Lecture Notes on Iteration Using For-Statement in C++

Introduction

• Topic: New control statement in C++ called the for statement for iteration by counting.
• Applications: Estimating the value of logarithm and solving the Him Jindal number problem.

Iterating by Counting: The For-Statement

Basic Concept

• In counting problems, the number of iterations required ( ) may be known in advance.
• While-loop alternative: While loops use a condition, but for counting for loops are more straightforward.

Mechanism

• Counting mechanism: Initialize count, execute a block, increment count, and repeat until the count equals n.
• Code Example (While Loop):
  • Initialize variables.
  • Set a count variable.
  • Loop while count <= n.
  • Increment the count at the end of each iteration.

Variations

• Changing increments: For example, count by 2 instead of 1, using count += 2.
• Handling even and odd n values.

Applying Counting to Problems

Sum of n Numbers

• Algorithm: Sum n given numbers by iterating n times using a loop.
• Key Idea: No need for artificial end-of-input conditions.
• Code Example: Initialize sum and count, loop for n times to read numbers and accumulate the sum.

Finding Maximum of n Numbers

• Similar to summing numbers but track the maximum value encountered.

Calculating Factorials

• Algorithm: Multiply successive terms iteratively, starting from 1.
• Code Example (While Loop): Initialize factorial to 1, loop with a counting mechanism, multiply in each iteration.
• Improvement: Use count starting from 2 to avoid unnecessary multiplication.

The for Statement in C++

Structure

• Syntax: for (initialization; condition; increment).
• Example: for (count = 1; count <= n; count++) { /* statements */ }
• Actions are executed in order: initialize, check condition, execute block, increment.
• Benefits: Consolidates initialization, condition check, and increment in one line.

Implementation Examples

• Sum of n Numbers: Using for-loop to read and sum n numbers.
• Finding Maximum: Uses similar structure to identify the largest number in n inputs.
• Sum of First n Natural Numbers: for loop adds integers from 1 to n.
• Calculating Factorials: Uses for loop to iterate and calculate the factorial.

Advanced For-Statement Use

• Multiple Initialization and Increment: Use commas to separate multiple statements.
• Infinite Loop: for (;;) is equivalent to while (1), and will loop indefinitely without any conditions.

Application: Estimating Logarithms Using Riemann Sums

Concept of Riemann Integral

• Function: Integration approximated by summing areas of rectangles under the curve.
• Use Case: Estimating log(a) to the base e by computing the area under y = 1/x from 1 to a.

Approach

• Width of Rectangles: (a - 1) / n if n rectangles are used.
• Height Calculation: Depends on x-coordinate, determined by iteration index.
• Code Example: Program to read a, divide the area, and sum areas of rectangles in a loop.

Generalizing the Program

• Input value of n for the number of rectangles instead of a fixed number.

Special Topic: Him Jindal Numbers

Problem Introduction

• Scenario: Different ways to construct a wall of length using bricks of length 1 and 2.
• Historical Context: Relation to Pingala and Him Chandra's work on poetic meters.

Recurrence Relations

• Definition: Number of ways to construct n lengths = ways to construct (n-1) lengths + (n-2) lengths.
• Recursive Nature: Solve using an iterative approach.
• Initialization: Base cases known; use iterative updates to compute further values.

Code Example

• Approach: Use an iteration to calculate subsequent values based on previous two values.
• Variables: Current, previous, and next values updated iteratively.

Key Takeaways

• Iteration by counting is a fundamental programming technique.
• The for-statement in C++ elegantly handles initialization, condition checking, and incrementation.
• Advanced applications: Estimating logarithms, solving combinatorial problems.

Final Thoughts

• Cultural Insight: Recognition of historical contributions in mathematics and algorithms.

Resources: Wikipedia articles on Indian mathematics and historical figures for further learning.