npm package: @stdlib/math-base-tools-normhermitepoly
Description: Evaluate a normalized Hermite polynomial using double-precision floating-point arithmetic.
Author: The Stdlib Authors
Latest version: 0.2.3

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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 double-precision floating-point arithmetic.

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<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-normhermitepoly
```

</section>

<section class="usage">

## Usage

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

#### normhermitepoly( n, x )

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

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

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

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

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

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

#### normhermitepoly.factory( n )

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

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

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

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<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 dmap = require( '@stdlib/strided-base-dmap' );
var logEach = require( '@stdlib/console-log-each' );
var normhermitepoly = require( '@stdlib/math-base-tools-normhermitepoly' );

// Generate random values at which to evaluate a polynomial:
var x = uniform( 10, -50.0, 50.0, {
    'dtype': 'float64'
});

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

// Allocate an output array:
var y = zeros( x.length, 'float64' );

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

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

// Allocate an output array:
y = zeros( x.length, 'float64' );

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

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

// Allocate an output array:
y = zeros( x.length, 'float64' );

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

</section>

<!-- /.examples -->

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<section class="references">

</section>

<!-- /.references -->

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<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-base/tools/evalpoly`][@stdlib/math/base/tools/evalpoly]</span><span class="delimiter">: </span><span class="description">evaluate a polynomial using double-precision floating-point arithmetic.</span>
-   <span class="package-name">[`@stdlib/math-base/tools/hermitepoly`][@stdlib/math/base/tools/hermitepoly]</span><span class="delimiter">: </span><span class="description">evaluate a physicist's Hermite polynomial.</span>

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## 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].

#### Community

[![Chat][chat-image]][chat-url]

---

## 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-normhermitepoly/main/LICENSE

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

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[@stdlib/math/base/tools/evalpoly]: https://www.npmjs.com/package/@stdlib/math-base-tools-evalpoly

[@stdlib/math/base/tools/hermitepoly]: https://www.npmjs.com/package/@stdlib/math-base-tools-hermitepoly

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