## Table of Contents

Higher-order functions are functions that can take other functions as arguments or return functions as results. In JavaScript, functions are first-class citizens, which means they can be treated like any other value. This allows for the creation and manipulation of higher-order functions. We have looked at some of these functions when we talked about Array functions and Closure. This article is part of a series on higher functional programming.

### Uses #

Higher-order functions are used practically in various scenarios to write cleaner, more modular, and reusable code. Here are some practical use cases for higher-order functions:

**Callback Functions:** Higher-order functions are often used to handle asynchronous operations and callbacks. For example, in event-driven programming or when making API calls, you can pass a callback function to a higher-order function that executes the callback once the operation is complete.

```
function fetchData(url, callback) {
// Perform asynchronous operation
// Call the callback function with the fetched data
callback(data);
}
function handleData(data) {
// Handle the fetched data
}
fetchData('https://example.com/api', handleData);
```

**Function Composition:** Higher-order functions allow you to compose multiple functions together to create more complex operations. This can enhance code readability and maintainability.

```
// Example functions
const square = arr=>arr.map(x=>x*x)
const sum = arr=>arr.reduce((accumulator, num) => accumulator + num, 0);
// Function composition
const compose = (f, g) => (x) => f(g(x));
// Compose functions to perform a series of operations
const sum_of_squares= compose(sum,square)
// Usage example
const result = sum_of_squares([1,2,3]);
console.log(result) //14
```

**Iteration and Transformation:** Higher-order functions like map(), filter(), and reduce() are commonly used for iterating over arrays, transforming data, and performing operations on array elements.

```
const numbers = [1, 2, 3, 4, 5];
const doubledNumbers = numbers.map(function (num) {
return num * 2;
});
const evenNumbers = numbers.filter(function (num) {
return num % 2 === 0;
});
const sum = numbers.reduce(function (acc, num) {
return acc + num;
}, 0);
console.log(doubledNumbers); // Output: [2, 4, 6, 8, 10]
console.log(evenNumbers); // Output: [2, 4]
console.log(sum); // Output: 15
```

**Currying and Partial Application:** Higher-order functions enable currying and partial application, allowing you to create new functions by pre-filling some arguments of an existing function.

```
// Currying using arrow functions
const curryMultiply = (a) => (b) => (c) => a * b * c;
console.log(curryMultiply(2)(3)(4)); // Output: 24
```

### Example of Solving Differential Equation using Higher Order Function #

Below an implementation of the Runge-Kutta method (specifically, the fourth-order Runge-Kutta method) in JavaScript. Experiment using this notebook on Scribbler: Runge-Kutta Method for Differential Equations

```
function runge_kutta(dydt, y0, t0, h) {
return t=>{
const numSteps = Math.floor((t - t0) / h);
let t1 = t0;
let y = y0;
for (let i = 0; i < numSteps; i++) {
const k1 = h * dydt(t1, y);
const k2 = h * dydt(t1 + h / 2, y + k1 / 2);
const k3 = h * dydt(t1 + h / 2, y + k2 / 2);
const k4 = h * dydt(t1 + h, y + k3);
y = y + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
t1 = t1 + h;
}
return y;
}
}
```

This function runge_kutta is a higher-order function that takes and input of a function representing dy/dt and returns another function that maps y as a function of t. Example usage: Solve the differential equation dy/dt = t^2 + y with initial condition y(0) = 1

```
const dydt = (t, y) => t * t + y; //Inline function definition
const y0 = 1;
const t0 = 0;
const h = 0.1;
const f = runge_kutta(dydt, y0, t0, h);
console.log("Value of y at 1",f(1));
```

### Example of Higher Order Function in Simulation #

Below is an example of a Monte Carlo simulation for football odds calculation using a simple scoring model. Experiment in this notebook: Monte-Carlo Simulation of Goals

```
// Function to perform Monte Carlo simulation for odds calculation
function monte_carlo_simulation( num_simulations, simulate_match) {
let home_wins = 0;
let away_wins = 0;
let draws = 0;
for (let i = 0; i < num_simulations; i++) {
const goal_difference = simulate_match();
if (goal_difference > 0) {
home_wins++;
} else if (goal_difference < 0) {
away_wins++;
} else {
draws++;
}
}
return {
"home_win_percentage":home_wins/num_simulations,
"away_win_precentage":away_wins/num_simulations,
"draw_percentage":draws/num_simulations,
};
}
```

This function takes a variable number of simulations and a function simulateMatch, that simulates a the scores each time it is called. The simulation can be customized and a sample is give below:

```
// Sample function to simulate a football match and return the goal difference
function simulate_match_on_strength(home_team_strength, away_team_strength) {
return ()=>{
// Generate random goal counts for each team
const home_goals = Math.round(Math.random() * home_team_strength);
const away_goals = Math.round(Math.random() * away_team_strength);
// Calculate the goal difference
const goal_difference = home_goals - away_goals;
return goal_difference;
}
}
```

This returns a function that encapsulates ‘Home Team Strength’ and ‘Away Team Strength’ using the concept of closure. This can be input into the simulation as below:

```
//Output win percentages on where home team is stronger than away team.
console.log(monte_carlo_simulation(1000,simulate_match_on_strength(2,1)))
```