feat: add fft/base/fftpack/rffti

lib/node_modules/@stdlib/fft/base/fftpack/rffti/README.md

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2026 The Stdlib Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+   http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+
+-->
+
+# rffti
+
+> Initialize a workspace array for performing a real-valued Fourier transform.
+
+<!-- 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">
+
+</section>
+
+<!-- /.intro -->
+
+<!-- Package usage documentation. -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+```
+
+#### rffti( N, workspace, strideW, offsetW )
+
+Initializes a workspace array for performing a real-valued Fourier transform.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var N = 8;
+var workspace = new Float64Array( ( 2*N ) + 34 );
+
+var out = rffti( N, workspace, 1, 0 );
+// returns <Float64Array>
+
+var bool = ( out === workspace );
+// returns true
+
+var twiddleFactors = workspace.slice( N, 2*N );
+// returns <Float64Array>[ 0, ~0.707, ~0.707, 0, 0, 0, 0, 0 ]
+
+var factors = workspace.slice( 2*N, ( 2*N ) + 4 );
+// returns <Float64Array>[ 8, 2, 2, 4 ]
+```
+
+The function accepts the following arguments:
+
+-   **N**: length of the sequence to transform.
+-   **workspace**: workspace array.
+-   **strideW**: stride length for `workspace`.
+-   **offsetW**: starting index for `workspace`.
+
+</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">
+
+## Notes
+
+-   The workspace array is divided into three sections:
+
+    ```text
+              size = N                 N            2+ceil(log2(N)/2)
+                   ↓                   ↓                  ↓
+        | scratch / workspace | twiddle factors | radix factor table |
+                   ↑                   ↑                  ↑
+    i = 0         ...         N       ...       2N       ...
+    ```
+
+    -   **scratch/workspace**: used as a scratch space when performing transforms. This section is not updated during initialization.
+    -   **twiddle factors**: a table of reusable complex-exponential constants stored as cosine/sine pairs.
+    -   **radix factor table**: a table containing the sequence length `N`, the number of factors into which `N` was decomposed, and the individual integer radix factors.
+
+-   In general, a workspace array should have `2N + 34` indexed elements (as `log2(N)/2 ≤ 32` for all `2^64`). During initialization, only the sections for storing twiddle factors and the factorization of `N` are updated.
+
+-   The radix factor table is comprised as follows:
+
+    ```text
+    | sequence_length | number_of_factors | integer_factors |
+    ```
+
+-   If `N` equals `1`, the function returns early without modifying the workspace, as a single data point is its own Fourier transform.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var zeroTo = require( '@stdlib/array/zero-to' );
+var logEach = require( '@stdlib/console/log-each' );
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+
+var N = 8;
+var workspace = new Float64Array( ( 2*N ) + 34 );
+
+rffti( N, workspace, 1, 0 );
+console.log( 'Sequence length: %d', N );
+
+console.log( 'Twiddle factors:' );
+var idx = zeroTo( N, 'generic' );
+logEach( '  workspace[ %d ] = %d', idx, workspace.slice( N, 2*N ) );
+
+console.log( 'Factorization:' );
+var nf = workspace[ (2*N)+1 ];
+
+console.log( '  number of factors: %d', nf );
+idx = zeroTo( nf, 'generic' );
+logEach( '  factor[ %d ]: %d', idx, workspace.slice( (2*N)+2, (2*N)+2+nf ) );
+```
+
+</section>
+
+<!-- /.examples -->
+
+<!-- Section to include cited references. If references are included, add a horizontal rule *before* the section. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="references">
+
+</section>
+
+<!-- /.references -->
+
+<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
+
+<section class="related">
+
+</section>
+
+<!-- /.related -->
+
+<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="links">
+
+</section>
+
+<!-- /.links -->

lib/node_modules/@stdlib/fft/base/fftpack/rffti/benchmark/benchmark.js

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+/*
+* @license Apache-2.0
+*
+* Copyright (c) 2026 The Stdlib Authors.
+*
+* Licensed under the Apache License, Version 2.0 (the "License");
+* you may not use this file except in compliance with the License.
+* You may obtain a copy of the License at
+*
+*    http://www.apache.org/licenses/LICENSE-2.0
+*
+* Unless required by applicable law or agreed to in writing, software
+* distributed under the License is distributed on an "AS IS" BASIS,
+* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+* See the License for the specific language governing permissions and
+* limitations under the License.
+*/
+
+'use strict';
+
+// MODULES //
+
+var bench = require( '@stdlib/bench' );
+var format = require( '@stdlib/string/format' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+var Float64Array = require( '@stdlib/array/float64' );
+var pkg = require( './../package.json' ).name;
+var rffti = require( './../lib' );
+
+
+// FUNCTIONS //
+
+/**
+* Creates a benchmark function.
+*
+* @private
+* @param {PositiveInteger} N - sequence length
+* @returns {Function} benchmark function
+*/
+function createBenchmark( N ) {
+	var workspace = new Float64Array( ( 2*N ) + 34 );
+	return benchmark;
+
+	/**
+	* Benchmark function.
+	*
+	* @private
+	* @param {Benchmark} b - benchmark instance
+	*/
+	function benchmark( b ) {
+		var i;
+
+		b.tic();
+		for ( i = 0; i < b.iterations; i++ ) {
+			rffti( N, workspace, 1, 0 );
+			if ( isnan( workspace[ N+1 ] ) ) {
+				b.fail( 'should not return NaN' );
+			}
+		}
+		b.toc();
+		if ( isnan( workspace[ N+1 ] ) ) {
+			b.fail( 'should not return NaN' );
+		}
+		b.pass( 'benchmark finished' );
+		b.end();
+	}
+}
+
+
+// MAIN //
+
+/**
+* Main execution sequence.
+*
+* @private
+*/
+function main() {
+	var lengths;
+	var N;
+	var f;
+	var i;
+
+	lengths = [
+		8,
+		16,
+		32,
+		64,
+		128,
+		256,
+		512,
+		1024
+	];
+
+	for ( i = 0; i < lengths.length; i++ ) {
+		N = lengths[ i ];
+		f = createBenchmark( N );
+		bench( format( '%s:N=%d', pkg, N ), f );
+	}
+}
+
+main();

lib/node_modules/@stdlib/fft/base/fftpack/rffti/docs/repl.txt

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+
+{{alias}}( N, workspace, strideW, offsetW )
+    Initializes a workspace array for performing a real-valued Fourier
+    transform.
+
+    The workspace array is divided into three sections:
+
+    1. scratch/workspace: the section ranges from indices 0 to N-1 and is used
+    while performing transforms. This section is not updated during
+    initialization.
+
+    2. twiddle factors: the section ranges from indices N to 2N-1 and stores a
+    table of reusable complex exponential constants as cosine/sine pairs.
+
+    3. radix factor table: the section starts at index 2N and stores the
+    sequence length N, the number of factors into which N was decomposed, and
+    the individual integer radix factors.
+
+    Any remaining array space remains as unused storage.
+
+    Parameters
+    ----------
+    N: integer
+        Length of the sequence.
+
+    workspace: ArrayLikeObject<number>
+        Workspace array.
+
+    strideW: integer
+        Stride length for `workspace`.
+
+    offsetW: integer
+        Starting index for `workspace`.
+
+    Returns
+    -------
+    out: ArrayLikeObject<number>
+        Workspace array.
+
+    Examples
+    --------
+    > var N = 8;
+    > var workspace = new {{alias:@stdlib/array/float64}}( ( 2*N ) + 34 );
+    > var out = {{alias}}( N, workspace, 1, 0 )
+    <Float64Array>
+    > var bool = ( out === workspace )
+    true
+    > var twiddleFactors = workspace.slice( N, 2*N )
+    <Float64Array>[ 0, ~0.707, ~0.707, 0, 0, 0, 0, 0 ]
+    > var factors = workspace.slice( 2*N, ( 2*N ) + 4 )
+    <Float64Array>[ 8, 2, 2, 4 ]
+
+    See Also
+    --------
+

lib/node_modules/@stdlib/fft/base/fftpack/rffti/docs/types/index.d.ts

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+/*
+* @license Apache-2.0
+*
+* Copyright (c) 2026 The Stdlib Authors.
+*
+* Licensed under the Apache License, Version 2.0 (the "License");
+* you may not use this file except in compliance with the License.
+* You may obtain a copy of the License at
+*
+*    http://www.apache.org/licenses/LICENSE-2.0
+*
+* Unless required by applicable law or agreed to in writing, software
+* distributed under the License is distributed on an "AS IS" BASIS,
+* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+* See the License for the specific language governing permissions and
+* limitations under the License.
+*/
+
+// TypeScript Version: 4.1
+
+/// <reference types="@stdlib/types"/>
+
+import { Collection } from '@stdlib/types/array';
+
+/**
+* Initializes a workspace array for performing a real-valued Fourier transform.
+*
+* ## Notes
+*
+* -   The workspace array should have a length of at least `( 2*N ) + 34` elements.
+* -   For single-point sequences (N=1), the function returns immediately as the FFT is the identity operation.
+*
+* @param N - length of the sequence
+* @param workspace - workspace array
+* @param strideW - stride length for `workspace`
+* @param offsetW - starting index for `workspace`
+* @returns workspace array
+*
+* @example
+* var Float64Array = require( '@stdlib/array/float64' );
+*
+* var N = 8;
+* var workspace = new Float64Array( ( 2*N ) + 34 );
+*
+* var out = rffti( N, workspace, 1, 0 );
+* // returns <Float64Array>
+*
+* var bool = ( out === workspace );
+* // returns true
+*
+* var twiddleFactors = workspace.slice( N, 2*N );
+* // returns <Float64Array>[ 0, ~0.707, ~0.707, 0, 0, 0, 0, 0 ]
+*
+* var factors = workspace.slice( 2*N, ( 2*N ) + 4 );
+* // returns <Float64Array>[ 8, 2, 2, 4 ]
+*/
+declare function rffti<T extends Collection<number>>( N: number, workspace: T, strideW: number, offsetW: number ): T;
+
+
+// EXPORTS //
+
+export = rffti;