19 Jul 2025

Algorithms, Flowcharts, and Program Structure

python programming-fundamentals algorithms flowcharts

A scientific program begins with a method, not with Python syntax. The method is first stated as an algorithm, checked as a flowchart or pseudocode, and only then translated into executable statements.

Algorithm

An algorithm is a finite, ordered sequence of unambiguous steps that transforms specified input into specified output. A usable numerical algorithm must state:

1. the input data and their allowed range;
2. the initialization of variables;
3. the order of calculations and decisions;
4. the condition that ends every repetition;
5. the output to be reported.

For example, the mean of $n>0$ readings $x_0,\ldots,x_{n-1}$ is

\[\bar x=\frac1n\sum_{i=0}^{n-1}x_i.\]

An algorithm that follows this definition is

START
READ n and x[0], x[1], ..., x[n-1]
SET s = 0 and i = 0
WHILE i < n
    SET s = s + x[i]
    SET i = i + 1
SET mean = s / n
PRINT mean
STOP

The initialization $s=0$ supplies the additive identity, and $i=0$ selects the first stored element. When the loop ends, $s$ equals the required sum, so the final division implements the displayed formula.

Flowchart

A flowchart represents the control path with standard symbols.

symbol	purpose
oval	start or stop
parallelogram	input or output
rectangle	calculation or assignment
diamond	decision with labelled branches
arrow	direction of execution

The figure can be revised from its editable TikZ source.

The decision $i<n$ has two exits. The yes branch updates the partial sum and index before returning to the decision. The no branch is the loop-termination condition and leads to the result.

Program structure

A small scientific program is easiest to inspect when it follows the same order as the algorithm:

description and assumptions
input
initialization
calculation and control flow
output

The mean algorithm translates directly to Python:

# Input
n = int(input("Number of readings: "))
values = []

for i in range(n):
    value = float(input(f"x[{i}] = "))
    values.append(value)

# Calculation
total = 0.0
for value in values:
    total = total + value

mean = total / n

# Output
print(f"Mean = {mean:.6g}")

Each name has one clear role, indentation shows which statements belong to a loop, and the output precision is stated explicitly. Later programs may place repeated calculations inside functions, but the input-process-output structure remains the same.