Up Numbers#

What if you want to add up all numbers from 1 to 10? Instead of typing 1 + 2 + 3 + ... + 10, use a loop:

total = 0

for number in range(1, 11):
    total = total + number
    print(f"Added {number}, total is now {total}")

print(f"\nFinal sum: {total}")

Output:

Added 1, total is now 1
Added 2, total is now 3
Added 3, total is now 6
...
Added 10, total is now 55

Final sum: 55

This pattern (start with zero, then add things in a loop) appears everywhere in numerical methods. You’ll use it for summing series, computing integrals, and more.

Euler’s method for falling parachutist#

The loop repeats the indented code for each value of t. Here, range(0, 14, 2) gives us: 0, 2, 4, 6, 8, 10, 12.

g = 9.8
c = 12.5
m = 68.1
dt = 2
v = 0

for t in range(0, 14, 2):
    print(f"t = {t:2d} s, v = {v:5.2f} m/s")
    v = v + (g - c/m * v) * dt

Output:

t =  0 s, v =  0.00 m/s
t =  2 s, v = 19.60 m/s
t =  4 s, v = 32.00 m/s
t =  6 s, v = 39.85 m/s
t =  8 s, v = 44.82 m/s
t = 10 s, v = 47.97 m/s
t = 12 s, v = 49.96 m/s

How range() works#

range(5)         # 0, 1, 2, 3, 4        (start defaults to 0, stop at 5)
range(2, 7)      # 2, 3, 4, 5, 6        (start at 2, stop before 7)
range(0, 10, 2)  # 0, 2, 4, 6, 8        (start at 0, stop before 10, step by 2)
range(10, 0, -1) # 10, 9, 8, ..., 1     (count backwards)

Looping over a list#

Read for compound in compounds as “for each compound in the list compounds, do…”

The loop variable (compound) is a new variable created by the loop. It takes on each value from the list, one at a time. You can name it anything you want.

compounds = ["methane", "ethane", "propane"]

for compound in compounds:
    print(compound)

Output:

methane
ethane
propane

While Loop#

Use while when you don’t know how many iterations you need. The loop keeps running as long as the condition is True.

This is exactly what you need for iterative numerical methods: you keep improving your answer until it’s “good enough.” But how many iterations will that take? You don’t know in advance, and that’s why you need while.

How while works#

count = 0
while count < 3:
    print(count)
    count = count + 1

Output: 0, 1, 2 (stops when count becomes 3)

Warning

Always update the variable in your condition! Otherwise the loop runs forever.

Example: Iterate Until a Condition is Met#

How long until the parachutist reaches 99% of terminal velocity? We don’t know in advance, so we use while.

g, c, m, dt = 9.8, 12.5, 68.1, 0.1
v_terminal = g * m / c

v = 0
t = 0

while v < 0.99 * v_terminal:
    v = v + (g - c/m * v) * dt
    t = t + dt

print(f"Reached 99% terminal velocity at t = {t:.1f} s")

Output: Reached 99% terminal velocity at t = 25.1 s

Decaying concentration#

concentration = 1.0
while concentration > 0.1:
    concentration = concentration * 0.5
    print(f"C = {concentration}")

Quick Reference#

Pattern

Syntax

Use When

For loop (count)

for i in range(n):

You know how many iterations

For loop (items)

for x in list:

Loop through each item

While loop

while condition:

Loop until condition is false

range(n)

range(5) → 0,1,2,3,4

Sequence starting from 0

range(start, stop)

range(2, 7) → 2,3,4,5,6

Sequence from start to stop-1

range(start, stop, step)

range(0, 10, 2) → 0,2,4,6,8

Sequence with custom step

Next Steps#

Continue to Data Structures to learn about lists, dictionaries, and more.

Tip

Want to see loops in numerical methods? After learning Functions, check out these interactive notebooks:

Tip

For advanced loop patterns like enumerate(), zip(), and list comprehensions, see Iterators and Looping Patterns after you’ve learned about data structures.

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