std::hermite, std::hermitef, std::hermitel in C++

Introduction

C++ is a powerful programming language for its rich standard library that consists of various functions and utilities to help math computations. Special mathematical functions are among these utilities, which include Hermite Polynomials. Hermite polynomials are important in quantum mechanics, probability theory, and numerical analysis fields. To facilitate computation of these polynomials C++ has introduced three functions- std::hermitef, std::hermitel, and std::hermite.

These three functions will be examined here in terms of their usage as well as what they aim at solving so that we can analyze their syntax and parameter specifications together with some common real-life examples making use of them.

Problem Statement

Hermite polynomials form an integral part of many mathematical models and calculations. However, the process of manually calculating these polynomials may prove to be overwhelming and prone to mistakes, especially in cases where high-order polynomials are involved. In C++, the need for an accurate method that can efficiently calculate hermite polynomials necessitated the introduction of std::hermite functions. These are functions that have been made to evaluate precise Hermite polynomials in C++, and they also consider different datatypes.

There are three available functions in C++ to compute Hermite polynomials. The first function uses default double precision and is called std::hermite, which is used by the C++ Standard Library for computing Hermite polynomials. These functions are divided into types based on their inputs and return a float value when invoked.

Brief overview of Hermite Polynomials

Physics, probability, and numerical analysis have utilized Hermite polynomials as well. They employ recursive formulae that solve ordinary differential equations in particular, those described by Hermite's equation. Hn(x) represents an nth degree polynomial of Hermite.

Therefore, these sets of orthogonal polynomials exhibit some specific features and iterative relations.

Understanding std::hermite, std::hermitef, and std::hermitel

C++ Standard Library offers three separate functions for computing Hermite polynomials:

• std::hermite: This function uses double precision by default to calculate hermite polynomials.
• std::hermitef: It is a variant of std::Hermite, but works with single precision numbers instead.
• std.hermiretel: It works with long double precision- which implies that it will be more accurate but less efficient computationally.

These functions allow templatization for flexibility in selecting different data types for great accuracy control during programming stage.

Syntax:

Each of these functions has the following basic syntax:

• double std::hermite(unsigned n, double x);
• float std::hermitef(unsigned n, float x);
• long double std::hermitel(unsigned n, long double x);

Parameters:

• n: The order of the Hermite polynomial. It must be a non-negative integer.
• x: The point at which the Hermite polynomial is evaluated.

Returns: The value of the Hermite polynomial Hn(x).

Examples Usage

Program 1:

Let's explore a simple example of how these functions can be used in a C++ program:

Different types of precision are employed in computing Hermite polynomials of order 3 at the point x=2.0x as shown in this code snippet. The results for each type of precision will be provided in the output.

Output:

 
Hermite polynomial (double precision): 40
Hermite polynomial (float precision): 40
Hermite polynomial (long double precision): 40   

Explanation:

1. Include Headers

• #include <iostream>: Includes the standard input-output stream library, necessary for using std::cout to output results to the console.
• #include <cmath>: Includes mathematical functions and constants. However, the <cmath> header does not provide functions for Hermite polynomials directly.

2. Main Function

• unsigned n = 3;: Initializes an unsigned integer n to 3, which represents the degree of the Hermite polynomial.
• double x = 2.0;: Initializes a double-precision floating-point variable x to 2.0, the point at which the Hermite polynomial is evaluated.

3. Calculations and Output

• Double Precision Calculation:
  
  • double result_d = std::hermite(n, x);
  • This is a line that should calculate the hermite polynomial of degree n at point x using double precision. Nevertheless; the standard C++ library does not contain the function std::hermite. To evaluate Hermite polynomials, you will have to employ either a special library or else write your own function.
• Float Precision Calculation:
  
  • float result_f = std::hermitef(n, static_cast<float>(x));
  • Here is a line that tries to calculate Hermite's polynomial in float precision. Again, this build will fail because it does not recognize std::hermitf as one of its functions.
• Long Double Precision Calculation:
  
  • long double result_l = std::hermitel(n, static_cast<long double>(x));
  • It is an attempt to compute the Hermite polynomial in long-double precision. At the same time, this function is not in standard library too.

Program 2:

Output:

 
Hermite polynomial (double precision): 40.0000000000
Hermite polynomial (float precision): 40.0000000000
Hermite polynomial (long double precision): 40.0000000000   

Explanation:

1. Function Definitions:
   
   • hermite(unsigned n, double x): Computes the Hermite polynomial of degree n at point x using the recurrence relation. This function handles double precision calculations.
   • hermitef(unsigned n, float x): Computes the Hermite polynomial in float precision by converting the result from the double-precision function.
   • hermitel(unsigned n, long double x): Computes the Hermite polynomial in long double precision by converting the result from the double-precision function.
2. Precision Handling:
   
   • std::fixed and std::setprecision(10): These I/O manipulators ensure that the output is in fixed-point notation with 10 decimal places of precision.
3. Main Function:
   
   • Defines the degree of the polynomial (n) and the point of evaluation (x).
   • Computes the Hermite polynomial using different precision levels and outputs the results.

Applications and Use Cases

Hermite Polynomials have various applications:

• In quantum mechanics, they are used to model wave functions for quantum harmonic oscillators.
• Numerical analysis: Numerical integration methods such as Gaussian quadrature employ hermite polynomials.
• Probability theory: In normal distribution, these polynomials are involved in the Edgeworth series expansion.

The std::hermite, std::hermitef, and std::hermitel functions are C++ language specific that can be used to program them everywhere with certainty that these programmed polynomials work correctly and provide precise results if necessary.

Conclusion

In conclusion, a powerful tool for calculating Hermite polynomials is offered by the C++ Standard Library's std::hermite, std::hermitef, and std::hermitel functions with different levels of accuracy. Developers who understand their usage and differences can add these features to their programs effectively using single, double, or extended precision when required.

Through these developers' aids, much time is saved from calculation mistakes that otherwise would occur repeatedly; hence important mainly where the use of Hermite polynomial has been made.