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feat: add benchmarks, examples, docs
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‎lib/node_modules/@stdlib/math/base/special/chebyshev-t-polynomial/README.md‎

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<!--
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@license Apache-2.0
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Copyright (c) 2025 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# Chebyshev Polynomial of the first kind
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> [Chebyshev Polynomial][chebyshev-polynomial] of the first kind.
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<section class="intro">
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The [Chebyshev Polynomial][chebyshev-polynomial] of the first kind is defined as
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<!-- <equation class="equation" label="eq:chebyshev_polynomial" align="center" raw="T_n(x) = \cos( n \cos^{-1}(x) )" alt="Chebyshev Polynomial of the first kind."> -->
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```math
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T_n(x) = \cos( n \cos^{-1}(x) )
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```
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<!-- </equation> -->
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where `n >= 0` is the polynomial degree.
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The [Chebyshev Polynomial][chebyshev-polynomial] of the first kind is related to the Gaussian hypergeometric function via the following equation
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<!-- <equation class="equation" label="eq:chebyshev_polynomial2" align="center" raw="T_n(x) = {}_2F_1\left( -n, n; \frac{1}{2}; \frac{1}{2}(1-x) \right)" alt="Chebyshev Polynomial of the first kind expressed in terms of the Gaussian hypergeometric function."> -->
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```math
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T_n(x) = {}_2F_1\left( -n, n; \frac{1}{2}; \frac{1}{2}(1-x) \right)
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```
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<!-- </equation> -->
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var chebyshevtpoly = require( '@stdlib/math/base/special/chebyshev-t-polynomial' );
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```
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#### chebyshevtpoly( n, x )
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Evaluates the [Chebyshev Polynomial][chebyshev-polynomial] of the first kind for a degree `n` at a value `x`.
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```javascript
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var y = chebyshevtpoly( 0.0, 0.5 );
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// returns 1.0
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y = chebyshevtpoly( 1.0, -0.5 );
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// returns -0.5
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y = chebyshevtpoly( 5.0, 0.5 );
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// returns 0.5
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```
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If provided `NaN` as any argument, the function returns `NaN`.
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```javascript
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var y = chebyshevtpoly( NaN, 1.0 );
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// returns NaN
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y = chebyshevtpoly( 0.0, NaN );
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// returns NaN
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```
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If provided a polynomial degree `n < 0`, the function returns `NaN`.
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```javascript
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var y = chebyshevtpoly( -2.0, 0.5 );
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// returns NaN
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y = chebyshevtpoly( -4.5, -0.5 );
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// returns NaN
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```
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If provided a polynomial degree `n` which is not a nonnegative integer, the function returns `NaN`.
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```javascript
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var y = chebyshevtpoly( 2.5, 0.5 );
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// returns NaN
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y = chebyshevtpoly( 0.5, -0.5 );
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// returns NaN
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var randu = require( '@stdlib/random/base/randu' );
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var round = require( '@stdlib/math/base/special/round' );
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var chebyshevtpoly = require( '@stdlib/math/base/special/chebyshev-t-polynomial' );
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var n;
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var x;
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var y;
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var i;
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for ( i = 0; i < 10; i++ ) {
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n = round( randu()*10.0 );
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x = ( randu()*2.0 ) - 1.0;
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y = chebyshevtpoly( n, x );
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console.log( 'n: %d, x: %d, T_n(x): %d', n.toFixed( 4 ), x.toFixed( 4 ), y.toFixed( 4 ) );
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}
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```
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</section>
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<!-- /.examples -->
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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
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<section class="related">
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</section>
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<!-- /.related -->
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<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
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<section class="links">
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[chebyshev-polynomial]: https://en.wikipedia.org/wiki/Chebyshev_polynomials#Definitions
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/math/base/special/chebyshev-t-polynomial/benchmark/benchmark.js‎

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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2025 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var uniform = require( '@stdlib/random/array/uniform' );
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var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
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var isnan = require( '@stdlib/math/base/assert/is-nan' );
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var pkg = require( './../package.json' ).name;
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var chebyshevtpoly = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( b ) {
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var n;
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var x;
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var y;
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var i;
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n = discreteUniform( 100, 0, 50 );
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x = uniform( 100, -1.0, 1.0 );
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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y = chebyshevtpoly( n[ i % n.length ], x[ i % x.length ] );
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( isnan( y ) ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

‎lib/node_modules/@stdlib/math/base/special/chebyshev-t-polynomial/docs/repl.txt‎

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{{alias}}( n, x )
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Evaluates the Chebyshev polynomial of the first kind
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for a degree `n` at a value `x`.
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Parameters
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----------
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n: number
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Degree of the polynomial.
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x: number
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Input value.
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Returns
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-------
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out: number
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Polynomial value.
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Examples
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--------
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> var y = {{alias}}( 0.0, 0.5 )
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1.0
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> y = {{alias}}( 1.0, -0.5 )
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-0.5
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> y = {{alias}}( 5.0, 0.5 )
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0.5
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> y = {{alias}}( 2.5, 0.5 )
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NaN
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> y = {{alias}}( -1.0, 0.5 )
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NaN
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> y = {{alias}}( NaN, 1.0 )
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NaN
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> y = {{alias}}( 1.0, NaN )
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NaN
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See Also
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--------
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‎lib/node_modules/@stdlib/math/base/special/chebyshev-t-polynomial/docs/types/index.d.ts‎

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/*
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* @license Apache-2.0
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*
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* Copyright (c) 2019 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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// TypeScript Version: 4.1
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/**
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* Evaluates the Chebyshev polynomial of the first kind for a degree `n` at a value `x`.
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*
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* @param n - degree of the polynomial
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* @param x - input value
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* @returns polynomial value
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*
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* @example
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* var y = chebyshevtpoly( 0.0, 0.5 );
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* // returns 1.0
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*
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* @example
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* var y = chebyshevtpoly( 1.0, -0.5 );
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* // returns -0.5
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*
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* @example
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* var y = chebyshevtpoly( 5.0, 0.5 );
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* // returns 0.5
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*
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* @example
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* var y = chebyshevtpoly( 2.5, 0.5 );
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* // returns NaN
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*
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* @example
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* var y = chebyshevtpoly( -1.0, 0.5 );
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* // returns NaN
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*
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* @example
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* var y = chebyshevtpoly( NaN, 1.0 );
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* // returns NaN
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*
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* @example
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* var y = chebyshevtpoly( 1.0, NaN );
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* // returns NaN
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*/
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declare function chebyshevtpoly( n: number, x: number ): number;
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// EXPORTS //
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export = chebyshevtpoly;

‎lib/node_modules/@stdlib/math/base/special/chebyshev-t-polynomial/docs/types/test.ts‎

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/*
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* @license Apache-2.0
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*
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* Copyright (c) 2019 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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import chebyshevtpoly = require( './index' );
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// TESTS //
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// The function returns a number...
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{
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chebyshevtpoly( 8.0, 1.0 ); // $ExpectType number
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}
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// The compiler throws an error if the function is provided values other than two numbers...
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{
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chebyshevtpoly( true, 3 ); // $ExpectError
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chebyshevtpoly( false, 2 ); // $ExpectError
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chebyshevtpoly( '5', 1 ); // $ExpectError
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chebyshevtpoly( [], 1 ); // $ExpectError
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chebyshevtpoly( {}, 2 ); // $ExpectError
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chebyshevtpoly( ( x: number ): number => x, 2 ); // $ExpectError
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chebyshevtpoly( 9, true ); // $ExpectError
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chebyshevtpoly( 9, false ); // $ExpectError
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chebyshevtpoly( 5, '5' ); // $ExpectError
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chebyshevtpoly( 8, [] ); // $ExpectError
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chebyshevtpoly( 9, {} ); // $ExpectError
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chebyshevtpoly( 8, ( x: number ): number => x ); // $ExpectError
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chebyshevtpoly( [], true ); // $ExpectError
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chebyshevtpoly( {}, false ); // $ExpectError
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chebyshevtpoly( false, '5' ); // $ExpectError
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chebyshevtpoly( {}, [] ); // $ExpectError
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chebyshevtpoly( '5', ( x: number ): number => x ); // $ExpectError
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}
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// The compiler throws an error if the function is provided insufficient arguments...
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{
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chebyshevtpoly(); // $ExpectError
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chebyshevtpoly( 3 ); // $ExpectError
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}

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