A query that evaluates a given equation over a range

Evaluating all equations within the interval [L, R] gives us a range of values ​​for these variables. Examples of how to use it include modeling, data analysis, and problem-solving scenarios.

In this case, we define equation variable values ​​for all points within the range. So this can be done by specifying the step size of the range and evaluating the equation for each variable value in the range.

Specification

This can be called a request to the database for information. When certain requirements are met, data is extracted using specific commands. To obtain, filter, sort, and summarize data from a database, queries are often written in programming languages. Queries can be as simple as possible, depending on the complexity of the data and information that must be extracted.

A computer program that accepts as input an equation range [L, R] and a step size and produces the result of the equation for each value of the variable within the range can be used to automate this process.

Problem handling methods

Finding the value of a given equation within a range is the goal of a query that evaluates any given equation in the range [L, R]. Here is a potential approach for a query like -

• Parses the provided equation and creates an expression tree from it. Binary trees can be used to visualize expression trees, with each node representing an operator or operand in an equation.

• Presort the expression tree and iterate over each subtree, evaluating the equation for each subtree. Each node of the expression tree should contain a result.

• Create a range query function that accepts the root of the expression tree, the lower bound L, and the upper bound R as input. The goal of this function is to iterate the expression tree and return the solution to the equation in the provided range [L, R].

• The equation solution for each subrange of the specified range [L,R] can be additionally precomputed and saved to improve the range query function.

• Finally, we can use the range query function to calculate equations in different ranges.

grammar

In C, you can use a loop to iterate over the values ​​in each range and then apply the provided equation to each value to determine its evaluation within the range [L, R]. Create the following loop to evaluate the equation for each value of x in the range [L, R] -

y = x2 + 2x + 1
// Define equation to evaluate
   int equation(int x) {
   return x*x + 2*x + 1;
}

// Evaluate equation for every value of x in range [L, R]
int L, R; // Define the range
for (int x = L; x <= R; x++) {
   int y = equation(x);

   // Do something with value of y like print it
   cout << "x = " << x << ", y = " << y << endl;
}

algorithm

This is the C algorithm for evaluating equations in the interval [L, R] -

• Step 1 - Give an example of how to define an equation as a function that takes a variable x and returns a value y -

double equation(double x) {
   return x*x + 2*x + 1;
}

• Step 2 - Write a function that accepts two integers L and R as arguments and outputs the solution to the equation for each integer value between L and R. You can use a loop to iterate over the range [L, R], evaluating the equation for each integer value -

vector<double> evaluate_equation(int L, int R) {
   vector<double> results;
   for (int x = L; x <= R; x++) {
      double y = equation(x);
      results.push_back(y);
   }
   return results;
}

• Step 3 - After evaluating the equation on the range [L, R], you can use this function to obtain the result, which is passed as a vector of double values ​​-

  vector<double> results = evaluate_equation(1, 10);
  

NOTE - You can change the process of calculating any equation by simply replacing the equation function with the desired equation.

Method to follow

method 1

In C, you can evaluate equations in the range [L, R] using a loop that loops over each value in the range and evaluates the equation within it.

The range evaluated in the example is [1, 10], and the equation evaluated is i*i 2*i 1. The for loop repeatedly evaluates the equation for each value in the range and prints the answer to the console. Equations and ranges can be changed as needed.

Example 1

#include <iostream>
using namespace std;
int main() {
   int L = 1, R = 10; // range of values to evaluate
   for (int i = L; i <= R; i++) {
      int result = i*i + 2*i + 1; // equation to evaluate
      cout << "Result at " << i << " = " << result << endl;
   }
   return 0;
}

Output

Result at 1 = 4
Result at 2 = 9
Result at 3 = 16
Result at 4 = 25
Result at 5 = 36
Result at 6 = 49
Result at 7 = 64
Result at 8 = 81
Result at 9 = 100
Result at 10 = 121

Method 2

Here is a description of a C query that can be used to analyze the range of values ​​between L and R for a given equation -

In this diagram, the equation that needs to be calculated is first defined as a function called an equation. We then create an evaluation function that accepts two parameters, L and R, which represent the range of values ​​over which we want to evaluate the equation.

We iteratively evaluate the equation for each value between L and R (inclusive) within the evaluation function. Then, using cout, we print the results for each value.

We specify in the main function the range over which the equation is to be calculated (in this case, L = 1 and R = 10) and call the evaluation function with these values. The programmer's output will be the solution to the problem for each number between 1 and 10.

Example 2

#include <bits/stdc++.h>
using namespace std;

// Define the equation you want to evaluate
int equation(int x) {
   return x * x + 2 * x + 1;
}

// Define a function to evaluate the equation for a given range [L, R]
void evaluate(int L, int R) {
   for (int i = L; i <= R; i++) {
      int result = equation(i);
      cout << "The result of equation for " << i << " is " << result << endl;
   }
}
int main() {
   int L = 1, R = 10;
   evaluate(L, R);
   return 0;
}

Output

The result of equation for 1 is 4
The result of equation for 2 is 9
The result of equation for 3 is 16
The result of equation for 4 is 25
The result of equation for 5 is 36
The result of equation for 6 is 49
The result of equation for 7 is 64
The result of equation for 8 is 81
The result of equation for 9 is 100
The result of equation for 10 is 121

in conclusion

In summary, we can apply the prefix sum or cumulative sum method to evaluate a given equation within the interval [L,R]. By precomputing the prefix sum of equation values ​​up to each index, each query can be answered in constant time. The time complexity of this strategy (where N is the size of the input array) is O(N) for precomputation and O(1) for each query.

In general, the size of the input array and the number of queries to be run determine which method should be used. If the number of queries is much larger than the size of the array, the prefix sum technique is more efficient. However, if the number of queries is small, a binary search strategy may be a better choice.

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