npm package: @stdlib/math-base-tools-normhermitepolyf
Description: Evaluate a normalized Hermite polynomial using single-precision floating-point arithmetic.
Author: The Stdlib Authors
Latest version: 0.0.3

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# Normalized Hermite Polynomial

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> Evaluate a normalized [Hermite polynomial][hermite-polynomial] using single-precision floating-point arithmetic.

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<section class="intro">

The normalized (aka "probabilist") [Hermite polynomials][hermite-polynomial] are given by

<!-- <equation class="equation" label="eq:normalized_hermite_polynomials" align="center" raw="He_{n}(x)=(-1)^{n} e^{\frac{x^2}{2}} \frac{\mathrm d^{n}}{\mathrm d x^{n}} e^{-\frac{x^2}{2}}" alt="Equation for normalized Hermite polynomials."> -->

<div class="equation" align="center" data-raw-text="He_{n}(x)=(-1)^{n} e^{\frac{x^2}{2}} \frac{\mathrm{d}^{n}}{\mathrm{d}x^n} e^{-\frac{x^2}{2}}" data-equation="eq:normalized_hermite_polynomials">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bea0101eb61892f160eec8d97dc79188fd937523/lib/node_modules/@stdlib/math/base/tools/normhermitepoly/docs/img/equation_normalized_hermite_polynomials.svg" alt="Equation for normalized Hermite polynomials.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/math-base-tools-normhermitepolyf
```

</section>

<section class="usage">

## Usage

```javascript
var normhermitepolyf = require( '@stdlib/math-base-tools-normhermitepolyf' );
```

#### normhermitepolyf( n, x )

Evaluates a normalized [Hermite polynomial][hermite-polynomial] of degree `n` using single-precision floating-point arithmetic.

```javascript
var v = normhermitepolyf( 1, 1.0 );
// returns 1.0

v = normhermitepolyf( 1, 0.5 );
// returns 0.5

v = normhermitepolyf( 0, 0.5 );
// returns 1.0

v = normhermitepolyf( 2, 0.5 );
// returns -0.75

v = normhermitepolyf( -1, 0.5 );
// returns NaN
```

#### normhermitepolyf.factory( n )

Returns a function for evaluating a normalized [Hermite polynomial][hermite-polynomial] of degree `n` using single-precision floating-point arithmetic.

```javascript
var polyval = normhermitepolyf.factory( 2 );

var v = polyval( 0.5 );
// returns -0.75
```

</section>

<!-- /.usage -->

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<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var zeros = require( '@stdlib/array-zeros' );
var smap = require( '@stdlib/strided-base-smap' );
var logEach = require( '@stdlib/console-log-each' );
var normhermitepolyf = require( '@stdlib/math-base-tools-normhermitepolyf' );

// Generate random values at which to evaluate a polynomial:
var x = uniform( 10, -50.0, 50.0, {
    'dtype': 'float32'
});

// Create a polynomial function of degree 1:
var f = normhermitepolyf.factory( 1 );

// Allocate an output array:
var y = zeros( x.length, 'float32' );

// Evaluate the polynomial:
smap( x.length, x, 1, y, 1, f );
logEach( 'He_%d(%.3f) = %.3f', 1, x, y );

// Create a polynomial function of degree 2:
f = normhermitepolyf.factory( 2 );

// Allocate an output array:
y = zeros( x.length, 'float32' );

// Evaluate the polynomial:
smap( x.length, x, 1, y, 1, f );
logEach( 'He_%d(%.3f) = %.3f', 2, x, y );

// Create a polynomial function of degree 3:
f = normhermitepolyf.factory( 3 );

// Allocate an output array:
y = zeros( x.length, 'float32' );

// Evaluate the polynomial:
smap( x.length, x, 1, y, 1, f );
logEach( 'He_%d(%.3f) = %.3f', 3, x, y );
```

</section>

<!-- /.examples -->

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</section>

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* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

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## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-tools-normhermitepolyf/main/LICENSE

[hermite-polynomial]: https://en.wikipedia.org/wiki/Hermite_polynomials

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