Question

Given an array Values with \(n\) elements Values[0], Values[1],.... Values \([n-1 \mid\) each containing a real number, the following algorithm finds the sum of all the positive values in Values. Write an invariant for the loop. rollingSum \(=0\) for \(i=\phi, 2, \ldots, n-1\) if Values \([t]>0\) rolling Sum \(=\) rollingSum \(+\) Values \([i]\) Output rollingSum.

Short answer

The invariant is that `rollingSum` equals the sum of positive numbers encountered so far.

Step-by-step solution

Understand the Problem

We have a list of numbers called 'Values', and we need to calculate the sum of only the positive numbers in this list using a loop.

Set Initial Variables

We start by defining the initial sum, `rollingSum`, as 0. This variable will be used to accumulate the sum of the positive numbers.

Loop Through Each Element

We iterate through each element in the array using a loop that goes from index 0 to n-1. For each element, we check if it is positive.

Check If Element Is Positive

In each iteration, check whether the current element `Values[i]` is greater than 0. If it is, add it to `rollingSum`.

Update rollingSum

If `Values[i]` is positive, update `rollingSum` by adding `Values[i]` to it. If not, continue to the next iteration.

Define the Loop Invariant

The invariant is that at the start and end of each loop iteration, `rollingSum` is the sum of all positive numbers in the subarray `Values[0]` to `Values[i-1]`.

Output the Result

After finishing the loop for all elements, the `rollingSum` contains the sum of all positive numbers in the array. Output this value.

Key concepts

Positive Numbers

Understanding positive numbers is crucial in this exercise. Positive numbers are any numbers greater than zero. These numbers lie to the right of zero on the number line. When dealing with arrays, identifying positive numbers helps in various calculations, like the one we have here.
In this problem, we're interested specifically in finding the sum of these positive numbers within an array. Filtering out and just focusing on numbers greater than zero can provide useful insights and results.
By checking if each number in the array is positive, we can target only those numbers for further operations, like summation. This knowledge is foundational for students learning how to manipulate and analyze numeric data effectively.

Array Traversal

Array traversal refers to the process of accessing each element within an array systematically. In most cases, this involves using a loop to iterate over elements from start to end.

• Start from the first element with an index of 0.
• Continue looping until the last element, which is at index \(n-1\).
• Check each element one by one.

In this exercise, array traversal enables us to check each number in "Values" and determine whether it's positive. This step-by-step inspection is necessary to implement operations like Summation or transform other data within the array.
By implementing a loop that effectively traverses an array, students learn not only how to iterate over data structures, but also to apply logical conditions – such as checking if a number is positive.

Sum Accumulation

Sum accumulation is the process of continuously adding elements to a cumulative total. In programming, especially with arrays, sum accumulation is often used to tally values under certain conditions.
In this scenario, we use a variable called `rollingSum` to keep track of the sum of all positive numbers. At the start, `rollingSum` is initialized to zero, because no numbers have been added yet.

• If a number is found to be positive, it is added to `rollingSum`.
• If it is not positive, nothing is added and the loop continues.
• The process continues until every array element has been checked and added where applicable.

This concept is pivotal in programming as it demonstrates not only how to accumulate totals but also how to conditionally evaluate which numbers to include in the sum. This approach teaches students the importance of selection in algorithms and how efficiently iterating over an array can lead to meaningful results, like finding the sum of positive numbers alone.