diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/README.md b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/README.md
new file mode 100644
index 000000000000..60f25003f647
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/README.md
@@ -0,0 +1,156 @@
+
+
+# rfftf
+
+> Compute the forward real-valued fast Fourier transform (FFT).
+
+
+
+
+
+
+
+
+
+
+
+## Usage
+
+```javascript
+var rfftf = require( '@stdlib/fft/base/fftpack/rfftf' );
+```
+
+#### rfftf( N, r, strideR, offsetR, workspace, strideW, offsetW )
+
+Computes the forward real-valued fast Fourier transform (FFT).
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+
+var N = 4;
+var workspace = new Float64Array( ( 2*N ) + 34 );
+
+rffti( N, workspace, 1, 0 );
+
+var r = new Float64Array( [ 1.0, 2.0, 3.0, 4.0 ] );
+
+rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+// r => [ 10.0, -2.0, 2.0, -2.0 ]
+```
+
+The function accepts the following arguments:
+
+- **N**: length of the sequence to transform.
+- **r**: input array.
+- **strideR**: stride length for `r`.
+- **offsetR**: starting index for `r`.
+- **workspace**: workspace array containing pre-computed values.
+- **strideW**: stride length for `workspace`.
+- **offsetW**: starting index for `workspace`.
+
+
+
+
+
+
+
+
+
+## Notes
+
+- Before calling this function, initialize the workspace by calling [`rffti`][@stdlib/fft/base/fftpack/rffti] with the same sequence length and workspace layout.
+
+- The function performs the transform in-place and returns the same input array reference.
+
+- For `N = 4`, the output
+
+ ```text
+ [ 10.0, -2.0, 2.0, -2.0 ]
+ ```
+
+ corresponds to a zero-frequency term `10.0`, a complex coefficient `-2.0 + 2.0i` at frequency `1`, and a Nyquist term `-2.0`.
+
+- If `N` equals `1`, the function returns early without modifying the input, as a single data point is its own Fourier transform. In that case, the function still returns the same input array reference.
+
+
+
+
+
+
+
+## Examples
+
+
+
+```javascript
+var zeros = require( '@stdlib/array/zeros' );
+var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+var rfftf = require( '@stdlib/fft/base/fftpack/rfftf' );
+
+var N = 4;
+var opts = {
+ 'dtype': 'float64'
+};
+var r = discreteUniform( N, -10, 10, opts );
+var workspace = zeros( ( 2*N ) + 34 );
+
+console.log( r );
+
+rffti( N, workspace, 1, 0 );
+rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+console.log( r );
+```
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+[@stdlib/fft/base/fftpack/rffti]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/fft/base/fftpack/rffti
+
+
+
+
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/benchmark/benchmark.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/benchmark/benchmark.js
new file mode 100644
index 000000000000..ec2747300307
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/benchmark/benchmark.js
@@ -0,0 +1,138 @@
+/**
+* @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 slice = require( '@stdlib/array/slice' );
+var uniform = require( '@stdlib/random/array/uniform' );
+var floor = require( '@stdlib/math/base/special/floor' );
+var pow = require( '@stdlib/math/base/special/pow' );
+var format = require( '@stdlib/string/format' );
+var isnan = require( '@stdlib/math/base/assert/is-nan' );
+var Float64Array = require( '@stdlib/array/float64' );
+var dcopy = require( '@stdlib/blas/base/dcopy' );
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+var pkg = require( './../package.json' ).name;
+var rfftf = require( './../lib' );
+
+
+// VARIABLES //
+
+var options = {
+ 'dtype': 'float64'
+};
+
+
+// FUNCTIONS //
+
+/**
+* Creates a benchmark function.
+*
+* @private
+* @param {PositiveInteger} iter - number of iterations
+* @param {PositiveInteger} N - sequence length
+* @returns {Function} benchmark function
+*/
+function createBenchmark( iter, N ) {
+ var workspace = new Float64Array( ( 2*N ) + 34 );
+ var x;
+ var i;
+
+ x = [];
+ for ( i = 0; i < iter; i++ ) {
+ x.push( uniform( N, -100.0, 100.0, options ) );
+ }
+
+ rffti( N, workspace, 1, 0 );
+ return benchmark;
+
+ /**
+ * Benchmark function.
+ *
+ * @private
+ * @param {Benchmark} b - benchmark instance
+ */
+ function benchmark( b ) {
+ var xc;
+ var y;
+ var i;
+
+ xc = slice( x );
+ for ( i = 0; i < iter; i++ ) {
+ xc[ i ] = dcopy( N, x[ i ], 1, new Float64Array( N ), 1 );
+ }
+ b.tic();
+ for ( i = 0; i < b.iterations; i++ ) {
+ y = rfftf( N, xc[ i ], 1, 0, workspace, 1, 0 );
+ if ( isnan( y[ i%N ] ) ) {
+ b.fail( 'should not return NaN' );
+ }
+ }
+ b.toc();
+ if ( isnan( y[ i%N ] ) ) {
+ b.fail( 'should not return NaN' );
+ }
+ b.pass( 'benchmark finished' );
+ b.end();
+ }
+}
+
+
+// MAIN //
+
+/**
+* Main execution sequence.
+*
+* @private
+*/
+function main() {
+ var lengths;
+ var opts;
+ var iter;
+ var N;
+ var f;
+ var i;
+
+ lengths = [
+ 8,
+ 16,
+ 32,
+ 64,
+ 128,
+ 256,
+ 512,
+ 1024
+ ];
+
+ iter = 1e6;
+
+ for ( i = 0; i < lengths.length; i++ ) {
+ N = lengths[ i ];
+ f = createBenchmark( iter, N );
+ opts = {
+ 'iterations': iter
+ };
+ bench( format( '%s:N=%d', pkg, N ), opts, f );
+ iter = floor( pow( iter, 3.0/4.0 ) );
+ }
+}
+
+main();
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/repl.txt b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/repl.txt
new file mode 100644
index 000000000000..c117bdc2cd0f
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/repl.txt
@@ -0,0 +1,54 @@
+
+{{alias}}( N, r, strideR, offsetR, workspace, strideW, offsetW )
+ Computes the forward real-valued fast Fourier transform (FFT).
+
+ Before calling this function, initialize the workspace by calling
+ `@stdlib/fft/base/fftpack/rffti` with the same sequence length and workspace
+ layout.
+
+ Parameters
+ ----------
+ N: integer
+ Length of the sequence to transform.
+
+ r: ArrayLikeObject
+ Input array.
+
+ strideR: integer
+ Stride length for `r`.
+
+ offsetR: integer
+ Starting index for `r`.
+
+ workspace: ArrayLikeObject
+ Workspace array containing pre-computed values.
+
+ strideW: integer
+ Stride length for `workspace`.
+
+ offsetW: integer
+ Starting index for `workspace`.
+
+ Returns
+ -------
+ r: ArrayLikeObject
+ Input array.
+
+ Examples
+ --------
+ > var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+ > var N = 4;
+ > var r = new {{alias:@stdlib/array/float64}}( [ 1.0, 2.0, 3.0, 4.0 ] );
+ > var workspace = new {{alias:@stdlib/array/float64}}( ( 2*N ) + 34 );
+ > rffti( N, workspace, 1, 0 )
+
+ > var out = {{alias}}( N, r, 1, 0, workspace, 1, 0 )
+
+ > var bool = ( out === r )
+ true
+ > r
+ [ 10.0, -2.0, 2.0, -2.0 ]
+
+ See Also
+ --------
+
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/types/index.d.ts b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/types/index.d.ts
new file mode 100644
index 000000000000..a95ed2ed1a44
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/types/index.d.ts
@@ -0,0 +1,56 @@
+/*
+* @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
+
+///
+
+import { Collection } from '@stdlib/types/array';
+
+/**
+* Computes the forward real-valued fast Fourier transform (FFT).
+*
+* @param N - length of the sequence to transform
+* @param r - input array
+* @param strideR - stride length for `r`
+* @param offsetR - starting index for `r`
+* @param workspace - workspace array containing pre-computed values
+* @param strideW - stride length for `workspace`
+* @param offsetW - starting index for `workspace`
+* @returns input array
+*
+* @example
+* var Float64Array = require( '@stdlib/array/float64' );
+* var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+*
+* var N = 4;
+*
+* var workspace = new Float64Array( ( 2*N ) + 34 );
+* rffti( N, workspace, 1, 0 );
+*
+* var r = new Float64Array( [ 1.0, 2.0, 3.0, 4.0 ] );
+*
+* rfftf( N, r, 1, 0, workspace, 1, 0 );
+* // r => [ 10.0, -2.0, 2.0, -2.0 ]
+*/
+declare function rfftf>( N: number, r: T, strideR: number, offsetR: number, workspace: Collection, strideW: number, offsetW: number ): T;
+
+
+// EXPORTS //
+
+export = rfftf;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/types/test.ts b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/types/test.ts
new file mode 100644
index 000000000000..135116bb4667
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/docs/types/test.ts
@@ -0,0 +1,148 @@
+/*
+* @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.
+*/
+
+import rfftf = require( './index' );
+
+
+// TESTS //
+
+// The function returns a collection...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( 4, r, 1, 0, workspace, 1, 0 ); // $ExpectType Float64Array
+}
+
+// The compiler throws an error if the function is provided a first argument which is not a number...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( '4', r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( true, r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( false, r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( null, r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( void 0, r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( [], r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( {}, r, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( ( x: number ): number => x, r, 1, 0, workspace, 1, 0 ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a second argument which is not a collection...
+{
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( 4, '4', 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, 4, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, true, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, false, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, null, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, void 0, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, {}, 1, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, ( x: number ): number => x, 1, 0, workspace, 1, 0 ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a third argument which is not a number...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( 4, r, '1', 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, true, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, false, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, null, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, void 0, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, [], 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, {}, 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, ( x: number ): number => x, 0, workspace, 1, 0 ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a fourth argument which is not a number...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( 4, r, 1, '0', workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, true, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, false, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, null, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, void 0, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, [], workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, {}, workspace, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, ( x: number ): number => x, workspace, 1, 0 ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a fifth argument which is not a collection...
+{
+ const r = new Float64Array( 4 );
+
+ rfftf( 4, r, 1, 0, '42', 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, 42, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, true, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, false, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, null, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, void 0, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, {}, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, ( x: number ): number => x, 1, 0 ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a sixth argument which is not a number...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( 4, r, 1, 0, workspace, '1', 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, true, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, false, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, null, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, void 0, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, [], 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, {}, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, ( x: number ): number => x, 0 ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided a seventh argument which is not a number...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf( 4, r, 1, 0, workspace, 1, '0' ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, true ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, false ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, null ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, void 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, [] ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, {} ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, ( x: number ): number => x ); // $ExpectError
+}
+
+// The compiler throws an error if the function is provided an unsupported number of arguments...
+{
+ const r = new Float64Array( 4 );
+ const workspace = new Float64Array( ( 2*4 ) + 34 );
+
+ rfftf(); // $ExpectError
+ rfftf( 4 ); // $ExpectError
+ rfftf( 4, r ); // $ExpectError
+ rfftf( 4, r, 1 ); // $ExpectError
+ rfftf( 4, r, 1, 0 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1 ); // $ExpectError
+ rfftf( 4, r, 1, 0, workspace, 1, 0, 123 ); // $ExpectError
+}
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/examples/index.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/examples/index.js
new file mode 100644
index 000000000000..b2b4f1218523
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/examples/index.js
@@ -0,0 +1,38 @@
+/**
+* @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';
+
+var zeros = require( '@stdlib/array/zeros' );
+var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+var rfftf = require( './../lib' );
+
+var N = 4;
+var opts = {
+ 'dtype': 'float64'
+};
+var r = discreteUniform( N, -10, 10, opts );
+var workspace = zeros( ( 2*N ) + 34 );
+
+console.log( r );
+
+rffti( N, workspace, 1, 0 );
+rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+console.log( r );
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/index.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/index.js
new file mode 100644
index 000000000000..d4e7f52daa0a
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/index.js
@@ -0,0 +1,49 @@
+/**
+* @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';
+
+/**
+* Compute the forward real-valued fast Fourier transform (FFT).
+*
+* @module @stdlib/fft/base/fftpack/rfftf
+*
+* @example
+* var Float64Array = require( '@stdlib/array/float64' );
+* var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+* var rfftf = require( '@stdlib/fft/base/fftpack/rfftf' );
+*
+* var N = 4;
+*
+* var workspace = new Float64Array( ( 2*N ) + 34 );
+* rffti( N, workspace, 1, 0 );
+*
+* var r = new Float64Array( [ 1.0, 2.0, 3.0, 4.0 ] );
+*
+* rfftf( N, r, 1, 0, workspace, 1, 0 );
+* // r => [ 10.0, -2.0, 2.0, -2.0 ]
+*/
+
+// MODULES //
+
+var main = require( './main.js' );
+
+
+// EXPORTS //
+
+module.exports = main;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/main.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/main.js
new file mode 100644
index 000000000000..1dbfbd084cfb
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/main.js
@@ -0,0 +1,115 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+'use strict';
+
+// MODULES //
+
+var rfftf1 = require( './rfftf1.js' );
+
+
+// MAIN //
+
+/**
+* Computes the forward real-valued fast Fourier transform (FFT).
+*
+* @param {NonNegativeInteger} N - length of the sequence to transform
+* @param {Collection} r - input array
+* @param {integer} strideR - stride length for `r`
+* @param {NonNegativeInteger} offsetR - starting index for `r`
+* @param {Collection} workspace - workspace array containing pre-computed values
+* @param {integer} strideW - stride length for `workspace`
+* @param {NonNegativeInteger} offsetW - starting index for `workspace`
+* @returns {Collection} input array
+*
+* @example
+* var Float64Array = require( '@stdlib/array/float64' );
+* var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+*
+* var N = 4;
+*
+* var workspace = new Float64Array( ( 2*N ) + 34 );
+* rffti( N, workspace, 1, 0 );
+*
+* var r = new Float64Array( [ 1.0, 2.0, 3.0, 4.0 ] );
+*
+* rfftf( N, r, 1, 0, workspace, 1, 0 );
+* // r => [ 10.0, -2.0, 2.0, -2.0 ]
+*/
+function rfftf( N, r, strideR, offsetR, workspace, strideW, offsetW ) {
+ var offsetT;
+ var offsetF;
+
+ // When a sub-sequence is a single data point, the FFT is the identity, so no transformation necessary...
+ if ( N === 1 ) {
+ return r;
+ }
+
+ // Resolve the starting indices for storing twiddle factors and factorization results:
+ offsetT = offsetW + ( N * strideW ); // index offset for twiddle factors
+ offsetF = offsetT + ( N * strideW ); // index offset for factors describing the sub-transforms
+
+ rfftf1( N, r, strideR, offsetR, workspace, strideW, offsetW, workspace, strideW, offsetT, workspace, strideW, offsetF ); // eslint-disable-line max-len
+
+ return r;
+}
+
+
+// EXPORTS //
+
+module.exports = rfftf;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf2.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf2.js
new file mode 100644
index 000000000000..7e7871ca0ebf
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf2.js
@@ -0,0 +1,366 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+/* eslint-disable max-len */
+
+'use strict';
+
+// FUNCTIONS //
+
+/**
+* Resolves an index into the input array.
+*
+* ## Notes
+*
+* In a forward real FFT, the previous stage writes its results as two "rows" per sub-sequence.
+*
+* Thus, when reading from an input array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having two "rows" (`even + odd` and `even - odd`, respectively) and where each "row" is arranged as `M*L` contiguous elements corresponding to interleaved real and imaginary components.
+*
+* Accordingly, the following is a logical view of an input array (zero-based indexing) which contains `L = 3` transforms and in which each sub-sequence has length `M = 4`:
+*
+* ```text
+* │ k = 0 k = 1 k = 2
+* │ ──────────────────────────────────────────────────────────────────────────→ k
+* j = 0 (even+odd) │ cc(0,0,0) ... cc(3,0,0) cc(0,1,0) ... cc(3,1,0) cc(0,2,0) ... cc(3,2,0)
+* │
+* j = 1 (even-odd) │ cc(0,0,1) ... cc(3,0,1) cc(0,1,1) ... cc(3,1,1) cc(0,2,1) ... cc(3,2,1)
+* └───────────────────────────────────────────────────────────────────────────→ i
+* ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 M-1 0 M-1 0 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short sub-sequence corresponding to either the `even + odd` or `even - odd` row.
+* - `j` selects between the `even + odd` and `even - odd` row.
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | cc(0,0,0)...cc(3,0,0) ... cc(0,2,0)...cc(3,2,0) | cc(0,0,1)...cc(3,0,1) ... cc(0,2,1)...cc(3,2,1) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 LM LM-1 (L+1)M (L+1)M-1 (2L-1)M 2LM-1
+* ```
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} j - input row
+* @param {NonNegativeInteger} L - number of sub-sequences
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the input array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the input array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = iptr( 0, 0, 0, L, M, stride, offset );
+* // returns 0
+*
+* idx = iptr( 1, 0, 0, L, M, stride, offset );
+* // returns 1
+*
+* idx = iptr( M-1, 0, 0, L, M, stride, offset );
+* // returns 3
+*
+* idx = iptr( 0, 1, 0, L, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = iptr( M-1, L-1, 1, L, M, stride, offset );
+* // returns 23
+*/
+function iptr( i, k, j, L, M, stride, offset ) {
+ var n = i + ( ( k+(j*L) ) * M );
+ return ( n*stride ) + offset;
+}
+
+/**
+* Resolves an index into the output array.
+*
+* ## Notes
+*
+* When writing to an output array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having two "columns" corresponding to the even and odd half-spectrum of a folded complex vector (with real and imaginary parts of each half-spectrum interleaved along each sub-sequence) and where each "column" has `M` elements.
+*
+* Accordingly, the following is a logical view of an output array (zero-based indexing) which contains `L = 3` transforms and in which each "column" sub-sequence has length `M = 4`:
+*
+* ```text
+* j = 0 ("even" column) j = 1 ("odd" column)
+* k = 0 ─┬───────────────────────────────────────┬───────────────────────────────────────┐
+* │ out(0,0,0) out(1,0,0) ... out(3,0,0) │ out(0,1,0) out(1,1,0) ... out(3,1,0) │
+* └───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 1 ─┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,1) out(1,0,1) ... out(3,0,1) │ out(0,1,1) out(1,1,1) ... out(3,1,1) │
+* └───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 2 ─┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,2) out(1,0,2) ... out(3,0,2) │ out(0,1,2) out(1,1,2) ... out(3,1,2) │
+* └───────────────────────────────────────┴───────────────────────────────────────┘
+* ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 1 M-1 0 1 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short "column" sub-sequence.
+* - `j` selects between the even (0) and odd (1) column.
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | out(0,0,0)...out(3,0,0) | out(0,1,0)...out(3,1,0) | out(0,0,1)...out(3,0,1) | ... | out(0,1,2)...out(3,1,2) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 M 2M-1 2M 3M-1 (2L-1)M 2LM-1
+* ```
+*
+* As may be observed, when resolving an index in the output array, the `j` and `k` dimensions are swapped relative index resolution in the input array. This stems from `radf2` being only one stage in a multi-stage driver which alternates between using `cc` and `out` as workspace buffers. After each stage, the next stage reads what the previous stage wrote.
+*
+* Each stage expects a transpose, and, in order to avoid explicit transposition between the stages, we swap the last two logical dimensions while still maintaining cache locality within the inner loop logical dimension, as indexed by `i`.
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} j - index specifying which of the two complex halves we are in (either `0` or `1`)
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the output array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the output array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = optr( 0, 0, 0, M, stride, offset );
+* // returns 0
+*
+* idx = optr( 1, 0, 0, M, stride, offset );
+* // returns 1
+*
+* idx = optr( M-1, 0, 0, M, stride, offset );
+* // returns 3
+*
+* idx = optr( 0, 1, 0, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = optr( M-1, 1, L-1, M, stride, offset );
+* // returns 23
+*/
+function optr( i, j, k, M, stride, offset ) {
+ var n = i + ( ( j+(k*2) ) * M );
+ return ( n*stride ) + offset;
+}
+
+
+// MAIN //
+
+/**
+* Performs one radix-2 stage within a forward Fourier transform for a real-valued sequence.
+*
+* @private
+* @param {NonNegativeInteger} M - number of elements in each sub-sequence to be transformed
+* @param {NonNegativeInteger} L - number of sub-sequences to be transformed
+* @param {Collection} cc - input array containing the sub-sequences to be transformed
+* @param {integer} sc - stride length for `cc`
+* @param {NonNegativeInteger} oc - index offset for `cc`
+* @param {Collection} out - output array containing transformed sub-sequences
+* @param {integer} so - stride length for `out`
+* @param {NonNegativeInteger} oo - index offset for `out`
+* @param {Collection} twiddles - array containing twiddle factors
+* @param {integer} st - stride length for `twiddles`
+* @param {NonNegativeInteger} ot - index offset for `twiddles`
+* @returns {void}
+*/
+function radf2( M, L, cc, sc, oc, out, so, oo, twiddles, st, ot ) { // eslint-disable-line max-params
+ var tr2;
+ var ti2;
+ var ip1;
+ var ip2;
+ var it1;
+ var it2;
+ var io;
+ var im;
+ var i;
+ var k;
+
+ /*
+ * First, perform the core butterfly for each sub-sequence being transformed.
+ *
+ * In the following loop, we handle harmonic `n = 0` for every transform. As described for `iptr`, the input array is interpreted as a three-dimensional array, containing two "rows" per sub-sequence.
+ *
+ * row0 = even + odd (j = 0)
+ * row1 = even - odd (j = 1)
+ *
+ * For a radix-2 forward FFT of a real-valued signal `x` and `n = 0`,
+ *
+ * x[0] = row0 + row1 = 2⋅even
+ * x[M] = row0 - row1 = 2⋅odd
+ *
+ * Because `W_2^0 = 1`, no twiddle multiplication is necessary.
+ */
+ for ( k = 0; k < L; k++ ) {
+ ip1 = iptr( 0, k, 0, L, M, sc, oc ); // real part row 0
+ ip2 = iptr( 0, k, 1, L, M, sc, oc ); // real part row 1
+
+ io = optr( 0, 0, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + cc[ ip2 ]; // even + odd
+
+ io = optr( M-1, 1, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] - cc[ ip2 ]; // even - odd
+ }
+ // When the number of elements in a sub-sequence is less than `2`, there is nothing more to do, as the above butterfly produced the full result...
+ if ( M < 2 ) {
+ return;
+ }
+ /*
+ * Next, apply the general case where we need to loop through the non-trivial harmonics.
+ *
+ * For each harmonic `n = 1, ..., M/2-1`, we need to
+ *
+ * - recover even/odd spectra from the two input rows.
+ * - multiply the odd part by the twiddle factor `W_n = cos(θ) - j sin(θ)`.
+ * - form the following
+ *
+ * x[2n] = E_n + W_n⋅O_n => column 0
+ * x[2n+1] = E_n - W_n⋅O_n => column 1
+ *
+ * The mirror index `im = M - i` selects the conjugate-symmetric partner, thus allowing the routine to read each symmetry pair only once.
+ */
+ if ( M >= 3 ) {
+ // Loop over each sub-sequence to be transformed...
+ for ( k = 0; k < L; k++ ) {
+ // Loop over the elements in each sub-sequence...
+ for ( i = 2; i < M; i += 2 ) {
+ im = M - i; // "mirror" index
+
+ // Resolve twiddle factor indices:
+ it1 = ( (i-2)*st ) + ot; // cos(θ)
+ it2 = ( (i-1)*st ) + ot; // sin(θ)
+
+ // Load the `even-odd` row...
+ ip1 = iptr( i-1, k, 1, L, M, sc, oc ); // c = Re(row1)
+ ip2 = iptr( i, k, 1, L, M, sc, oc ); // d = Im(row1)
+
+ // eslint-disable-next-line stdlib/capitalized-comments, stdlib/empty-line-before-comment
+ // tmp = W_n ⋅ (c + j⋅d)
+ tr2 = ( twiddles[ it1 ] * cc[ ip1 ] ) + ( twiddles[ it2 ] * cc[ ip2 ] ); // Re(tmp)
+ ti2 = ( twiddles[ it1 ] * cc[ ip2 ] ) - ( twiddles[ it2 ] * cc[ ip1 ] ); // Im(tmp)
+
+ // Load the `even+odd` row...
+ ip1 = iptr( i, k, 0, L, M, sc, oc ); // b = Im(row0)
+ io = optr( i, 0, k, M, so, oo ); // Im(x[2n])
+ out[ io ] = cc[ ip1 ] + ti2;
+
+ io = optr( im, 1, k, M, so, oo ); // Im(x[2n+1])
+ out[ io ] = ti2 - cc[ ip1 ];
+
+ ip1 = iptr( i-1, k, 0, L, M, sc, oc ); // a = Re(row0)
+ io = optr( i-1, 0, k, M, so, oo ); // Re(x[2n])
+ out[ io ] = cc[ ip1 ] + tr2;
+
+ io = optr( im-1, 1, k, M, so, oo ); // Re(x[2n+1])
+ out[ io ] = cc[ ip1 ] - tr2;
+ }
+ }
+ // When `M` is odd, there is no Nyquist pair to process, so we do not need to perform any further transformations...
+ if ( M%2 === 1 ) {
+ return;
+ }
+ }
+ /*
+ * Lastly, handle the Nyquist frequency where `i = M-1` (i.e., the last element of each sub-sequence).
+ *
+ * When `M` is even, the Nyquist index is `i = M/2`. In this stage, we've stored that element at the end of each sub-sequence (i.e., `i = M-1` in the packed layout).
+ *
+ * At this point, the cosine term is -1 and the sine term is 0, so the twiddle multiplication collapses to simple addition/subtraction.
+ *
+ * In this case,
+ *
+ * W_n⋅O_n = ±Re(O_n)
+ *
+ * where
+ *
+ * row0 = Re(E_n)
+ * row1 = -Re(O_n)
+ */
+ for ( k = 0; k < L; k++ ) {
+ ip1 = iptr( M-1, k, 1, L, M, sc, oc ); // -Re(O_n)
+ io = optr( 0, 1, k, M, so, oo );
+ out[ io ] = -cc[ ip1 ]; // -(-Re(O_n)) = Re(O_n)
+
+ ip1 = iptr( M-1, k, 0, L, M, sc, oc ); // Re(E_n)
+ io = optr( M-1, 0, k, M, so, oo );
+ out[ io ] = cc[ ip1 ];
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = radf2;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf3.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf3.js
new file mode 100644
index 000000000000..e0361c31120c
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf3.js
@@ -0,0 +1,401 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+/* eslint-disable max-len */
+
+'use strict';
+
+// VARIABLES //
+
+// cos( 2π/3 ):
+var TAUR = -0.5;
+
+// sin( 2π/3 ):
+var TAUI = 0.866025403784439;
+
+
+// FUNCTIONS //
+
+/**
+* Resolves an index into the input array.
+*
+* ## Notes
+*
+* In a forward real FFT, the previous stage writes its results as three "rows" per sub-sequence.
+*
+* Thus, when reading from an input array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having three "rows" (components 0, 1, and 2) and where each "row" is arranged as `M*L` contiguous elements corresponding to interleaved real and imaginary components.
+*
+* Accordingly, the following is a logical view of an input array (zero-based indexing) which contains `L = 3` transforms and in which each sub-sequence has length `M = 4`:
+*
+* ```text
+* │ k = 0 k = 1 k = 2
+* │ ──────────────────────────────────────────────────────────────────────────→ k
+* j = 0 │ cc(0,0,0) ... cc(3,0,0) cc(0,1,0) ... cc(3,1,0) cc(0,2,0) ... cc(3,2,0)
+* │
+* j = 1 │ cc(0,0,1) ... cc(3,0,1) cc(0,1,1) ... cc(3,1,1) cc(0,2,1) ... cc(3,2,1)
+* │
+* j = 2 │ cc(0,0,2) ... cc(3,0,2) cc(0,1,2) ... cc(3,1,2) cc(0,2,2) ... cc(3,2,2)
+* └───────────────────────────────────────────────────────────────────────────→ i
+* ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 M-1 0 M-1 0 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short sub-sequence corresponding to one of the three component rows.
+* - `j` selects between the three component rows (0, 1, or 2).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | cc(0,0,0)...cc(3,0,0) ... cc(0,2,0)...cc(3,2,0) | cc(0,0,1)...cc(3,0,1) ... cc(0,2,1)...cc(3,2,1) | cc(0,0,2)...cc(3,0,2) ... cc(0,2,2)...cc(3,2,2) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 LM LM-1 (L+1)M (L+1)M-1 (2L-1)M 2LM-1 2LM (2L+1)M-1 (3L-1)M 3LM-1
+* ```
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} j - input row
+* @param {NonNegativeInteger} L - number of sub-sequences
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the input array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the input array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = iptr( 0, 0, 0, L, M, stride, offset );
+* // returns 0
+*
+* idx = iptr( 1, 0, 0, L, M, stride, offset );
+* // returns 1
+*
+* idx = iptr( M-1, 0, 0, L, M, stride, offset );
+* // returns 3
+*
+* idx = iptr( 0, 1, 0, L, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = iptr( M-1, L-1, 1, L, M, stride, offset );
+* // returns 23
+*/
+function iptr( i, k, j, L, M, stride, offset ) {
+ var n = i + ( ( k+(j*L) ) * M );
+ return ( n*stride ) + offset;
+}
+
+/**
+* Resolves an index into the output array.
+*
+* ## Notes
+*
+* When writing to an output array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having three "columns" corresponding to the three components of a radix-3 stage (with real and imaginary parts of each component interleaved along each sub-sequence) and where each "column" has `M` elements.
+*
+* Accordingly, the following is a logical view of an output array (zero-based indexing) which contains `L = 3` transforms and in which each "column" sub-sequence has length `M = 4`:
+*
+* ```text
+* j = 0 (component 0) j = 1 (component 1) j = 2 (component 2)
+* k = 0 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┐
+* │ out(0,0,0) out(1,0,0) ... out(3,0,0) │ out(0,1,0) out(1,1,0) ... out(3,1,0) │ out(0,2,0) out(1,2,0) ... out(3,2,0) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 1 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,1) out(1,0,1) ... out(3,0,1) │ out(0,1,1) out(1,1,1) ... out(3,1,1) │ out(0,2,1) out(1,2,1) ... out(3,2,1) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 2 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,2) out(1,0,2) ... out(3,0,2) │ out(0,1,2) out(1,1,2) ... out(3,1,2) │ out(0,2,2) out(1,2,2) ... out(3,2,2) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┘
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 1 M-1 0 1 M-1 0 1 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short "column" sub-sequence.
+* - `j` selects between one of the three components (0, 1, or 2).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | out(0,0,0)...out(3,0,0) | out(0,1,0)...out(3,1,0) | out(0,2,0)...out(3,2,0) | out(0,0,1)...out(3,0,1) | ... | out(0,2,2)...out(3,2,2) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 M 2M-1 2M 3M-1 3M 4M-1 (3L-1)M 3LM-1
+* ```
+*
+* As may be observed, when resolving an index in the output array, the `j` and `k` dimensions are swapped relative index resolution in the input array. This stems from `radf3` being only one stage in a multi-stage driver which alternates between using `cc` and `out` as workspace buffers. After each stage, the next stage reads what the previous stage wrote.
+*
+* Each stage expects a transpose, and, in order to avoid explicit transposition between the stages, we swap the last two logical dimensions while still maintaining cache locality within the inner loop logical dimension, as indexed by `i`.
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} j - index specifying which of the three complex components we are in (0, 1, or 2)
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the output array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the output array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = optr( 0, 0, 0, M, stride, offset );
+* // returns 0
+*
+* idx = optr( 1, 0, 0, M, stride, offset );
+* // returns 1
+*
+* idx = optr( M-1, 0, 0, M, stride, offset );
+* // returns 3
+*
+* idx = optr( 0, 1, 0, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = optr( M-1, 2, L-1, M, stride, offset );
+* // returns 35
+*/
+function optr( i, j, k, M, stride, offset ) {
+ var n = i + ( ( j+(k*3) ) * M );
+ return ( n*stride ) + offset;
+}
+
+
+// MAIN //
+
+/**
+* Performs one radix-3 stage within a forward Fourier transform for a real-valued sequence.
+*
+* @private
+* @param {NonNegativeInteger} M - number of elements in each sub-sequence to be transformed
+* @param {NonNegativeInteger} L - number of sub-sequences to be transformed
+* @param {Collection} cc - input array containing the sub-sequences to be transformed
+* @param {integer} sc - stride length for `cc`
+* @param {NonNegativeInteger} oc - index offset for `cc`
+* @param {Collection} out - output array containing transformed sub-sequences
+* @param {integer} so - stride length for `out`
+* @param {NonNegativeInteger} oo - index offset for `out`
+* @param {Collection} twiddles1 - first array containing twiddle factors
+* @param {integer} st1 - stride length for `twiddles1`
+* @param {NonNegativeInteger} ot1 - index offset for `twiddles1`
+* @param {Collection} twiddles2 - second array containing twiddle factors
+* @param {integer} st2 - stride length for `twiddles2`
+* @param {NonNegativeInteger} ot2 - index offset for `twiddles2`
+* @returns {void}
+*/
+function radf3( M, L, cc, sc, oc, out, so, oo, twiddles1, st1, ot1, twiddles2, st2, ot2 ) { // eslint-disable-line max-params
+ var it11;
+ var it12;
+ var it21;
+ var it22;
+ var tr2;
+ var ti2;
+ var tr3;
+ var ti3;
+ var cr2;
+ var ci2;
+ var dr2;
+ var di2;
+ var dr3;
+ var di3;
+ var ip1;
+ var ip2;
+ var ip3;
+ var io;
+ var im;
+ var i;
+ var k;
+
+ /*
+ * First, perform the core butterfly for each sub-sequence being transformed.
+ *
+ * In the following loop, we handle harmonic `n = 0` for every transform. As described for `iptr`, the input array is interpreted as a three-dimensional array, containing three "rows" per sub-sequence.
+ *
+ * row0 = input component 0 (j = 0)
+ * row1 = input component 1 (j = 1)
+ * row2 = input component 2 (j = 2)
+ *
+ * For a radix-3 forward FFT of a real-valued signal `x` and `n = 0`,
+ *
+ * taur = cos(2π/3)
+ * taui = sin(2π/3)
+ *
+ * x[0] = row0 + row1 + row2
+ * x[2M] = taui * ( row2-row1 )
+ * x[M] = row0 + ( taur * ( row1+row2 ) )
+ *
+ * Because `W_3^0 = 1`, no twiddle multiplication is necessary beyond these constant factors (`taur` and `taui`).
+ */
+ for ( k = 0; k < L; k++ ) {
+ ip1 = iptr( 0, k, 0, L, M, sc, oc ); // real part row 0
+ ip2 = iptr( 0, k, 1, L, M, sc, oc ); // real part row 1
+ ip3 = iptr( 0, k, 2, L, M, sc, oc ); // real part row 2
+
+ cr2 = cc[ ip2 ] + cc[ ip3 ];
+
+ io = optr( 0, 0, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + cr2; // row0 + row1 + row2
+
+ io = optr( 0, 2, k, M, so, oo );
+ out[ io ] = TAUI * ( cc[ ip3 ] - cc[ ip2 ] ); // taui * ( row2-row1 )
+
+ io = optr( M-1, 1, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + ( TAUR * cr2 ); // row0 + ( taur * ( row1+row2 ) )
+ }
+ // When the number of elements in a sub-sequence is less than `2`, there is nothing more to do, as the above butterfly produced the full result...
+ if ( M < 2 ) {
+ return;
+ }
+ /*
+ * Next, apply the general case where we need to loop through the non-trivial harmonics.
+ *
+ * For each harmonic `n = 1, ..., M/2-1`, we need to
+ *
+ * - recover three spectra from the three input rows.
+ * - apply radix-3 twiddle factors to rows 1 and 2, then combine with row 0 to form three output columns.
+ * - form the following
+ *
+ * x[3n] = X₀ + (W₁⋅X₁ + W₂⋅X₂) => column 0
+ * x[3n+1] = X₀ + taur⋅(W₁⋅X₁ + W₂⋅X₂) + taui⋅i⋅(W₁⋅X₁ - W₂⋅X₂) => column 2
+ * x[3n+2] = X₀ + taur⋅(W₁⋅X₁ + W₂⋅X₂) - taui⋅i⋅(W₁⋅X₁ - W₂⋅X₂) => column 1
+ *
+ * The mirror index `im = M - i` selects the conjugate-symmetric partner, thus allowing the routine to read each symmetry pair only once.
+ */
+ // Loop over each sub-sequence to be transformed...
+ for ( k = 0; k < L; k++ ) {
+ // Loop over the elements in each sub-sequence...
+ for ( i = 2; i < M; i += 2 ) {
+ im = M - i; // "mirror" index
+
+ // Resolve twiddle factor indices for component 1:
+ it11 = ( (i-2)*st1 ) + ot1; // cos(θ) for component 1
+ it12 = ( (i-1)*st1 ) + ot1; // sin(θ) for component 1
+
+ // Resolve twiddle factor indices for component 2:
+ it21 = ( (i-2)*st2 ) + ot2; // cos(θ) for component 2
+ it22 = ( (i-1)*st2 ) + ot2; // sin(θ) for component 2
+
+ // Load component 1 data...
+ ip1 = iptr( i-1, k, 1, L, M, sc, oc ); // real part component 1
+ ip2 = iptr( i, k, 1, L, M, sc, oc ); // imag part component 1
+
+ // Apply twiddle factor for component 1
+ dr2 = ( twiddles1[ it11 ] * cc[ ip1 ] ) + ( twiddles1[ it12 ] * cc[ ip2 ] );
+ di2 = ( twiddles1[ it11 ] * cc[ ip2 ] ) - ( twiddles1[ it12 ] * cc[ ip1 ] );
+
+ // Load component 2 data...
+ ip1 = iptr( i-1, k, 2, L, M, sc, oc ); // real part component 2
+ ip2 = iptr( i, k, 2, L, M, sc, oc ); // imag part component 2
+
+ // Apply twiddle factor for component 2
+ dr3 = ( twiddles2[ it21 ] * cc[ ip1 ] ) + ( twiddles2[ it22 ] * cc[ ip2 ] );
+ di3 = ( twiddles2[ it21 ] * cc[ ip2 ] ) - ( twiddles2[ it22 ] * cc[ ip1 ] );
+
+ // Combine the twiddled components
+ cr2 = dr2 + dr3; // sum of real parts
+ ci2 = di2 + di3; // sum of imag parts
+
+ // Load component 0 data...
+ ip1 = iptr( i-1, k, 0, L, M, sc, oc ); // real part component 0
+ io = optr( i-1, 0, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + cr2;
+
+ ip2 = iptr( i, k, 0, L, M, sc, oc ); // imag part component 0
+ io = optr( i, 0, k, M, so, oo );
+ out[ io ] = cc[ ip2 ] + ci2;
+
+ // Intermediate results for components 1 and 2
+ tr2 = cc[ ip1 ] + ( TAUR * cr2 );
+ ti2 = cc[ ip2 ] + ( TAUR * ci2 );
+ tr3 = TAUI * ( di2 - di3 );
+ ti3 = TAUI * ( dr3 - dr2 );
+
+ // Output component 1
+ io = optr( i-1, 2, k, M, so, oo );
+ out[ io ] = tr2 + tr3;
+
+ io = optr( im-1, 1, k, M, so, oo );
+ out[ io ] = tr2 - tr3;
+
+ // Output component 2
+ io = optr( i, 2, k, M, so, oo );
+ out[ io ] = ti2 + ti3;
+
+ io = optr( im, 1, k, M, so, oo );
+ out[ io ] = ti3 - ti2;
+ }
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = radf3;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf4.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf4.js
new file mode 100644
index 000000000000..65fa22ef79c3
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf4.js
@@ -0,0 +1,492 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+/* eslint-disable max-len, max-statements */
+
+'use strict';
+
+// MODULES //
+
+var SQRT_HALF = require( '@stdlib/constants/float64/sqrt-half' );
+
+
+// FUNCTIONS //
+
+/**
+* Resolves an index into the input array.
+*
+* ## Notes
+*
+* In a forward real FFT, the previous stage writes its results as four "rows" per sub-sequence.
+*
+* Thus, when reading from an input array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having four "rows" (components 0, 1, 2, and 3) and where each "row" is arranged as `M*L` contiguous elements corresponding to interleaved real and imaginary components.
+*
+* Accordingly, the following is a logical view of an input array (zero-based indexing) which contains `L = 3` transforms and in which each sub-sequence has length `M = 4`:
+*
+* ```text
+* │ k = 0 k = 1 k = 2
+* │ ──────────────────────────────────────────────────────────────────────────→ k
+* j = 0 │ cc(0,0,0) ... cc(3,0,0) cc(0,1,0) ... cc(3,1,0) cc(0,2,0) ... cc(3,2,0)
+* │
+* j = 1 │ cc(0,0,1) ... cc(3,0,1) cc(0,1,1) ... cc(3,1,1) cc(0,2,1) ... cc(3,2,1)
+* │
+* j = 2 │ cc(0,0,2) ... cc(3,0,2) cc(0,1,2) ... cc(3,1,2) cc(0,2,2) ... cc(3,2,2)
+* │
+* j = 3 │ cc(0,0,3) ... cc(3,0,3) cc(0,1,3) ... cc(3,1,3) cc(0,2,3) ... cc(3,2,3)
+* └───────────────────────────────────────────────────────────────────────────→ i
+* ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 M-1 0 M-1 0 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short sub-sequence corresponding to one of the four component rows.
+* - `j` selects between the four component rows (0, 1, 2, or 3).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | cc(0,0,0)...cc(3,0,0) ... cc(0,2,0)...cc(3,2,0) | cc(0,0,1)...cc(3,0,1) ... cc(0,2,1)...cc(3,2,1) | ... | cc(0,0,3)...cc(3,0,3) ... cc(0,2,3)...cc(3,2,3) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 LM LM-1 (L+1)M (L+1)M-1 (2L-1)M 2LM-1 3LM (3L+1)M-1 (4L-1)M 4LM-1
+* ```
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} j - input row
+* @param {NonNegativeInteger} L - number of sub-sequences
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the input array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the input array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = iptr( 0, 0, 0, L, M, stride, offset );
+* // returns 0
+*
+* idx = iptr( 1, 0, 0, L, M, stride, offset );
+* // returns 1
+*
+* idx = iptr( M-1, 0, 0, L, M, stride, offset );
+* // returns 3
+*
+* idx = iptr( 0, 1, 0, L, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = iptr( M-1, L-1, 1, L, M, stride, offset );
+* // returns 23
+*/
+function iptr( i, k, j, L, M, stride, offset ) {
+ var n = i + ( ( k+(j*L) ) * M );
+ return ( n*stride ) + offset;
+}
+
+/**
+* Resolves an index into the output array.
+*
+* ## Notes
+*
+* When writing to an output array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having four "columns" corresponding to the four components of a radix-4 stage (with real and imaginary parts of each component interleaved along each sub-sequence) and where each "column" has `M` elements.
+*
+* Accordingly, the following is a logical view of an output array (zero-based indexing) which contains `L = 3` transforms and in which each "column" sub-sequence has length `M = 4`:
+*
+* ```text
+* j = 0 (component 0) j = 1 (component 1) j = 2 (component 2) j = 3 (component 3)
+* k = 0 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┐
+* │ out(0,0,0) out(1,0,0) ... out(3,0,0) │ out(0,1,0) out(1,1,0) ... out(3,1,0) │ out(0,2,0) out(1,2,0) ... out(3,2,0) │ out(0,3,0) out(1,3,0) ... out(3,3,0) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 1 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,1) out(1,0,1) ... out(3,0,1) │ out(0,1,1) out(1,1,1) ... out(3,1,1) │ out(0,2,1) out(1,2,1) ... out(3,2,1) │ out(0,3,1) out(1,3,1) ... out(3,3,1) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 2 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,2) out(1,0,2) ... out(3,0,2) │ out(0,1,2) out(1,1,2) ... out(3,1,2) │ out(0,2,2) out(1,2,2) ... out(3,2,2) │ out(0,3,2) out(1,3,2) ... out(3,3,2) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┘
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 1 M-1 0 1 M-1 0 1 M-1 0 1 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short "column" sub-sequence.
+* - `j` selects which of the four components (0, 1, 2, or 3).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | out(0,0,0)...out(3,0,0) | out(0,1,0)...out(3,1,0) | out(0,2,0)...out(3,2,0) | out(0,3,0)...out(3,3,0) | out(0,0,1)...out(3,0,1) | ... | out(0,3,2)...out(3,3,2) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 M 2M-1 2M 3M-1 3M 4M-1 4M 5M-1 (4L-1)M 4LM-1
+* ```
+*
+* As may be observed, when resolving an index in the output array, the `j` and `k` dimensions are swapped relative index resolution in the input array. This stems from `radf4` being only one stage in a multi-stage driver which alternates between using `cc` and `out` as workspace buffers. After each stage, the next stage reads what the previous stage wrote.
+*
+* Each stage expects a transpose, and, in order to avoid explicit transposition between the stages, we swap the last two logical dimensions while still maintaining cache locality within the inner loop logical dimension, as indexed by `i`.
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} j - index specifying which of the four complex components we are in (0, 1, 2, or 3)
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the output array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the output array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = optr( 0, 0, 0, M, stride, offset );
+* // returns 0
+*
+* idx = optr( 1, 0, 0, M, stride, offset );
+* // returns 1
+*
+* idx = optr( M-1, 0, 0, M, stride, offset );
+* // returns 3
+*
+* idx = optr( 0, 1, 0, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = optr( M-1, 3, L-1, M, stride, offset );
+* // returns 47
+*/
+function optr( i, j, k, M, stride, offset ) {
+ var n = i + ( ( j+(k*4) ) * M );
+ return ( n*stride ) + offset;
+}
+
+
+// MAIN //
+
+/**
+* Performs one radix-4 stage within a forward Fourier transform for a real-valued sequence.
+*
+* @private
+* @param {NonNegativeInteger} M - number of elements in each sub-sequence to be transformed
+* @param {NonNegativeInteger} L - number of sub-sequences to be transformed
+* @param {Collection} cc - input array containing the sub-sequences to be transformed
+* @param {integer} sc - stride length for `cc`
+* @param {NonNegativeInteger} oc - index offset for `cc`
+* @param {Collection} out - output array containing transformed sub-sequences
+* @param {integer} so - stride length for `out`
+* @param {NonNegativeInteger} oo - index offset for `out`
+* @param {Collection} twiddles1 - first array containing twiddle factors
+* @param {integer} st1 - stride length for `twiddles1`
+* @param {NonNegativeInteger} ot1 - index offset for `twiddles1`
+* @param {Collection} twiddles2 - second array containing twiddle factors
+* @param {integer} st2 - stride length for `twiddles2`
+* @param {NonNegativeInteger} ot2 - index offset for `twiddles2`
+* @param {Collection} twiddles3 - third array containing twiddle factors
+* @param {integer} st3 - stride length for `twiddles3`
+* @param {NonNegativeInteger} ot3 - index offset for `twiddles3`
+* @returns {void}
+*/
+function radf4( M, L, cc, sc, oc, out, so, oo, twiddles1, st1, ot1, twiddles2, st2, ot2, twiddles3, st3, ot3 ) { // eslint-disable-line max-params
+ var it11;
+ var it12;
+ var it21;
+ var it22;
+ var it31;
+ var it32;
+ var tr1;
+ var tr2;
+ var tr3;
+ var tr4;
+ var ti1;
+ var ti2;
+ var ti3;
+ var ti4;
+ var cr2;
+ var cr3;
+ var cr4;
+ var ci2;
+ var ci3;
+ var ci4;
+ var ip1;
+ var ip2;
+ var ip3;
+ var ip4;
+ var io;
+ var im;
+ var i;
+ var k;
+
+ /*
+ * First, perform the core butterfly for each sub-sequence being transformed.
+ *
+ * In the following loop, we handle harmonic `n = 0` for every transform. As described for `iptr`, the input array is interpreted as a three-dimensional array, containing four "rows" per sub-sequence.
+ *
+ * row0 = input component 0 (j = 0)
+ * row1 = input component 1 (j = 1)
+ * row2 = input component 2 (j = 2)
+ * row3 = input component 3 (j = 3)
+ *
+ * For a radix-4 forward FFT of a real-valued signal `x` and `n = 0`,
+ *
+ * tr1 = row1 + row3
+ * tr2 = row0 + row2
+ *
+ * x[0] = tr1 + tr2
+ * x[2M-1] = row0 - row2
+ * x[2M] = row3 - row1
+ * x[4M-1] = tr2 - tr1
+ *
+ * Because `W_4^0 = 1`, no twiddle multiplication is necessary.
+ */
+ for ( k = 0; k < L; k++ ) {
+ ip1 = iptr( 0, k, 0, L, M, sc, oc ); // real part row 0
+ ip2 = iptr( 0, k, 1, L, M, sc, oc ); // real part row 1
+ ip3 = iptr( 0, k, 2, L, M, sc, oc ); // real part row 2
+ ip4 = iptr( 0, k, 3, L, M, sc, oc ); // real part row 3
+
+ tr1 = cc[ ip2 ] + cc[ ip4 ]; // row1 + row3
+ tr2 = cc[ ip1 ] + cc[ ip3 ]; // row0 + row2
+
+ io = optr( 0, 0, k, M, so, oo );
+ out[ io ] = tr1 + tr2;
+
+ io = optr( M-1, 3, k, M, so, oo );
+ out[ io ] = tr2 - tr1;
+
+ io = optr( M-1, 1, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] - cc[ ip3 ];
+
+ io = optr( 0, 2, k, M, so, oo );
+ out[ io ] = cc[ ip4 ] - cc[ ip2 ];
+ }
+ // When the number of elements in a sub-sequence is equal to `1`, there is nothing more to do, as the above butterfly produced the full result...
+ if ( M < 2 ) {
+ return;
+ }
+ /*
+ * Next, apply the general case where we need to loop through the non-trivial harmonics.
+ *
+ * For each harmonic `n = 1, ..., M/2-1`, we need to
+ *
+ * - recover four spectra from the four input rows.
+ * - apply radix-4 twiddle factors to rows 1, 2, and 3, then combine with row 0 to form four output columns.
+ * - form the following
+ *
+ * x[4n] = X₀ + W₂·X₂ + (W₁·X₁ + W₃·X₃) => column 0
+ * x[4n+1] = X₀ + W₂·X₂ - (W₁·X₁ + W₃·X₃) => column 3
+ * x[4n+2] = X₀ - W₂·X₂ + i·(W₁·X₁ - W₃·X₃) => column 2
+ * x[4n+3] = X₀ - W₂·X₂ - i·(W₁·X₁ - W₃·X₃) => column 1
+ *
+ * The mirror index `im = M - i` selects the conjugate-symmetric partner, thus allowing the routine to read each symmetry pair only once.
+ */
+ if ( M >= 3 ) {
+ // Loop over each sub-sequence to be transformed...
+ for ( k = 0; k < L; k++ ) {
+ // Loop over the elements in each sub-sequence...
+ for ( i = 2; i < M; i += 2 ) {
+ im = M - i; // "mirror" index
+
+ // Resolve twiddle factor indices for component 1:
+ it11 = ( (i-2)*st1 ) + ot1; // cos(θ) for component 1
+ it12 = ( (i-1)*st1 ) + ot1; // sin(θ) for component 1
+
+ // Resolve twiddle factor indices for component 2:
+ it21 = ( (i-2)*st2 ) + ot2; // cos(θ) for component 2
+ it22 = ( (i-1)*st2 ) + ot2; // sin(θ) for component 2
+
+ // Resolve twiddle factor indices for component 3:
+ it31 = ( (i-2)*st3 ) + ot3; // cos(θ) for component 3
+ it32 = ( (i-1)*st3 ) + ot3; // sin(θ) for component 3
+
+ // Load component 1 data and apply twiddle...
+ ip1 = iptr( i-1, k, 1, L, M, sc, oc ); // real part component 1
+ ip2 = iptr( i, k, 1, L, M, sc, oc ); // imag part component 1
+
+ // Apply the twiddle factors for component 1:
+ cr2 = ( twiddles1[ it11 ] * cc[ ip1 ] ) + ( twiddles1[ it12 ] * cc[ ip2 ] );
+ ci2 = ( twiddles1[ it11 ] * cc[ ip2 ] ) - ( twiddles1[ it12 ] * cc[ ip1 ] );
+
+ // Load component 2 data and apply twiddle...
+ ip1 = iptr( i-1, k, 2, L, M, sc, oc ); // real part component 2
+ ip2 = iptr( i, k, 2, L, M, sc, oc ); // imag part component 2
+
+ // Apply the twiddle factors for component 2:
+ cr3 = ( twiddles2[ it21 ] * cc[ ip1 ] ) + ( twiddles2[ it22 ] * cc[ ip2 ] );
+ ci3 = ( twiddles2[ it21 ] * cc[ ip2 ] ) - ( twiddles2[ it22 ] * cc[ ip1 ] );
+
+ // Load component 3 data and apply twiddle...
+ ip1 = iptr( i-1, k, 3, L, M, sc, oc ); // real part component 3
+ ip2 = iptr( i, k, 3, L, M, sc, oc ); // imag part component 3
+
+ // Apply the twiddle factors for component 3:
+ cr4 = ( twiddles3[ it31 ] * cc[ ip1 ] ) + ( twiddles3[ it32 ] * cc[ ip2 ] );
+ ci4 = ( twiddles3[ it31 ] * cc[ ip2 ] ) - ( twiddles3[ it32 ] * cc[ ip1 ] );
+
+ // Radix-4 butterfly intermediate computations:
+ tr1 = cr2 + cr4;
+ tr4 = cr4 - cr2;
+ ti1 = ci2 + ci4;
+ ti4 = ci2 - ci4;
+
+ // Load component 0 data and compute intermediates...
+ ip1 = iptr( i-1, k, 0, L, M, sc, oc ); // real part component 0
+ tr2 = cc[ ip1 ] + cr3;
+ tr3 = cc[ ip1 ] - cr3;
+
+ ip2 = iptr( i, k, 0, L, M, sc, oc ); // imag part component 0
+ ti2 = cc[ ip2 ] + ci3;
+ ti3 = cc[ ip2 ] - ci3;
+
+ // Output column 0 (real part)
+ io = optr( i-1, 0, k, M, so, oo );
+ out[ io ] = tr1 + tr2;
+
+ // Output column 3 (real part)
+ io = optr( im-1, 3, k, M, so, oo );
+ out[ io ] = tr2 - tr1;
+
+ // Output column 0 (imag part)
+ io = optr( i, 0, k, M, so, oo );
+ out[ io ] = ti1 + ti2;
+
+ // Output column 3 (imag part)
+ io = optr( im, 3, k, M, so, oo );
+ out[ io ] = ti1 - ti2;
+
+ // Output column 2 (real part)
+ io = optr( i-1, 2, k, M, so, oo );
+ out[ io ] = ti4 + tr3;
+
+ // Output column 1 (real part)
+ io = optr( im-1, 1, k, M, so, oo );
+ out[ io ] = tr3 - ti4;
+
+ // Output column 2 (imag part)
+ io = optr( i, 2, k, M, so, oo );
+ out[ io ] = tr4 + ti3;
+
+ // Output column 1 (imag part)
+ io = optr( im, 1, k, M, so, oo );
+ out[ io ] = tr4 - ti3;
+ }
+ }
+
+ // When `M` is odd, there is no Nyquist pair to process, so we do not need to perform any further transformations...
+ if ( M%2 === 1 ) {
+ return;
+ }
+ }
+
+ /*
+ * Lastly, handle the Nyquist frequency where `i = M-1` (i.e., the last element of each sub-sequence).
+ *
+ * When `M` is even, the Nyquist index is `i = M/2`. In this stage, we've stored that element at the end of each sub-sequence (i.e., `i = M-1` in the packed layout).
+ *
+ * At this point, only real components exist for rows 1 and 3 (imaginary parts are zero due to symmetry), and the twiddle multiplication simplifies to involve √(1/2) factors.
+ *
+ * In this case,
+ *
+ * ti1 = -√(1/2) ⋅ (row1 + row3)
+ * tr1 = √(1/2) ⋅ (row1 - row3)
+ *
+ * where
+ *
+ * row0 = Re(X₀)
+ * row1 = Re(X₁) (imaginary part is zero)
+ * row2 = Re(X₂)
+ * row3 = Re(X₃) (imaginary part is zero)
+ */
+ for ( k = 0; k < L; k++ ) {
+ ip1 = iptr( M-1, k, 1, L, M, sc, oc ); // Re(X₁)
+ ip2 = iptr( M-1, k, 3, L, M, sc, oc ); // Re(X₃)
+
+ ti1 = -SQRT_HALF * ( cc[ ip1 ] + cc[ ip2 ] );
+ tr1 = SQRT_HALF * ( cc[ ip1 ] - cc[ ip2 ] );
+
+ ip3 = iptr( M-1, k, 0, L, M, sc, oc ); // Re(X₀)
+ ip4 = iptr( M-1, k, 2, L, M, sc, oc ); // Re(X₂)
+
+ io = optr( M-1, 0, k, M, so, oo );
+ out[ io ] = cc[ ip3 ] + tr1;
+
+ io = optr( M-1, 2, k, M, so, oo );
+ out[ io ] = cc[ ip3 ] - tr1;
+
+ io = optr( 0, 1, k, M, so, oo );
+ out[ io ] = ti1 - cc[ ip4 ];
+
+ io = optr( 0, 3, k, M, so, oo );
+ out[ io ] = cc[ ip4 ] + ti1;
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = radf4;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf5.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf5.js
new file mode 100644
index 000000000000..dec092981317
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radf5.js
@@ -0,0 +1,511 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+/* eslint-disable max-len, max-statements */
+
+'use strict';
+
+// VARIABLES //
+
+// cos( 2π/5 ):
+var TR11 = 0.309016994374947;
+
+// sin( 2π/5 ):
+var TI11 = 0.951056516295154;
+
+// cos( 4π/5 ):
+var TR12 = -0.809016994374947;
+
+// sin( 4π/5 ):
+var TI12 = 0.587785252292473;
+
+
+// FUNCTIONS //
+
+/**
+* Resolves an index into the input array.
+*
+* ## Notes
+*
+* In a forward real FFT, the previous stage writes its results as five "rows" per sub-sequence.
+*
+* Thus, when reading from an input array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having five "rows" (components 0, 1, 2, 3, and 4) and where each "row" is arranged as `M*L` contiguous elements corresponding to interleaved real and imaginary components.
+*
+* Accordingly, the following is a logical view of an input array (zero-based indexing) which contains `L = 3` transforms and in which each sub-sequence has length `M = 4`:
+*
+* ```text
+* │ k = 0 k = 1 k = 2
+* │ ──────────────────────────────────────────────────────────────────────────→ k
+* j = 0 │ cc(0,0,0) ... cc(3,0,0) cc(0,1,0) ... cc(3,1,0) cc(0,2,0) ... cc(3,2,0)
+* │
+* j = 1 │ cc(0,0,1) ... cc(3,0,1) cc(0,1,1) ... cc(3,1,1) cc(0,2,1) ... cc(3,2,1)
+* │
+* j = 2 │ cc(0,0,2) ... cc(3,0,2) cc(0,1,2) ... cc(3,1,2) cc(0,2,2) ... cc(3,2,2)
+* │
+* j = 3 │ cc(0,0,3) ... cc(3,0,3) cc(0,1,3) ... cc(3,1,3) cc(0,2,3) ... cc(3,2,3)
+* │
+* j = 4 │ cc(0,0,4) ... cc(3,0,4) cc(0,1,4) ... cc(3,1,4) cc(0,2,4) ... cc(3,2,4)
+* └───────────────────────────────────────────────────────────────────────────→ i
+* ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 M-1 0 M-1 0 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short sub-sequence corresponding to one of the five component rows.
+* - `j` selects between the five component rows (0, 1, 2, 3, or 4).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | cc(0,0,0)...cc(3,0,0) ... cc(0,2,0)...cc(3,2,0) | cc(0,0,1)...cc(3,0,1) ... cc(0,2,1)...cc(3,2,1) | ... | cc(0,0,4)...cc(3,0,4) ... cc(0,2,4)...cc(3,2,4) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 LM LM-1 (L+1)M (L+1)M-1 (2L-1)M 2LM-1 4LM (4L+1)M-1 (5L-1)M 5LM-1
+* ```
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} j - input row
+* @param {NonNegativeInteger} L - number of sub-sequences
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the input array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the input array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = iptr( 0, 0, 0, L, M, stride, offset );
+* // returns 0
+*
+* idx = iptr( 1, 0, 0, L, M, stride, offset );
+* // returns 1
+*
+* idx = iptr( M-1, 0, 0, L, M, stride, offset );
+* // returns 3
+*
+* idx = iptr( 0, 1, 0, L, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = iptr( M-1, L-1, 1, L, M, stride, offset );
+* // returns 23
+*/
+function iptr( i, k, j, L, M, stride, offset ) {
+ var n = i + ( ( k+(j*L) ) * M );
+ return ( n*stride ) + offset;
+}
+
+/**
+* Resolves an index into the output array.
+*
+* ## Notes
+*
+* When writing to an output array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having five "columns" corresponding to the five components of a radix-5 stage (with real and imaginary parts of each component interleaved along each sub-sequence) and where each "column" has `M` elements.
+*
+* Accordingly, the following is a logical view of an output array (zero-based indexing) which contains `L = 3` transforms and in which each "column" sub-sequence has length `M = 4`:
+*
+* ```text
+* j = 0 (component 0) j = 1 (component 1) j = 2 (component 2) j = 3 (component 3) j = 4 (component 4)
+* k = 0 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┐
+* │ out(0,0,0) out(1,0,0) ... out(3,0,0) │ out(0,1,0) out(1,1,0) ... out(3,1,0) │ out(0,2,0) out(1,2,0) ... out(3,2,0) │ out(0,3,0) out(1,3,0) ... out(3,3,0) │ out(0,4,0) out(1,4,0) ... out(3,4,0) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 1 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,1) out(1,0,1) ... out(3,0,1) │ out(0,1,1) out(1,1,1) ... out(3,1,1) │ out(0,2,1) out(1,2,1) ... out(3,2,1) │ out(0,3,1) out(1,3,1) ... out(3,3,1) │ out(0,4,1) out(1,4,1) ... out(3,4,1) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┤
+* k = 2 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┤
+* │ out(0,0,2) out(1,0,2) ... out(3,0,2) │ out(0,1,2) out(1,1,2) ... out(3,1,2) │ out(0,2,2) out(1,2,2) ... out(3,2,2) │ out(0,3,2) out(1,3,2) ... out(3,3,2) │ out(0,4,2) out(1,4,2) ... out(3,4,2) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┘
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 1 M-1 0 1 M-1 0 1 M-1 0 1 M-1 0 1 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short "column" sub-sequence.
+* - `j` selects which of the five components (0, 1, 2, 3, or 4).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | out(0,0,0)...out(3,0,0) | out(0,1,0)...out(3,1,0) | out(0,2,0)...out(3,2,0) | out(0,3,0)...out(3,3,0) | out(0,4,0)...out(3,4,0) | out(0,0,1)...out(3,0,1) | ... | out(0,4,2)...out(3,4,2) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 M 2M-1 2M 3M-1 3M 4M-1 4M 5M-1 5M 6M-1 (5L-1)M 5LM-1
+* ```
+*
+* As may be observed, when resolving an index in the output array, the `j` and `k` dimensions are swapped relative index resolution in the input array. This stems from `radf5` being only one stage in a multi-stage driver which alternates between using `cc` and `out` as workspace buffers. After each stage, the next stage reads what the previous stage wrote.
+*
+* Each stage expects a transpose, and, in order to avoid explicit transposition between the stages, we swap the last two logical dimensions while still maintaining cache locality within the inner loop logical dimension, as indexed by `i`.
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} j - index specifying which of the five complex components we are in (0, 1, 2, 3, or 4)
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the output array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the output array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = optr( 0, 0, 0, M, stride, offset );
+* // returns 0
+*
+* idx = optr( 1, 0, 0, M, stride, offset );
+* // returns 1
+*
+* idx = optr( M-1, 0, 0, M, stride, offset );
+* // returns 3
+*
+* idx = optr( 0, 1, 0, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = optr( M-1, 4, L-1, M, stride, offset );
+* // returns 59
+*/
+function optr( i, j, k, M, stride, offset ) {
+ var n = i + ( ( j+(k*5) ) * M );
+ return ( n*stride ) + offset;
+}
+
+
+// MAIN //
+
+/**
+* Performs one radix-5 stage within a forward Fourier transform for a real-valued sequence.
+*
+* @private
+* @param {NonNegativeInteger} M - number of elements in each sub-sequence to be transformed
+* @param {NonNegativeInteger} L - number of sub-sequences to be transformed
+* @param {Collection} cc - input array containing the sub-sequences to be transformed
+* @param {integer} sc - stride length for `cc`
+* @param {NonNegativeInteger} oc - index offset for `cc`
+* @param {Collection} out - output array containing transformed sub-sequences
+* @param {integer} so - stride length for `out`
+* @param {NonNegativeInteger} oo - index offset for `out`
+* @param {Collection} twiddles1 - first array containing twiddle factors
+* @param {integer} st1 - stride length for `twiddles1`
+* @param {NonNegativeInteger} ot1 - index offset for `twiddles1`
+* @param {Collection} twiddles2 - second array containing twiddle factors
+* @param {integer} st2 - stride length for `twiddles2`
+* @param {NonNegativeInteger} ot2 - index offset for `twiddles2`
+* @param {Collection} twiddles3 - third array containing twiddle factors
+* @param {integer} st3 - stride length for `twiddles3`
+* @param {NonNegativeInteger} ot3 - index offset for `twiddles3`
+* @param {Collection} twiddles4 - fourth array containing twiddle factors
+* @param {integer} st4 - stride length for `twiddles4`
+* @param {NonNegativeInteger} ot4 - index offset for `twiddles4`
+* @returns {void}
+*/
+function radf5( M, L, cc, sc, oc, out, so, oo, twiddles1, st1, ot1, twiddles2, st2, ot2, twiddles3, st3, ot3, twiddles4, st4, ot4 ) { // eslint-disable-line max-params
+ var it11;
+ var it12;
+ var it21;
+ var it22;
+ var it31;
+ var it32;
+ var it41;
+ var it42;
+ var ci2;
+ var ci3;
+ var ci4;
+ var ci5;
+ var cr2;
+ var cr3;
+ var cr4;
+ var cr5;
+ var dr2;
+ var dr3;
+ var dr4;
+ var dr5;
+ var di2;
+ var di3;
+ var di4;
+ var di5;
+ var ti2;
+ var ti3;
+ var ti4;
+ var ti5;
+ var tr2;
+ var tr3;
+ var tr4;
+ var tr5;
+ var ip1;
+ var ip2;
+ var ip3;
+ var ip4;
+ var ip5;
+ var io;
+ var im;
+ var i;
+ var k;
+
+ /*
+ * First, perform the core butterfly for each sub-sequence being transformed.
+ *
+ * In the following loop, we handle harmonic `n = 0` for every transform. As described for `iptr`, the input array is interpreted as a three-dimensional array, containing five "rows" per sub-sequence.
+ *
+ * row0 = input component 0 (j = 0)
+ * row1 = input component 1 (j = 1)
+ * row2 = input component 2 (j = 2)
+ * row3 = input component 3 (j = 3)
+ * row4 = input component 4 (j = 4)
+ *
+ * For a radix-5 forward FFT of a real-valued signal `x` and `n = 0`,
+ *
+ * cr2 = row4 + row1
+ * ci5 = row4 - row1
+ * cr3 = row3 + row2
+ * ci4 = row3 - row2
+ *
+ * x[0] = row0 + cr2 + cr3
+ * x[M] = row0 + ( TR11*cr2 ) + ( TR12*cr3 )
+ * x[2M] = ( TI11*ci5 ) + ( TI12*ci4 )
+ * x[3M] = row0 + ( TR12*cr2 ) + ( TR11*cr3 )
+ * x[4M] = ( TI12*ci5 ) - ( TI11*ci4 )
+ *
+ * Because `W_5^0 = 1`, no frequency-dependent twiddle multiplication is necessary beyond these constant factors.
+ */
+ for ( k = 0; k < L; k++ ) {
+ ip1 = iptr( 0, k, 0, L, M, sc, oc ); // row 0
+ ip2 = iptr( 0, k, 1, L, M, sc, oc ); // row 1
+ ip3 = iptr( 0, k, 2, L, M, sc, oc ); // row 2
+ ip4 = iptr( 0, k, 3, L, M, sc, oc ); // row 3
+ ip5 = iptr( 0, k, 4, L, M, sc, oc ); // row 4
+
+ cr2 = cc[ ip5 ] + cc[ ip2 ]; // row4 + row1
+ ci5 = cc[ ip5 ] - cc[ ip2 ]; // row4 - row1
+ cr3 = cc[ ip4 ] + cc[ ip3 ]; // row3 + row2
+ ci4 = cc[ ip4 ] - cc[ ip3 ]; // row3 - row2
+
+ io = optr( 0, 0, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + cr2 + cr3;
+
+ io = optr( M-1, 1, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + ( TR11 * cr2 ) + ( TR12 * cr3 );
+
+ io = optr( 0, 2, k, M, so, oo );
+ out[ io ] = ( TI11 * ci5 ) + ( TI12 * ci4 );
+
+ io = optr( M-1, 3, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + ( TR12 * cr2 ) + ( TR11 * cr3 );
+
+ io = optr( 0, 4, k, M, so, oo );
+ out[ io ] = ( TI12 * ci5 ) - ( TI11 * ci4 );
+ }
+ // When the number of elements in a sub-sequence is less than `2`, there is nothing more to do, as the above butterfly produced the full result...
+ if ( M < 2 ) {
+ return;
+ }
+ /*
+ * Next, apply the general case where we need to loop through the non-trivial harmonics.
+ *
+ * For each harmonic `n = 1, ..., M/2-1`, we need to
+ *
+ * - recover five spectra from the five input rows.
+ * - apply radix-5 twiddle factors to rows 1, 2, 3, and 4, then combine with row 0 to form five output columns.
+ * - form the following
+ *
+ * x[5n] = X₀ + (W₁⋅X₁ + W₄⋅X₄) + (W₂⋅X₂ + W₃⋅X₃) => column 0
+ * x[5n+1] = X₀ + TR11⋅(W₁⋅X₁ + W₄⋅X₄) + TR12⋅(W₂⋅X₂ + W₃⋅X₃) - i⋅(TI11⋅(W₁⋅X₁ - W₄⋅X₄) + TI12⋅(W₂⋅X₂ - W₃⋅X₃)) => column 1
+ * x[5n+2] = X₀ + TR11⋅(W₁⋅X₁ + W₄⋅X₄) + TR12⋅(W₂⋅X₂ + W₃⋅X₃) + i⋅(TI11⋅(W₁⋅X₁ - W₄⋅X₄) + TI12⋅(W₂⋅X₂ - W₃⋅X₃)) => column 2
+ * x[5n+3] = X₀ + TR12⋅(W₁⋅X₁ + W₄⋅X₄) + TR11⋅(W₂⋅X₂ + W₃⋅X₃) - i⋅(TI12⋅(W₁⋅X₁ - W₄⋅X₄) - TI11⋅(W₂⋅X₂ - W₃⋅X₃)) => column 3
+ * x[5n+4] = X₀ + TR12⋅(W₁⋅X₁ + W₄⋅X₄) + TR11⋅(W₂⋅X₂ + W₃⋅X₃) + i⋅(TI12⋅(W₁⋅X₁ - W₄⋅X₄) - TI11⋅(W₂⋅X₂ - W₃⋅X₃)) => column 4
+ *
+ * The mirror index `im = M - i` selects the conjugate-symmetric partner, thus allowing the routine to read each symmetry pair only once.
+ */
+ // Loop over each sub-sequence to be transformed...
+ for ( k = 0; k < L; k++ ) {
+ // Loop over the elements in each sub-sequence...
+ for ( i = 2; i < M; i += 2 ) {
+ im = M - i; // "mirror" index
+
+ // Resolve twiddle factor indices for component 1:
+ it11 = ( (i-2)*st1 ) + ot1; // cos(θ) for component 1
+ it12 = ( (i-1)*st1 ) + ot1; // sin(θ) for component 1
+
+ // Resolve twiddle factor indices for component 2:
+ it21 = ( (i-2)*st2 ) + ot2; // cos(θ) for component 2
+ it22 = ( (i-1)*st2 ) + ot2; // sin(θ) for component 2
+
+ // Resolve twiddle factor indices for component 3:
+ it31 = ( (i-2)*st3 ) + ot3; // cos(θ) for component 3
+ it32 = ( (i-1)*st3 ) + ot3; // sin(θ) for component 3
+
+ // Resolve twiddle factor indices for component 4:
+ it41 = ( (i-2)*st4 ) + ot4; // cos(θ) for component 4
+ it42 = ( (i-1)*st4 ) + ot4; // sin(θ) for component 4
+
+ // Load component 1 data and apply twiddle...
+ ip1 = iptr( i-1, k, 1, L, M, sc, oc ); // real part component 1
+ ip2 = iptr( i, k, 1, L, M, sc, oc ); // imag part component 1
+
+ // Apply the twiddle factors for component 1:
+ dr2 = ( twiddles1[ it11 ] * cc[ ip1 ] ) + ( twiddles1[ it12 ] * cc[ ip2 ] ); // Re(dr2)
+ di2 = ( twiddles1[ it11 ] * cc[ ip2 ] ) - ( twiddles1[ it12 ] * cc[ ip1 ] ); // Im(di2)
+
+ // Load component 2 data and apply twiddle...
+ ip1 = iptr( i-1, k, 2, L, M, sc, oc ); // real part component 2
+ ip2 = iptr( i, k, 2, L, M, sc, oc ); // imag part component 2
+
+ // Apply the twiddle factors for component 2:
+ dr3 = ( twiddles2[ it21 ] * cc[ ip1 ] ) + ( twiddles2[ it22 ] * cc[ ip2 ] ); // Re(dr3)
+ di3 = ( twiddles2[ it21 ] * cc[ ip2 ] ) - ( twiddles2[ it22 ] * cc[ ip1 ] ); // Im(di3)
+
+ // Load component 3 data and apply twiddle...
+ ip1 = iptr( i-1, k, 3, L, M, sc, oc ); // real part component 3
+ ip2 = iptr( i, k, 3, L, M, sc, oc ); // imag part component 3
+
+ // Apply the twiddle factors for component 3:
+ dr4 = ( twiddles3[ it31 ] * cc[ ip1 ] ) + ( twiddles3[ it32 ] * cc[ ip2 ] ); // Re(dr4)
+ di4 = ( twiddles3[ it31 ] * cc[ ip2 ] ) - ( twiddles3[ it32 ] * cc[ ip1 ] ); // Im(di4)
+
+ // Load component 4 data and apply twiddle...
+ ip1 = iptr( i-1, k, 4, L, M, sc, oc ); // real part component 4
+ ip2 = iptr( i, k, 4, L, M, sc, oc ); // imag part component 4
+
+ // Apply the twiddle factors for component 4:
+ dr5 = ( twiddles4[ it41 ] * cc[ ip1 ] ) + ( twiddles4[ it42 ] * cc[ ip2 ] ); // Re(dr5)
+ di5 = ( twiddles4[ it41 ] * cc[ ip2 ] ) - ( twiddles4[ it42 ] * cc[ ip1 ] ); // Im(di5)
+
+ // Radix-5 butterfly intermediate computations (symmetric/antisymmetric combinations):
+ cr2 = dr2 + dr5;
+ ci5 = dr5 - dr2;
+ cr5 = di2 - di5;
+ ci2 = di5 + di2;
+ cr3 = dr3 + dr4;
+ ci4 = dr4 - dr3;
+ cr4 = di3 - di4;
+ ci3 = di3 + di4;
+
+ // Output column 0 (real part)
+ ip1 = iptr( i-1, k, 0, L, M, sc, oc );
+ io = optr( i-1, 0, k, M, so, oo );
+ out[ io ] = cc[ ip1 ] + cr2 + cr3;
+
+ // Output column 0 (imag part)
+ ip2 = iptr( i, k, 0, L, M, sc, oc );
+ io = optr( i, 0, k, M, so, oo );
+ out[ io ] = cc[ ip2 ] + ci2 + ci3;
+
+ // Load component 0 data and compute intermediates...
+ tr2 = cc[ ip1 ] + ( TR11 * cr2 ) + ( TR12 * cr3 );
+ ti2 = cc[ ip2 ] + ( TR11 * ci2 ) + ( TR12 * ci3 );
+ tr3 = cc[ ip1 ] + ( TR12 * cr2 ) + ( TR11 * cr3 );
+ ti3 = cc[ ip2 ] + ( TR12 * ci2 ) + ( TR11 * ci3 );
+
+ // Additional intermediates:
+ tr5 = ( TI11 * cr5 ) + ( TI12 * cr4 );
+ ti5 = ( TI11 * ci5 ) + ( TI12 * ci4 );
+ tr4 = ( TI12 * cr5 ) - ( TI11 * cr4 );
+ ti4 = ( TI12 * ci5 ) - ( TI11 * ci4 );
+
+ // Output column 2 (real part)
+ io = optr( i-1, 2, k, M, so, oo );
+ out[ io ] = tr2 + tr5;
+
+ // Output column 1 (real part)
+ io = optr( im-1, 1, k, M, so, oo );
+ out[ io ] = tr2 - tr5;
+
+ // Output column 2 (imag part)
+ io = optr( i, 2, k, M, so, oo );
+ out[ io ] = ti2 + ti5;
+
+ // Output column 1 (imag part)
+ io = optr( im, 1, k, M, so, oo );
+ out[ io ] = ti5 - ti2;
+
+ // Output column 4 (real part)
+ io = optr( i-1, 4, k, M, so, oo );
+ out[ io ] = tr3 + tr4;
+
+ // Output column 3 (real part)
+ io = optr( im-1, 3, k, M, so, oo );
+ out[ io ] = tr3 - tr4;
+
+ // Output column 4 (imag part)
+ io = optr( i, 4, k, M, so, oo );
+ out[ io ] = ti3 + ti4;
+
+ // Output column 3 (imag part)
+ io = optr( im, 3, k, M, so, oo );
+ out[ io ] = ti4 - ti3;
+ }
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = radf5;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radfg.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radfg.js
new file mode 100644
index 000000000000..8e00218e2186
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/radfg.js
@@ -0,0 +1,765 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+/* eslint-disable max-len, max-statements, max-lines-per-function */
+
+'use strict';
+
+// MODULES //
+
+var TWO_PI = require( '@stdlib/constants/float64/two-pi' );
+var cos = require( '@stdlib/math/base/special/cos' );
+var sin = require( '@stdlib/math/base/special/sin' );
+var floor = require( '@stdlib/math/base/special/floor' );
+
+
+// FUNCTIONS //
+
+/**
+* Resolves an index into the input array.
+*
+* ## Notes
+*
+* In a forward real FFT, the previous stage writes its results as `R` "rows" per sub-sequence.
+*
+* Thus, when reading from an input array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having `R` "rows" (components 0, 1, 2, ..., R-1) and where each "row" is arranged as `M*L` contiguous elements corresponding to interleaved real and imaginary components.
+*
+* Accordingly, the following is a logical view of an input array (zero-based indexing) which contains `L = 3` transforms and in which each sub-sequence has length `M = 4`:
+*
+* ```text
+* │ k = 0 k = 1 k = 2
+* │ ─────────────────────────────────────────────────────────────────────────────────────→ k
+* j = 0 │ cc(0,0,0) ... cc(3,0,0) cc(0,1,0) ... cc(3,1,0) cc(0,2,0) ... cc(3,2,0)
+* │
+* j = 1 │ cc(0,0,1) ... cc(3,0,1) cc(0,1,1) ... cc(3,1,1) cc(0,2,1) ... cc(3,2,1)
+* │
+* j = 2 │ cc(0,0,2) ... cc(3,0,2) cc(0,1,2) ... cc(3,1,2) cc(0,2,2) ... cc(3,2,2)
+* │
+* j = 3 │ cc(0,0,3) ... cc(3,0,3) cc(0,1,3) ... cc(3,1,3) cc(0,2,3) ... cc(3,2,3)
+* │
+* ... │ ... ... ...
+* │
+* j=R-1 │ cc(0,0,R-1) ... cc(3,0,R-1) cc(0,1,R-1) ... cc(3,1,R-1) cc(0,2,R-1) ... cc(3,2,R-1)
+* └──────────────────────────────────────────────────────────────────────────────────────→ i
+* ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 M-1 0 M-1 0 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short sub-sequence corresponding to one of the R component rows.
+* - `j` selects between the R component rows (0, 1, 2, ..., R-1).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | cc(0,0,0)...cc(3,0,0) ... cc(0,2,0)...cc(3,2,0) | cc(0,0,1)...cc(3,0,1) ... cc(0,2,1)...cc(3,2,1) | ... | cc(0,0,R-1)...cc(3,0,R-1) ... cc(0,2,R-1)...cc(3,2,R-1) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 LM LM-1 (L+1)M (L+1)M-1 (2L-1)M 2LM-1 (R-1)LM ((R-1)L+1)M-1 (RL-1)M RLM-1
+* ```
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} j - input row
+* @param {NonNegativeInteger} L - number of sub-sequences
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the input array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the input array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = iptr( 0, 0, 0, L, M, stride, offset );
+* // returns 0
+*
+* idx = iptr( 1, 0, 0, L, M, stride, offset );
+* // returns 1
+*
+* idx = iptr( M-1, 0, 0, L, M, stride, offset );
+* // returns 3
+*
+* idx = iptr( 0, 1, 0, L, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = iptr( M-1, L-1, 6, L, M, stride, offset );
+* // returns 83
+*/
+function iptr( i, k, j, L, M, stride, offset ) {
+ var n = i + ( ( k+(j*L) ) * M );
+ return ( n*stride ) + offset;
+}
+
+/**
+* Resolves an index into the output array.
+*
+* ## Notes
+*
+* When writing to an output array stored in linear memory, we can reinterpret the array as a three-dimensional logical view containing `L` independent sub-sequences having `R` "columns" corresponding to the `R` components of a radix-R stage (with real and imaginary parts of each component interleaved along each sub-sequence) and where each "column" has `M` elements.
+*
+* Accordingly, the following is a logical view of an output array (zero-based indexing) which contains `L = 3` transforms and in which each "column" sub-sequence has length `M = 4` for an arbitrary radix `R`:
+*
+* ```text
+* j = 0 (component 0) j = 1 (component 1) j = 2 (component 2) ... j = R-1 (component R-1)
+* k = 0 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────────────┐
+* │ out(0,0,0) out(1,0,0) ... out(3,0,0) │ out(0,1,0) out(1,1,0) ... out(3,1,0) │ out(0,2,0) out(1,2,0) ... out(3,2,0) │ ... │ out(0,R-1,0) out(1,R-1,0) ... out(3,R-1,0) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────────────┤
+* k = 1 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────────────┤
+* │ out(0,0,1) out(1,0,1) ... out(3,0,1) │ out(0,1,1) out(1,1,1) ... out(3,1,1) │ out(0,2,1) out(1,2,1) ... out(3,2,1) │ ... │ out(0,R-1,1) out(1,R-1,1) ... out(3,R-1,1) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────────────┤
+* k = 2 ─┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────┬───────────────────────────────────────────────┤
+* │ out(0,0,2) out(1,0,2) ... out(3,0,2) │ out(0,1,2) out(1,1,2) ... out(3,1,2) │ out(0,2,2) out(1,2,2) ... out(3,2,2) │ ... │ out(0,R-1,2) out(1,R-1,2) ... out(3,R-1,2) │
+* └───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────┴───────────────────────────────────────────────┘
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* i = 0 1 M-1 0 1 M-1 0 1 M-1 0 1 M-1
+* ```
+*
+* In the above,
+*
+* - `i` is the fastest varying index, which walks within one short "column" sub-sequence.
+* - `j` selects which of the R components we are in (0, 1, 2, ..., R-1).
+* - `k` specifies the index of one of the `L` independent transforms we are processing.
+*
+* In linear memory, the three-dimensional logical view is arranged as follows:
+*
+* ```text
+* | out(0,0,0)...out(3,0,0) | out(0,1,0)...out(3,1,0) | out(0,2,0)...out(3,2,0) | ... | out(0,R-1,0)...out(3,R-1,0) | out(0,0,1)...out(3,0,1) | ... | out(0,R-1,2)...out(3,R-1,2) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 M-1 M 2M-1 2M 3M-1 (R-1)M RM-1 RM (R+1)M-1 (RL-1)M RLM-1
+* ```
+*
+* As may be observed, when resolving an index in the output array, the `j` and `k` dimensions are swapped relative index resolution in the input array. This stems from `radfg` being only one stage in a multi-stage driver which alternates between using `cc` and `out` as workspace buffers. After each stage, the next stage reads what the previous stage wrote.
+*
+* Each stage expects a transpose, and, in order to avoid explicit transposition between the stages, we swap the last two logical dimensions while still maintaining cache locality within the inner loop logical dimension, as indexed by `i`.
+*
+* @private
+* @param {NonNegativeInteger} i - index of an element within a sub-sequence
+* @param {NonNegativeInteger} j - index specifying which of the R complex components we are in (0, 1, ..., R-1)
+* @param {NonNegativeInteger} k - index of the sub-sequence being transformed
+* @param {NonNegativeInteger} R - radix
+* @param {NonNegativeInteger} M - sub-sequence length
+* @param {integer} stride - stride length of the output array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the output array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+*
+* var idx = optr( 0, 0, 0, 7, M, stride, offset );
+* // returns 0
+*
+* idx = optr( 1, 0, 0, 7, M, stride, offset );
+* // returns 1
+*
+* idx = optr( M-1, 0, 0, 7, M, stride, offset );
+* // returns 3
+*
+* idx = optr( 0, 1, 0, 7, M, stride, offset );
+* // returns 4
+*
+* // ...
+*
+* idx = optr( M-1, 6, L-1, 7, M, stride, offset );
+* // returns 83
+*/
+function optr( i, j, k, R, M, stride, offset ) {
+ var n = i + ( ( j+(k*R) ) * M );
+ return ( n*stride ) + offset;
+}
+
+/**
+* Resolves an index into a flattened workspace array.
+*
+* ## Notes
+*
+* During the general radix stage, flattened workspace arrays store `ML = M * L` elements for each of the `R` component columns. Both the flattened input and output workspace arrays share the same `[ML, R]` logical layout.
+*
+* Thus, when accessing a flattened workspace array stored in linear memory, we can reinterpret the array as a two-dimensional logical view having `R` component columns, where each column has `ML` elements.
+*
+* Accordingly, the following is a logical view of a flattened workspace array (zero-based indexing) having flattened dimension `ML` and arbitrary radix `R`:
+*
+* ```text
+* │ j = 0 j = 1 j = 2 ... j = R-1
+* │ ────────────────────────────────────────────────────────────────────────→ j
+* ik = 0 │ ws(0,0) ws(0,1) ws(0,2) ... ws(0,R-1)
+* │
+* ik = 1 │ ws(1,0) ws(1,1) ws(1,2) ... ws(1,R-1)
+* │
+* ik = 2 │ ws(2,0) ws(2,1) ws(2,2) ... ws(2,R-1)
+* │
+* ik = 3 │ ws(3,0) ws(3,1) ws(3,2) ... ws(3,R-1)
+* │
+* ... │ ... ... ... ... ...
+* │
+* ik = ML-1 │ ws(ML-1,0) ws(ML-1,1) ws(ML-1,2) ... ws(ML-1,R-1)
+* ```
+*
+* In the above,
+*
+* - `ik` is the fastest varying index, which walks along the flattened sub-sequence dimension.
+* - `j` selects between the `R` workspace component columns (0, 1, 2, ..., R-1).
+*
+* In linear memory, the two-dimensional logical view is arranged as follows:
+*
+* ```text
+* | ws(0,0)...ws(ML-1,0) | ws(0,1)...ws(ML-1,1) | ws(0,2)...ws(ML-1,2) | ... | ws(0,R-1)...ws(ML-1,R-1) |
+* ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
+* 0 ML-1 ML 2ML-1 2ML 3ML-1 (R-1)ML RML-1
+* ```
+*
+* Here, the original `M` and `L` dimensions have been collapsed into the single flattened `ik` dimension. Each workspace component `j` therefore occupies one contiguous block of `ML` elements in linear memory.
+*
+* @private
+* @param {NonNegativeInteger} ik - index of an element within a flattened sub-sequence
+* @param {NonNegativeInteger} j - index specifying which of the R workspace component columns we are in (0, 1, ..., R-1)
+* @param {NonNegativeInteger} ML - flattened sub-sequence length
+* @param {integer} stride - stride length of the flattened workspace array
+* @param {NonNegativeInteger} offset - index specifying the first indexed element in the flattened workspace array
+* @returns {NonNegativeInteger} computed index
+*
+* @example
+* var stride = 1;
+* var offset = 0;
+*
+* var M = 4; // sub-sequence length
+* var L = 3; // number of sub-sequences
+* var ML = M * L; // flattened dimension ( 4*3 = 12 )
+*
+* var idx = fptr( 0, 0, ML, stride, offset );
+* // returns 0
+*
+* idx = fptr( 1, 0, ML, stride, offset );
+* // returns 1
+*
+* idx = fptr( ML-1, 0, ML, stride, offset );
+* // returns 11
+*
+* idx = fptr( 0, 1, ML, stride, offset );
+* // returns 12
+*
+* // ...
+*
+* idx = fptr( ML-1, 6, ML, stride, offset );
+* // returns 83
+*/
+function fptr( ik, j, ML, stride, offset ) {
+ var n = ik + ( j*ML );
+ return ( n*stride ) + offset;
+}
+
+
+// MAIN //
+
+/**
+* Performs one general radix stage within a forward Fourier transform for a real-valued sequence.
+*
+* @private
+* @param {NonNegativeInteger} M - number of elements in each sub-sequence to be transformed
+* @param {NonNegativeInteger} R - radix of the transform
+* @param {NonNegativeInteger} L - number of sub-sequences to be transformed
+* @param {NonNegativeInteger} ML - number of elements in each flattened sub-sequence (`M*L`)
+* @param {Collection} cc - input array containing the sub-sequences to be transformed
+* @param {integer} sc - stride length for `cc`
+* @param {NonNegativeInteger} oc - index offset for `cc`
+* @param {Collection} c1 - input workspace array containing intermediate sub-sequences to be transformed
+* @param {integer} sc1 - stride length for `c1`
+* @param {NonNegativeInteger} oc1 - index offset for `c1`
+* @param {Collection} c2 - flattened input workspace array containing intermediate sub-sequences
+* @param {integer} sc2 - stride length for `c2`
+* @param {NonNegativeInteger} oc2 - index offset for `c2`
+* @param {Collection} ch - output array containing transformed sequences
+* @param {integer} sch - stride length for `ch`
+* @param {NonNegativeInteger} och - index offset for `ch`
+* @param {Collection} ch2 - flattened output workspace array containing intermediate sub-sequences
+* @param {integer} sch2 - stride length for `ch2`
+* @param {NonNegativeInteger} och2 - index offset for `ch2`
+* @param {Collection} twiddles - array of twiddle factors
+* @param {integer} stw - stride length for `twiddles`
+* @param {NonNegativeInteger} otw - index offset for `twiddles`
+* @returns {void}
+*/
+function radfg( M, R, L, ML, cc, sc, oc, c1, sc1, oc1, c2, sc2, oc2, ch, sch, och, ch2, sch2, och2, twiddles, stw, otw ) { // eslint-disable-line max-params
+ var ar1h;
+ var ar2h;
+ var idij;
+ var ic2o;
+ var ic1o;
+ var ich;
+ var icc;
+ var Rph;
+ var Rp1;
+ var ih1;
+ var ih2;
+ var ih3;
+ var ih4;
+ var ic1;
+ var ic2;
+ var ic3;
+ var ic4;
+ var it1;
+ var it2;
+ var io1;
+ var io2;
+ var io3;
+ var io4;
+ var dc2;
+ var ai1;
+ var ai2;
+ var ar1;
+ var ar2;
+ var ds2;
+ var nbd;
+ var dcp;
+ var dsp;
+ var im;
+ var ik;
+ var is;
+ var jc;
+ var lc;
+ var j2;
+ var i;
+ var j;
+ var k;
+ var l;
+
+ // Compute the basic rotation angle for the radix-R DFT matrix:
+ dcp = cos( TWO_PI / R ); // cos(2π/R)
+ dsp = sin( TWO_PI / R ); // sin(2π/R)
+
+ // Compute half the radix, rounded up, which is used as the exclusive upper bound for conjugate-symmetric component loops:
+ Rph = floor( ( R + 1 ) / 2 );
+ Rp1 = R + 1;
+
+ // Compute the number of non-DC complex harmonics in each sub-sequence:
+ nbd = floor( ( M - 1 ) / 2 );
+
+ /*
+ * First, initialize the work arrays for the general-radix butterfly.
+ *
+ * At this stage, each sub-sequence has already been split into `R` radix components. The `j = 0` component is carried in the flattened work arrays, while the remaining components `j = 1, ..., R-1` are carried in the three-dimensional work arrays.
+ *
+ * For harmonic `n = 0`, we only copy the DC terms.
+ *
+ * For harmonics `n = 1, ..., floor((M-1)/2)`, each nonzero radix component is multiplied by its stage twiddle factor and stored in `ch`.
+ */
+ if ( M === 1 ) {
+ // Copy the DC column from the flattened output workspace back to the flattened input workspace...
+ for ( ik = 0; ik < ML; ik++ ) {
+ ic2o = fptr( ik, 0, ML, sc2, oc2 );
+ ich = fptr( ik, 0, ML, sch2, och2 );
+ c2[ ic2o ] = ch2[ ich ];
+ }
+ } else {
+ // Copy the DC column into the flattened output workspace...
+ for ( ik = 0; ik < ML; ik++ ) {
+ ich = fptr( ik, 0, ML, sch2, och2 );
+ ic2o = fptr( ik, 0, ML, sc2, oc2 );
+ ch2[ ich ] = c2[ ic2o ];
+ }
+
+ // Copy the DC terms for the remaining radix components...
+ for ( j = 1; j < R; j++ ) {
+ for ( k = 0; k < L; k++ ) {
+ ih1 = optr( 0, k, j, L, M, sch, och );
+ ic1 = iptr( 0, k, j, L, M, sc1, oc1 );
+ ch[ ih1 ] = c1[ ic1 ];
+ }
+ }
+
+ // Apply twiddle factors to the non-DC harmonics of the remaining radix components...
+ if ( nbd <= L ) {
+ // Loop over harmonics before sub-sequences...
+ is = 0;
+ for ( j = 1; j < R; j++ ) {
+ idij = is;
+ for ( i = 2; i < M; i += 2 ) {
+ for ( k = 0; k < L; k++ ) {
+ // Resolve output indices in ch:
+ ih1 = optr( i-1, k, j, L, M, sch, och ); // real part
+ ih2 = optr( i, k, j, L, M, sch, och ); // imaginary part
+
+ // Resolve indices in the input workspace:
+ ic1 = iptr( i-1, k, j, L, M, sc1, oc1 ); // real part
+ ic2 = iptr( i, k, j, L, M, sc1, oc1 ); // imaginary part
+
+ // Resolve twiddle factor indices:
+ it1 = ( idij * stw ) + otw; // cos(θ)
+ it2 = ( (idij+1) * stw ) + otw; // sin(θ)
+
+ // Apply the twiddle factor:
+ ch[ ih1 ] = ( twiddles[ it1 ] * c1[ ic1 ] ) + ( twiddles[ it2 ] * c1[ ic2 ] ); // Re(ch)
+ ch[ ih2 ] = ( twiddles[ it1 ] * c1[ ic2 ] ) - ( twiddles[ it2 ] * c1[ ic1 ] ); // Im(ch)
+ }
+ idij += 2;
+ }
+ is += M;
+ }
+ } else {
+ // Loop over sub-sequences before harmonics...
+ is = 0;
+ for ( j = 1; j < R; j++ ) {
+ for ( k = 0; k < L; k++ ) {
+ idij = is;
+ for ( i = 2; i < M; i += 2 ) {
+ // Resolve output indices in ch:
+ ih1 = optr( i-1, k, j, L, M, sch, och ); // real part
+ ih2 = optr( i, k, j, L, M, sch, och ); // imaginary part
+
+ // Resolve indices in the input workspace:
+ ic1 = iptr( i-1, k, j, L, M, sc1, oc1 ); // real part
+ ic2 = iptr( i, k, j, L, M, sc1, oc1 ); // imaginary part
+
+ // Resolve twiddle factor indices:
+ it1 = ( idij * stw ) + otw; // cos(θ)
+ it2 = ( (idij+1) * stw ) + otw; // sin(θ)
+
+ // Apply the twiddle factor:
+ ch[ ih1 ] = ( twiddles[ it1 ] * c1[ ic1 ] ) + ( twiddles[ it2 ] * c1[ ic2 ] ); // Re(ch)
+ ch[ ih2 ] = ( twiddles[ it1 ] * c1[ ic2 ] ) - ( twiddles[ it2 ] * c1[ ic1 ] ); // Im(ch)
+
+ idij += 2;
+ }
+ }
+ is += M;
+ }
+ }
+
+ /*
+ * Next, combine mirrored radix components, storing their sums and differences in c1.
+ */
+ if ( nbd >= L ) {
+ // Loop over sub-sequences before harmonics...
+ for ( j = 1; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+ for ( k = 0; k < L; k++ ) {
+ for ( i = 2; i < M; i += 2 ) {
+ // Resolve ch indices for component j:
+ ih1 = optr( i-1, k, j, L, M, sch, och ); // Re(ch[j])
+ ih2 = optr( i, k, j, L, M, sch, och ); // Im(ch[j])
+
+ // Resolve ch indices for conjugate component jc:
+ ih3 = optr( i-1, k, jc, L, M, sch, och ); // Re(ch[jc])
+ ih4 = optr( i, k, jc, L, M, sch, och ); // Im(ch[jc])
+
+ // Resolve input-workspace indices for component j:
+ ic1 = iptr( i-1, k, j, L, M, sc1, oc1 ); // Re(component j)
+ ic2 = iptr( i, k, j, L, M, sc1, oc1 ); // Im(component j)
+
+ // Resolve input-workspace indices for component jc:
+ ic3 = iptr( i-1, k, jc, L, M, sc1, oc1 ); // Re(component jc)
+ ic4 = iptr( i, k, jc, L, M, sc1, oc1 ); // Im(component jc)
+
+ // Form the sum and difference combinations:
+ c1[ ic1 ] = ch[ ih1 ] + ch[ ih3 ]; // Re(j) + Re(jc)
+ c1[ ic3 ] = ch[ ih2 ] - ch[ ih4 ]; // Im(j) - Im(jc)
+ c1[ ic2 ] = ch[ ih2 ] + ch[ ih4 ]; // Im(j) + Im(jc)
+ c1[ ic4 ] = ch[ ih3 ] - ch[ ih1 ]; // Re(jc) - Re(j)
+ }
+ }
+ }
+ } else {
+ // Loop over harmonics before sub-sequences...
+ for ( j = 1; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+ for ( i = 2; i < M; i += 2 ) {
+ for ( k = 0; k < L; k++ ) {
+ // Resolve ch indices for component j:
+ ih1 = optr( i-1, k, j, L, M, sch, och ); // Re(ch[j])
+ ih2 = optr( i, k, j, L, M, sch, och ); // Im(ch[j])
+
+ // Resolve ch indices for conjugate component jc:
+ ih3 = optr( i-1, k, jc, L, M, sch, och ); // Re(ch[jc])
+ ih4 = optr( i, k, jc, L, M, sch, och ); // Im(ch[jc])
+
+ // Resolve input-workspace indices for component j:
+ ic1 = iptr( i-1, k, j, L, M, sc1, oc1 ); // Re(component j)
+ ic2 = iptr( i, k, j, L, M, sc1, oc1 ); // Im(component j)
+
+ // Resolve input-workspace indices for component jc:
+ ic3 = iptr( i-1, k, jc, L, M, sc1, oc1 ); // Re(component jc)
+ ic4 = iptr( i, k, jc, L, M, sc1, oc1 ); // Im(component jc)
+
+ // Form the sum and difference combinations:
+ c1[ ic1 ] = ch[ ih1 ] + ch[ ih3 ]; // Re(j) + Re(jc)
+ c1[ ic3 ] = ch[ ih2 ] - ch[ ih4 ]; // Im(j) - Im(jc)
+ c1[ ic2 ] = ch[ ih2 ] + ch[ ih4 ]; // Im(j) + Im(jc)
+ c1[ ic4 ] = ch[ ih3 ] - ch[ ih1 ]; // Re(jc) - Re(j)
+ }
+ }
+ }
+ }
+ }
+
+ // Combine the DC terms for each conjugate-symmetric component pair...
+ for ( j = 1; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+ for ( k = 0; k < L; k++ ) {
+ ih1 = optr( 0, k, j, L, M, sch, och );
+ ih2 = optr( 0, k, jc, L, M, sch, och );
+
+ ic1 = iptr( 0, k, j, L, M, sc1, oc1 );
+ ic2 = iptr( 0, k, jc, L, M, sc1, oc1 );
+
+ c1[ ic1 ] = ch[ ih1 ] + ch[ ih2 ];
+ c1[ ic2 ] = ch[ ih2 ] - ch[ ih1 ];
+ }
+ }
+
+ /*
+ * Next, apply the radix-`R` DFT matrix.
+ *
+ * For each mirrored output pair, accumulate the contributions of the radix components in the flattened work array.
+ *
+ * The required rotation factors are generated recursively from the previous ones:
+ *
+ * W_l = W_{l-1} ⋅ W_1
+ * W_{lj} = W_{l(j-1)} ⋅ W_l
+ *
+ * Here, `W_1` is the basic radix-`R` rotation, `W_{l-1}` and `W_l` are the previous and current outer factors, and `W_{l(j-1)}` and `W_{lj}` are the previous and current inner factors.
+ */
+ ar1 = 1.0; // Re(W_0)
+ ai1 = 0.0; // Im(W_0)
+ for ( l = 1; l < Rph; l++ ) {
+ lc = Rp1 - l - 1; // "mirror" index
+
+ // Advance the rotation factor by one step: W_l = W_{l-1} ⋅ W_1
+ ar1h = ( dcp * ar1 ) - ( dsp * ai1 ); // Re(W_l)
+ ai1 = ( dcp * ai1 ) + ( dsp * ar1 ); // Im(W_l)
+ ar1 = ar1h;
+
+ // Accumulate the contributions from components 1 and R-1:
+ for ( ik = 0; ik < ML; ik++ ) {
+ ich = fptr( ik, l, ML, sch2, och2 );
+ ic1o = fptr( ik, lc, ML, sch2, och2 );
+
+ ic2o = fptr( ik, 0, ML, sc2, oc2 );
+ ic1 = fptr( ik, 1, ML, sc2, oc2 );
+ ic2 = fptr( ik, R-1, ML, sc2, oc2 );
+
+ ch2[ ich ] = c2[ ic2o ] + ( ar1 * c2[ ic1 ] );
+ ch2[ ic1o ] = ai1 * c2[ ic2 ];
+ }
+
+ // Save W_l for the inner recurrence:
+ dc2 = ar1;
+ ds2 = ai1;
+
+ // Initialize W_{l·1}:
+ ar2 = ar1;
+ ai2 = ai1;
+
+ // Accumulate the remaining component pairs:
+ for ( j = 2; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+
+ // Advance the nested rotation factor by one step: W_{l*j} = W_{l*(j-1)} ⋅ W_l
+ ar2h = ( dc2 * ar2 ) - ( ds2 * ai2 ); // Re(W_{l*j})
+ ai2 = ( dc2 * ai2 ) + ( ds2 * ar2 ); // Im(W_{l*j})
+ ar2 = ar2h;
+
+ // Accumulate the contributions from components j and jc:
+ for ( ik = 0; ik < ML; ik++ ) {
+ ich = fptr( ik, l, ML, sch2, och2 );
+ ic1o = fptr( ik, lc, ML, sch2, och2 );
+
+ ic1 = fptr( ik, j, ML, sc2, oc2 );
+ ic2 = fptr( ik, jc, ML, sc2, oc2 );
+
+ ch2[ ich ] += ar2 * c2[ ic1 ];
+ ch2[ ic1o ] += ai2 * c2[ ic2 ];
+ }
+ }
+ }
+
+ // Accumulate the nonzero radix components into the DC output...
+ for ( j = 1; j < Rph; j++ ) {
+ for ( ik = 0; ik < ML; ik++ ) {
+ ich = fptr( ik, 0, ML, sch2, och2 );
+ ic1 = fptr( ik, j, ML, sc2, oc2 );
+ ch2[ ich ] += c2[ ic1 ];
+ }
+ }
+
+ // Copy the first output column from ch to cc...
+ if ( M >= L ) {
+ for ( k = 0; k < L; k++ ) {
+ for ( i = 0; i < M; i++ ) {
+ icc = optr( i, 0, k, R, M, sc, oc );
+ ih1 = optr( i, k, 0, L, M, sch, och );
+ cc[ icc ] = ch[ ih1 ];
+ }
+ }
+ } else {
+ for ( i = 0; i < M; i++ ) {
+ for ( k = 0; k < L; k++ ) {
+ icc = optr( i, 0, k, R, M, sc, oc );
+ ih1 = optr( i, k, 0, L, M, sch, och );
+ cc[ icc ] = ch[ ih1 ];
+ }
+ }
+ }
+
+ /*
+ * `cc` is stored in output order, where the component index comes before the sub-sequence index. Accordingly, writes to `cc` use `optr`, not `iptr`.
+ */
+ // Store DC harmonics for conjugate pairs in the output array...
+ for ( j = 1; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+ j2 = 2 * j; // output column index
+ for ( k = 0; k < L; k++ ) {
+ io1 = optr( M-1, j2-1, k, R, M, sc, oc );
+ io2 = optr( 0, j2, k, R, M, sc, oc );
+
+ ih1 = optr( 0, k, j, L, M, sch, och );
+ ih2 = optr( 0, k, jc, L, M, sch, och );
+
+ cc[ io1 ] = ch[ ih1 ];
+ cc[ io2 ] = ch[ ih2 ];
+ }
+ }
+
+ // When M = 1, there are no non-DC harmonics to process, so we're done...
+ if ( M === 1 ) {
+ return;
+ }
+
+ /*
+ * Finally, store the non-DC harmonics in folded format.
+ *
+ * For each mirror pair, write the non-DC harmonics from `ch` to the two output columns as the required sum and difference combinations.
+ */
+ if ( nbd >= L ) {
+ // Loop over sub-sequences before harmonics...
+ for ( j = 1; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+ j2 = 2 * j; // output column index
+ for ( k = 0; k < L; k++ ) {
+ for ( i = 2; i < M; i += 2 ) {
+ im = M - i; // "mirror" harmonic index
+
+ // Resolve ch indices for component j:
+ ih1 = optr( i-1, k, j, L, M, sch, och ); // Re(ch[j])
+ ih2 = optr( i, k, j, L, M, sch, och ); // Im(ch[j])
+
+ // Resolve ch indices for conjugate component jc:
+ ih3 = optr( i-1, k, jc, L, M, sch, och ); // Re(ch[jc])
+ ih4 = optr( i, k, jc, L, M, sch, och ); // Im(ch[jc])
+
+ // Resolve cc output indices:
+ io1 = optr( i-1, j2, k, R, M, sc, oc );
+ io2 = optr( im-1, j2-1, k, R, M, sc, oc );
+ io3 = optr( i, j2, k, R, M, sc, oc );
+ io4 = optr( im, j2-1, k, R, M, sc, oc );
+
+ // Store the sum and difference combinations:
+ cc[ io1 ] = ch[ ih1 ] + ch[ ih3 ];
+ cc[ io2 ] = ch[ ih1 ] - ch[ ih3 ];
+ cc[ io3 ] = ch[ ih2 ] + ch[ ih4 ];
+ cc[ io4 ] = ch[ ih4 ] - ch[ ih2 ];
+ }
+ }
+ }
+ } else {
+ // Loop over harmonics before sub-sequences...
+ for ( j = 1; j < Rph; j++ ) {
+ jc = Rp1 - j - 1; // "mirror" index
+ j2 = 2 * j; // output column index
+ for ( i = 2; i < M; i += 2 ) {
+ im = M - i; // "mirror" harmonic index
+ for ( k = 0; k < L; k++ ) {
+ // Resolve ch indices for component j:
+ ih1 = optr( i-1, k, j, L, M, sch, och ); // Re(ch[j])
+ ih2 = optr( i, k, j, L, M, sch, och ); // Im(ch[j])
+
+ // Resolve ch indices for conjugate component jc:
+ ih3 = optr( i-1, k, jc, L, M, sch, och ); // Re(ch[jc])
+ ih4 = optr( i, k, jc, L, M, sch, och ); // Im(ch[jc])
+
+ // Resolve cc output indices:
+ io1 = optr( i-1, j2, k, R, M, sc, oc );
+ io2 = optr( im-1, j2-1, k, R, M, sc, oc );
+ io3 = optr( i, j2, k, R, M, sc, oc );
+ io4 = optr( im, j2-1, k, R, M, sc, oc );
+
+ // Store the sum and difference combinations:
+ cc[ io1 ] = ch[ ih1 ] + ch[ ih3 ];
+ cc[ io2 ] = ch[ ih1 ] - ch[ ih3 ];
+ cc[ io3 ] = ch[ ih2 ] + ch[ ih4 ];
+ cc[ io4 ] = ch[ ih4 ] - ch[ ih2 ];
+ }
+ }
+ }
+ }
+}
+
+
+// EXPORTS //
+
+module.exports = radfg;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/rfftf1.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/rfftf1.js
new file mode 100644
index 000000000000..01854653d35d
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/lib/rfftf1.js
@@ -0,0 +1,182 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [PFFFT library]{@link https://github.com/marton78/pffft/blob/0b4ee12c4ba45a4a8e567550c16d96d1679f50ce/src/fftpack.c}. The implementation follows the original, but has been modified for JavaScript.
+*
+* ```text
+* Copyright (c) 2004 the University Corporation for Atmospheric
+* Research ("UCAR"). All rights reserved. Developed by NCAR's
+* Computational and Information Systems Laboratory, UCAR,
+* www.cisl.ucar.edu.
+*
+* Redistribution and use of the Software in source and binary forms,
+* with or without modification, is permitted provided that the
+* following conditions are met:
+*
+* - Neither the names of NCAR's Computational and Information Systems
+* Laboratory, the University Corporation for Atmospheric Research,
+* nor the names of its sponsors or contributors may be used to
+* endorse or promote products derived from this Software without
+* specific prior written permission.
+*
+* - Redistributions of source code must retain the above copyright
+* notices, this list of conditions, and the disclaimer below.
+*
+* - Redistributions in binary form must reproduce the above copyright
+* notice, this list of conditions, and the disclaimer below in the
+* documentation and/or other materials provided with the
+* distribution.
+*
+* THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+* EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+* NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+* HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+* EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+* SOFTWARE.
+* ```
+*/
+
+/* eslint-disable max-len */
+
+'use strict';
+
+// MODULES //
+
+var floor = require( '@stdlib/math/base/special/floor' );
+var dcopy = require( '@stdlib/blas/base/dcopy' ).ndarray;
+var radf2 = require( './radf2.js' );
+var radf3 = require( './radf3.js' );
+var radf4 = require( './radf4.js' );
+var radf5 = require( './radf5.js' );
+var radfg = require( './radfg.js' );
+
+
+// MAIN //
+
+/**
+* Performs the forward real-valued Fourier transform.
+*
+* @private
+* @param {NonNegativeInteger} N - length of the sequence to transform
+* @param {Collection} c - input array containing the sequence to be transformed
+* @param {integer} sc - stride length for `c`
+* @param {NonNegativeInteger} oc - starting index for `c`
+* @param {Collection} ch - working array for intermediate results
+* @param {integer} sch - stride length for `ch`
+* @param {NonNegativeInteger} och - starting index for `ch`
+* @param {Collection} wa - workspace array for storing twiddle factors
+* @param {integer} swa - stride length for `wa`
+* @param {NonNegativeInteger} owa - starting index for `wa`
+* @param {Collection} ifac - workspace array for storing factorization results
+* @param {integer} si - stride length for `ifac`
+* @param {NonNegativeInteger} oi - starting index for `ifac`
+* @returns {void}
+*/
+function rfftf1( N, c, sc, oc, ch, sch, och, wa, swa, owa, ifac, si, oi ) { // eslint-disable-line max-params
+ var factor;
+ var idl1;
+ var FLG;
+ var ido;
+ var ix4;
+ var ix3;
+ var ix2;
+ var nf;
+ var l2;
+ var l1;
+ var kh;
+ var iw;
+ var k1;
+
+ // Resolve the number of factors:
+ nf = ifac[ oi + si ];
+
+ FLG = 1; // Flag to track whether the latest stage output resides in `c` or `ch`.
+ l2 = N;
+
+ // Initialize an index offset to the last element in each workspace:
+ iw = N - 1;
+
+ for ( k1 = 1; k1 <= nf; k1++ ) {
+ kh = nf - k1;
+
+ // Resolve the next factor:
+ factor = ifac[ oi + ( ( kh + 2 ) * si ) ];
+
+ // Compute a sub-transform length:
+ l1 = floor( l2 / factor );
+
+ ido = floor( N / l2 );
+ idl1 = ido * l1;
+
+ // Adjust the index offset
+ iw -= ( ( factor - 1 ) * ido );
+
+ // Toggle the flag to swap the input and output work buffers for the next stage:
+ FLG = 1 - FLG;
+
+ switch ( factor ) {
+ case 4:
+ ix2 = iw + ido;
+ ix3 = ix2 + ido;
+ radf4( ido, l1, ( FLG ) ? ch : c, ( FLG ) ? sch : sc, ( FLG ) ? och : oc, ( FLG ) ? c : ch, ( FLG ) ? sc : sch, ( FLG ) ? oc : och, wa, swa, owa + ( ( iw + 1 ) * swa ), wa, swa, owa + ( ( ix2 + 1 ) * swa ), wa, swa, owa + ( ( ix3 + 1 ) * swa ) );
+ break;
+ case 2:
+ radf2( ido, l1, ( FLG ) ? ch : c, ( FLG ) ? sch : sc, ( FLG ) ? och : oc, ( FLG ) ? c : ch, ( FLG ) ? sc : sch, ( FLG ) ? oc : och, wa, swa, owa + ( ( iw + 1 ) * swa ) );
+ break;
+ case 3:
+ ix2 = iw + ido;
+ radf3( ido, l1, ( FLG ) ? ch : c, ( FLG ) ? sch : sc, ( FLG ) ? och : oc, ( FLG ) ? c : ch, ( FLG ) ? sc : sch, ( FLG ) ? oc : och, wa, swa, owa + ( ( iw + 1 ) * swa ), wa, swa, owa + ( ( ix2 + 1 ) * swa ) );
+ break;
+ case 5:
+ ix2 = iw + ido;
+ ix3 = ix2 + ido;
+ ix4 = ix3 + ido;
+ radf5( ido, l1, ( FLG ) ? ch : c, ( FLG ) ? sch : sc, ( FLG ) ? och : oc, ( FLG ) ? c : ch, ( FLG ) ? sc : sch, ( FLG ) ? oc : och, wa, swa, owa + ( ( iw + 1 ) * swa ), wa, swa, owa + ( ( ix2 + 1 ) * swa ), wa, swa, owa + ( ( ix3 + 1 ) * swa ), wa, swa, owa + ( ( ix4 + 1 ) * swa ) );
+ break;
+ default:
+ if ( ido === 1 ) {
+ FLG = 1 - FLG;
+ }
+ if ( FLG === 0 ) {
+ radfg( ido, factor, l1, idl1, c, sc, oc, c, sc, oc, c, sc, oc, ch, sch, och, ch, sch, och, wa, swa, owa + ( ( iw + 1 ) * swa ) );
+ FLG = 1;
+ } else {
+ radfg( ido, factor, l1, idl1, ch, sch, och, ch, sch, och, ch, sch, och, c, sc, oc, c, sc, oc, wa, swa, owa + ( ( iw + 1 ) * swa ) );
+ FLG = 0;
+ }
+ break;
+ }
+ l2 = l1;
+ }
+ if ( FLG === 1 ) {
+ return;
+ }
+
+ // Now that we've finished computing the transforms, copy over the final results to the input array...
+ dcopy( N, ch, sch, och, c, sc, oc );
+}
+
+
+// EXPORTS //
+
+module.exports = rfftf1;
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/package.json b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/package.json
new file mode 100644
index 000000000000..1ef3db1b49a7
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/package.json
@@ -0,0 +1,62 @@
+{
+ "name": "@stdlib/fft/base/fftpack/rfftf",
+ "version": "0.0.0",
+ "description": "Compute the forward real-valued fast Fourier transform (FFT).",
+ "license": "Apache-2.0",
+ "author": {
+ "name": "The Stdlib Authors",
+ "url": "https://github.com/stdlib-js/stdlib/graphs/contributors"
+ },
+ "contributors": [
+ {
+ "name": "The Stdlib Authors",
+ "url": "https://github.com/stdlib-js/stdlib/graphs/contributors"
+ }
+ ],
+ "main": "./lib",
+ "directories": {
+ "benchmark": "./benchmark",
+ "doc": "./docs",
+ "example": "./examples",
+ "lib": "./lib",
+ "test": "./test"
+ },
+ "types": "./docs/types",
+ "scripts": {},
+ "homepage": "https://github.com/stdlib-js/stdlib",
+ "repository": {
+ "type": "git",
+ "url": "git://github.com/stdlib-js/stdlib.git"
+ },
+ "bugs": {
+ "url": "https://github.com/stdlib-js/stdlib/issues"
+ },
+ "dependencies": {},
+ "devDependencies": {},
+ "engines": {
+ "node": ">=0.10.0",
+ "npm": ">2.7.0"
+ },
+ "os": [
+ "aix",
+ "darwin",
+ "freebsd",
+ "linux",
+ "macos",
+ "openbsd",
+ "sunos",
+ "win32",
+ "windows"
+ ],
+ "keywords": [
+ "stdlib",
+ "fft",
+ "fftpack",
+ "rfftf",
+ "fourier",
+ "transform",
+ "forward",
+ "real"
+ ],
+ "__stdlib__": {}
+}
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/Makefile b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/Makefile
new file mode 100644
index 000000000000..4b8cac6acbd2
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/Makefile
@@ -0,0 +1,138 @@
+#/
+# @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.
+#/
+
+# VARIABLES #
+
+ifndef VERBOSE
+ QUIET := @
+endif
+
+# Specify the path to FFTPACK:
+FFTPACK ?=
+
+# Specify a list of FFTPACK source files:
+FFTPACK_SRC ?=
+
+# Determine the OS:
+#
+# [1]: https://en.wikipedia.org/wiki/Uname#Examples
+# [2]: http://stackoverflow.com/a/27776822/2225624
+OS ?= $(shell uname)
+ifneq (, $(findstring MINGW,$(OS)))
+ OS := WINNT
+else
+ifneq (, $(findstring MSYS,$(OS)))
+ OS := WINNT
+else
+ifneq (, $(findstring CYGWIN,$(OS)))
+ OS := WINNT
+endif
+endif
+endif
+
+# Define the program used for compiling C source files:
+ifdef C_COMPILER
+ CC := $(C_COMPILER)
+else
+ CC := gcc
+endif
+
+# Define the command-line options when compiling C files:
+CFLAGS ?= \
+ -std=c99 \
+ -O3 \
+ -ffp-contract=off \
+ -Wall \
+ -pedantic
+
+# Determine whether to generate [position independent code][1]:
+#
+# [1]: https://gcc.gnu.org/onlinedocs/gcc/Code-Gen-Options.html#Code-Gen-Options
+# [2]: http://stackoverflow.com/questions/5311515/gcc-fpic-option
+ifeq ($(OS), WINNT)
+ fPIC ?=
+else
+ fPIC ?= -fPIC
+endif
+
+# List of C targets:
+c_targets := runner.out
+
+
+# RULES #
+
+#/
+# Compiles C source files.
+#
+# @param {string} [C_COMPILER] - C compiler
+# @param {string} [CFLAGS] - C compiler flags
+# @param {(string|void)} [fPIC] - flag indicating whether to generate position independent code
+#
+# @example
+# make
+#
+# @example
+# make all
+#/
+all: $(c_targets)
+
+.PHONY: all
+
+#/
+# Compiles C source files.
+#
+# @private
+# @param {string} CC - C compiler
+# @param {string} CFLAGS - C compiler flags
+# @param {(string|void)} fPIC - flag indicating whether to generate position independent code
+#/
+$(c_targets): %.out: %.c
+ $(QUIET) $(CC) $(CFLAGS) -DFFTPACK_DOUBLE_PRECISION $(fPIC) -o $@ $(FFTPACK_SRC) $< -lm
+
+#/
+# Generates test fixtures.
+#
+# @example
+# make run
+#/
+run: $(c_targets)
+ $(QUIET) ./$<
+
+.PHONY: run
+
+#/
+# Removes generated files.
+#
+# @example
+# make clean
+#/
+clean:
+ $(QUIET) -rm -f *.o *.out
+
+.PHONY: clean
+
+#/
+# Removes generated test fixtures.
+#
+# @example
+# make clean-fixtures
+#/
+clean-fixtures:
+ $(QUIET) -rm -f *.json
+
+.PHONY: clean-fixtures
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/large.json b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/large.json
new file mode 100644
index 000000000000..bb503d1af6b9
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/large.json
@@ -0,0 +1 @@
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diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/medium.json b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/medium.json
new file mode 100644
index 000000000000..6f956976e745
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/medium.json
@@ -0,0 +1 @@
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diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/runner.c b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/runner.c
new file mode 100644
index 000000000000..83735d958f19
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/runner.c
@@ -0,0 +1,318 @@
+/**
+* @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.
+*/
+
+/**
+* Generate FFTPACK test fixtures.
+*
+* ## Notes
+*
+* - Run this script from the directory in which fixtures should be written.
+*
+*/
+
+#include
+#include
+#include
+
+/**
+* Define prototypes for external functions.
+*/
+extern void rffti( int n, double *wsave );
+extern void rfftf( int n, double *r, double *wsave );
+
+/**
+* Generates a random number on the interval [0,1].
+*
+* @return random number
+*/
+double rand_double( void ) {
+ int r = rand();
+ return (double)r / ( (double)RAND_MAX + 1.0 );
+}
+
+/**
+* Generates an array of pseudorandom integers drawn from a uniform distribution.
+*
+* ## Notes
+*
+* - WARNING: the method used here is not particularly robust, as some integer values may be sampled more frequently than others.
+*
+*
+* @param out output array
+* @param len array length
+* @param a lower bound (inclusive)
+* @param b upper bound (exclusive)
+*/
+void rand_array_i32( int *out, const unsigned int len, const int a, const int b ) {
+ unsigned int i;
+ unsigned int r;
+ double delta;
+
+ delta = (double)b - (double)a;
+
+ for ( i = 0; i < len; i++ ) {
+ r = (unsigned int)( delta * rand_double() ); // truncation
+ out[ i ] = (int)( a + r );
+ }
+}
+
+/**
+* Generates an array of pseudorandom numbers drawn from a uniform distribution.
+*
+* @param out output array
+* @param len array length
+* @param a lower bound (inclusive)
+* @param b upper bound (exclusive)
+*/
+void rand_array_f64( double *out, const unsigned int len, const double a, const double b ) {
+ unsigned int i;
+ double delta;
+
+ delta = b - a;
+
+ for ( i = 0; i < len; i++ ) {
+ out[ i ] = a + ( rand_double()*delta );
+ }
+}
+
+/**
+* Writes an array of doubles to a file as a series of comma-separated values.
+*
+* @param f file to write to
+* @param x array of doubles
+* @param len array length
+*/
+void write_array_f64( FILE *f, const double *x, const unsigned int len ) {
+ unsigned int i;
+
+ for ( i = 0; i < len; i++ ) {
+ fprintf( f, "%.17g", x[ i ] );
+ if ( i < len-1 ) {
+ fprintf( f, "," );
+ }
+ }
+}
+
+/**
+* Writes an array of integers to a file as a series of comma-separated values.
+*
+* @param f file to write to
+* @param x array of integers
+* @param len array length
+*/
+void write_array_i32( FILE *f, const int *x, const unsigned int len ) {
+ unsigned int i;
+
+ for ( i = 0; i < len; i++ ) {
+ fprintf( f, "%d", x[ i ] );
+ if ( i < len-1 ) {
+ fprintf( f, "," );
+ }
+ }
+}
+
+/**
+* Writes a named array of doubles to a file as JSON.
+*
+* @param f file to write to
+* @param name array name
+* @param x data
+* @param len array length
+*/
+void write_named_array_f64( FILE *f, const char *name, const double *x, const unsigned int len ) {
+ fprintf( f, "\"%s\":[", name );
+ write_array_f64( f, x, len );
+ fprintf( f, "]" );
+}
+
+/**
+* Writes a named array of integers to a file as JSON.
+*
+* @param f file to write to
+* @param name array name
+* @param x data
+* @param len array length
+*/
+void write_named_array_i32( FILE *f, const char *name, const int *x, const unsigned int len ) {
+ fprintf( f, "\"%s\":[", name );
+ write_array_i32( f, x, len );
+ fprintf( f, "]" );
+}
+
+/**
+* Writes data to a file as JSON.
+*
+* ## Notes
+*
+* - This function SHOULD be tailored to the input data (e.g., input types, output types, number of arguments, etc) and may vary from use case to use case.
+*
+*
+* @param f file to write to
+* @param lengths sequence lengths
+* @param offsets offsets
+* @param input input values
+* @param expected results
+* @param num number of sequence lengths
+* @param total total array length
+*/
+void write_data_as_json( FILE *f, const int *lengths, const int *offsets, const double *input, const double *expected, const unsigned int num, const unsigned int total ) {
+ fprintf( f, "{" );
+ write_named_array_i32( f, "lengths", lengths, num );
+ fprintf( f, "," );
+ write_named_array_i32( f, "offsets", offsets, num );
+ fprintf( f, "," );
+ write_named_array_f64( f, "input", input, total );
+ fprintf( f, "," );
+ write_named_array_f64( f, "expected", expected, total );
+ fprintf( f, "}\n" );
+}
+
+/**
+* Generates test fixtures.
+*
+* @param lengths sequence lengths
+* @param offsets input offsets into flat arrays
+* @param num number of sequence lengths
+* @param total total number of input and output values
+* @param name output filename
+*/
+void generate( const int *lengths, const int *offsets, const unsigned int num, const unsigned int total, const char *name ) {
+ unsigned int i;
+ unsigned int j;
+ double *expected;
+ double *input;
+ double *wsave;
+ FILE *f;
+ int off;
+ int n;
+
+ // Allocate output arrays:
+ input = (double*) malloc( total * sizeof(double) );
+ if ( input == NULL ) {
+ printf( "Error allocating memory.\n" );
+ exit( 1 );
+ }
+ expected = (double*) malloc( total * sizeof(double) );
+ if ( expected == NULL ) {
+ printf( "Error allocating memory.\n" );
+ exit( 1 );
+ }
+
+ // Generate fixture data:
+ for ( i = 0; i < num; i++ ) {
+ n = lengths[ i ];
+ off = offsets[ i ];
+
+ rand_array_f64( input + off, (unsigned int)n, -10.0, 10.0 );
+ for ( j = 0; j < (unsigned int)n; j++ ) {
+ expected[ off + j ] = input[ off + j ];
+ }
+
+ wsave = (double*) calloc( 2*n + 15, sizeof(double) );
+ if ( wsave == NULL ) {
+ printf( "Error allocating memory.\n" );
+ exit( 1 );
+ }
+ rffti( n, wsave );
+ rfftf( n, expected + off, wsave );
+ free( wsave );
+ }
+ // Open a new file:
+ f = fopen( name, "w" );
+ if ( f == NULL ) {
+ printf( "Error opening file.\n" );
+ exit( 1 );
+ }
+
+ // Write data as JSON:
+ write_data_as_json( f, lengths, offsets, input, expected, num, total );
+
+ // Close the file:
+ fclose( f );
+
+ // Free allocated memory:
+ free( input );
+ free( expected );
+}
+
+/**
+* Computes offsets into flat input and output arrays for each sequence length.
+*
+* @param offsets output array of offsets
+* @param lengths sequence lengths
+* @param num number of sequence lengths
+* @return total number of values
+*/
+unsigned int compute_offsets( int *offsets, const int *lengths, const unsigned int num ) {
+ unsigned int total;
+ unsigned int i;
+
+ total = 0;
+ for ( i = 0; i < num; i++ ) {
+ offsets[ i ] = total;
+ total += lengths[ i ];
+ }
+ return total;
+}
+
+/**
+* Main execution sequence.
+*/
+int main( void ) {
+ unsigned int total;
+ unsigned int num;
+ int *lengths;
+ int *offsets;
+
+ // Use the current time to seed the random number generator:
+ srand( time( NULL ) );
+
+ // Define the number of sequence lengths per range:
+ num = 10;
+
+ // Allocate arrays:
+ lengths = (int*) malloc( num * sizeof(int) );
+ if ( lengths == NULL ) {
+ printf( "Error allocating memory.\n" );
+ exit( 1 );
+ }
+ offsets = (int*) malloc( num * sizeof(int) );
+ if ( offsets == NULL ) {
+ printf( "Error allocating memory.\n" );
+ exit( 1 );
+ }
+
+ // Generate fixture data:
+ rand_array_i32( lengths, num, 2, 16 );
+ total = compute_offsets( offsets, lengths, num );
+ generate( lengths, offsets, num, total, "small.json" );
+
+ rand_array_i32( lengths, num, 16, 128 );
+ total = compute_offsets( offsets, lengths, num );
+ generate( lengths, offsets, num, total, "medium.json" );
+
+ rand_array_i32( lengths, num, 128, 1024 );
+ total = compute_offsets( offsets, lengths, num );
+ generate( lengths, offsets, num, total, "large.json" );
+
+ // Free allocated memory:
+ free( lengths );
+ free( offsets );
+
+ return 0;
+}
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/small.json b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/small.json
new file mode 100644
index 000000000000..8d557d1163b5
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/fixtures/c/fftpack/small.json
@@ -0,0 +1 @@
+{"lengths":[4,13,5,5,15,10,10,2,10,15],"offsets":[0,4,17,22,27,42,52,62,64,74],"input":[-9.116728724911809,-4.8596726823598146,3.4812678024172783,9.668060727417469,-8.9032003656029701,3.9114638883620501,-0.026319427415728569,-2.3505385220050812,-5.5008794739842415,6.7187159508466721,1.4591167215257883,3.3748283609747887,0.74036757461726665,3.3579106442630291,-3.5956973303109407,7.115019578486681,2.1341895684599876,9.3241720646619797,-8.6399579886347055,8.2260956522077322,-4.010230703279376,0.05261685699224472,4.3315941374748945,1.1027806997299194,-5.564692746847868,-5.7909615617245436,-8.6909349635243416,-8.5439217090606689,2.3078472074121237,7.9881112929433584,-3.8133587222546339,8.8800034765154123,6.2185775488615036,-4.3670093547552824,3.6738187074661255,5.8711233921349049,-4.0290241781622171,4.1906843520700932,-7.1679836977273226,7.6980144530534744,0.52905097603797913,-8.2401633262634277,7.5749892555177212,-7.1554449666291475,-1.5635318774729967,1.7198013328015804,4.7010921128094196,-8.7447449564933777,7.0715260319411755,-8.8618475571274757,-1.0718837380409241,4.8500846140086651,-4.6277761366218328,0.96651383675634861,4.1981401853263378,-1.8577941041439772,-3.9454446267336607,8.9122058637440205,7.4440999515354633,-7.0119780208915472,9.685426251962781,2.9591707978397608,-5.2162992861121893,9.6579357422888279,0.92617449350655079,6.2147978693246841,-7.8920833580195904,-2.2449817415326834,8.5919307451695204,4.5801795646548271,-0.92194274067878723,4.9084284529089928,-4.0428752824664116,-8.6048257909715176,-1.3070579431951046,-7.7227832470089197,3.1819853372871876,-0.37233305163681507,2.1984764840453863,9.7943628113716841,-6.1440743599087,-3.4577368106693029,5.8174742758274078,-5.7097223773598671,-3.3039627131074667,-9.7012667916715145,-9.1909652855247259,7.446452509611845,-7.4725344125181437],"expected":[-0.8270728774368763,-12.597996527329087,14.527733409777284,-10.443848967552185,8.4349771682173014,-9.7567153511884861,8.5923802120769253,4.57425773203913,4.0866333108592272,-14.22715060271884,-6.6576554982603335,-26.372765919953665,2.8296022954701328,-8.980587583926571,-6.8431158070115705,-7.3253292347795274,-19.352006438403265,4.952695881947875,3.259831328816762,1.0747977763238268,17.574250891864249,16.746805214608187,-14.612214434891939,11.173642831905873,-9.4473745727180862,6.9614497292273327,-5.541407201183806,11.195770418271422,-14.991257708546126,-15.946231614575851,-17.300471915627686,-4.6130790288011587,-19.788622037389842,-4.3067661197842559,-5.5157193718121711,-30.002457820459789,-25.180834475904703,-12.913945229888764,18.854087307931454,19.383661310646723,-5.7544798257416492,19.640630053430741,-1.47995974868536,6.3230358066860246,-1.1461816305236983,9.6659206372041417,15.672241202115009,17.024128064244636,19.603107401239232,-11.850330015300589,-4.8679641623852099,34.904343318194151,16.722563998773694,-6.1633397371209089,8.1826596782807925,2.5224321416795537,-2.68207208227156,-32.079778889875975,12.531437445164677,-0.17263982326092409,-15.643991329488166,8.7863272521644831,4.4416364561766386,-14.874235028401017,1.5148022118955851,-16.303884263575817,-3.8387589572180141,14.638762167581284,-6.9882295533190071,11.265068834141644,-29.609707780796953,-1.6302781281256387,0.023841251436460542,-8.1923944968730211,-25.943685574457049,-6.2112734554560713,-26.110422946311445,-2.2624743686374584,8.8755320203980546,-8.2645672629549143,25.11030762650708,-12.000561912417531,-15.582412082646586,-21.114386739209294,5.5941560454209212,34.194765437856233,-13.8671250199453,18.827406514084274,-0.55489016140835545]}
diff --git a/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/test.js b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/test.js
new file mode 100644
index 000000000000..344e70a53fef
--- /dev/null
+++ b/lib/node_modules/@stdlib/fft/base/fftpack/rfftf/test/test.js
@@ -0,0 +1,387 @@
+/**
+* @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 tape = require( 'tape' );
+var Float64Array = require( '@stdlib/array/float64' );
+var isAlmostSameValue = require( '@stdlib/number/float64/base/assert/is-almost-same-value' );
+var rffti = require( '@stdlib/fft/base/fftpack/rffti' );
+var rfftf = require( './../lib' );
+
+
+// FIXTURES //
+
+var small = require( './fixtures/c/fftpack/small.json' );
+var medium = require( './fixtures/c/fftpack/medium.json' );
+var large = require( './fixtures/c/fftpack/large.json' );
+
+
+// TESTS //
+
+tape( 'main export is a function', function test( t ) {
+ t.ok( true, __filename );
+ t.strictEqual( typeof rfftf, 'function', 'main export is a function' );
+ t.end();
+});
+
+tape( 'the function has an arity of 7', function test( t ) {
+ t.strictEqual( rfftf.length, 7, 'returns expected value' );
+ t.end();
+});
+
+tape( 'the function returns a reference to the input array', function test( t ) {
+ var workspace;
+ var out;
+ var N;
+ var r;
+
+ N = 4;
+ r = new Float64Array( [ 1.0, 2.0, 3.0, 4.0 ] );
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ out = rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ t.strictEqual( out, r, 'same reference' );
+ t.end();
+});
+
+tape( 'the function returns all zeros for a zero sequence', function test( t ) {
+ var workspace;
+ var expected;
+ var N;
+ var r;
+
+ N = 4;
+ r = new Float64Array( [ 0.0, 0.0, 0.0, 0.0 ] );
+ expected = new Float64Array( [ 0.0, 0.0, 0.0, 0.0 ] );
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ t.deepEqual( r, expected, 'returns expected value' );
+ t.end();
+});
+
+tape( 'the function returns only a zero-frequency term for a constant sequence', function test( t ) {
+ var workspace;
+ var expected;
+ var N;
+ var r;
+
+ N = 4;
+ r = new Float64Array( [ 1.0, 1.0, 1.0, 1.0 ] );
+ expected = new Float64Array( [ 4.0, 0.0, 0.0, 0.0 ] );
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ t.deepEqual( r, expected, 'returns expected value' );
+ t.end();
+});
+
+tape( 'the function computes the exact two-point real FFT', function test( t ) {
+ var workspace;
+ var expected;
+ var N;
+ var r;
+
+ N = 2;
+ r = new Float64Array( [ 3.0, -1.0 ] );
+
+ // For [ a, b ], output is [ a+b, a-b ]...
+ expected = new Float64Array( [ 2.0, 4.0 ] );
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ t.deepEqual( r, expected, 'returns expected value' );
+ t.end();
+});
+
+tape( 'the function computes a forward real-valued FFT (small sequence lengths)', function test( t ) {
+ var workspace;
+ var expected;
+ var lengths;
+ var offsets;
+ var input;
+ var ulps;
+ var off;
+ var y;
+ var e;
+ var N;
+ var i;
+ var k;
+ var r;
+
+ lengths = small.lengths;
+ offsets = small.offsets;
+ input = small.input;
+ expected = small.expected;
+
+ ulps = 20;
+ for ( k = 0; k < lengths.length; k++ ) {
+ N = lengths[ k ];
+ off = offsets[ k ];
+ r = new Float64Array( N );
+ for ( i = 0; i < N; i++ ) {
+ r[ i ] = input[ off + i ];
+ }
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ for ( i = 0; i < N; i++ ) {
+ y = r[ i ];
+ e = expected[ off + i ];
+ t.strictEqual( isAlmostSameValue( y, e, ulps ), true, 'within tolerance. N: '+N+'. index: '+i+'. y: '+y+'. E: '+e+'.' );
+ }
+ }
+ t.end();
+});
+
+tape( 'the function computes a forward real-valued FFT (medium sequence lengths)', function test( t ) {
+ var workspace;
+ var expected;
+ var lengths;
+ var offsets;
+ var input;
+ var ulps;
+ var off;
+ var y;
+ var e;
+ var N;
+ var i;
+ var k;
+ var r;
+
+ lengths = medium.lengths;
+ offsets = medium.offsets;
+ input = medium.input;
+ expected = medium.expected;
+
+ ulps = 130;
+ for ( k = 0; k < lengths.length; k++ ) {
+ N = lengths[ k ];
+ off = offsets[ k ];
+ r = new Float64Array( N );
+ for ( i = 0; i < N; i++ ) {
+ r[ i ] = input[ off + i ];
+ }
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ for ( i = 0; i < N; i++ ) {
+ y = r[ i ];
+ e = expected[ off + i ];
+ t.strictEqual( isAlmostSameValue( y, e, ulps ), true, 'within tolerance. N: '+N+'. index: '+i+'. y: '+y+'. E: '+e+'.' );
+ }
+ }
+ t.end();
+});
+
+tape( 'the function computes a forward real-valued FFT (large sequence lengths)', function test( t ) {
+ var workspace;
+ var expected;
+ var lengths;
+ var offsets;
+ var input;
+ var ulps;
+ var off;
+ var y;
+ var e;
+ var N;
+ var i;
+ var k;
+ var r;
+
+ lengths = large.lengths;
+ offsets = large.offsets;
+ input = large.input;
+ expected = large.expected;
+
+ ulps = 800;
+ for ( k = 0; k < lengths.length; k++ ) {
+ N = lengths[ k ];
+ off = offsets[ k ];
+ r = new Float64Array( N );
+ for ( i = 0; i < N; i++ ) {
+ r[ i ] = input[ off + i ];
+ }
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ rfftf( N, r, 1, 0, workspace, 1, 0 );
+
+ for ( i = 0; i < N; i++ ) {
+ y = r[ i ];
+ e = expected[ off + i ];
+ t.strictEqual( isAlmostSameValue( y, e, ulps ), true, 'within tolerance. N: '+N+'. index: '+i+'. y: '+y+'. E: '+e+'.' );
+ }
+ }
+ t.end();
+});
+
+tape( 'the function leaves a length-one sequence unchanged', function test( t ) {
+ var workspace;
+ var out;
+ var N;
+ var r;
+
+ N = 1;
+ r = new Float64Array( [ 5.0 ] );
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+
+ rffti( N, workspace, 1, 0 );
+ out = rfftf( N, r, 1, 0, workspace, 1, 0 );
+ t.strictEqual( out, r, 'same reference' );
+ t.strictEqual( r[ 0 ], 5.0, 'returns expected value' );
+ t.end();
+});
+
+tape( 'the function supports specifying a stride', function test( t ) {
+ var workspace;
+ var expected;
+ var v;
+ var i;
+ var N;
+ var r;
+
+ N = 4;
+ r = new Float64Array([
+ 1.0, // 0
+ -999.0,
+ 2.0, // 1
+ -999.0,
+ 3.0, // 2
+ -999.0,
+ 4.0, // 3
+ -999.0
+ ]);
+ expected = new Float64Array([
+ 10.0, // 0
+ -999.0,
+ -2.0, // 1
+ -999.0,
+ 2.0, // 2
+ -999.0,
+ -2.0, // 3
+ -999.0
+ ]);
+
+ workspace = new Float64Array( ( ( 2 * N ) + 34 ) * 2 );
+ rffti( N, workspace, 2, 0 );
+ rfftf( N, r, 2, 0, workspace, 2, 0 );
+
+ for ( i = 0; i < r.length; i++ ) {
+ v = r[ i ];
+ t.strictEqual( isAlmostSameValue( v, expected[ i ], 1 ), true, 'within tolerance. index: '+i+'. y: '+v+'. E: '+expected[ i ]+'.' );
+ }
+ t.end();
+});
+
+tape( 'the function supports specifying a negative stride', function test( t ) {
+ var workspace;
+ var expected;
+ var v;
+ var i;
+ var N;
+ var r;
+
+ N = 4;
+ r = new Float64Array([
+ 4.0, // 3
+ -999.0,
+ 3.0, // 2
+ -999.0,
+ 2.0, // 1
+ -999.0,
+ 1.0 // 0
+ ]);
+ expected = new Float64Array([
+ -2.0, // 3
+ -999.0,
+ 2.0, // 2
+ -999.0,
+ -2.0, // 1
+ -999.0,
+ 10.0 // 0
+ ]);
+
+ workspace = new Float64Array( ( 2 * N ) + 34 );
+ rffti( N, workspace, -1, workspace.length-1 );
+ rfftf( N, r, -2, r.length-1, workspace, -1, workspace.length-1 );
+
+ for ( i = 0; i < r.length; i++ ) {
+ v = r[ i ];
+ t.strictEqual( isAlmostSameValue( v, expected[ i ], 1 ), true, 'within tolerance. index: '+i+'. y: '+v+'. E: '+expected[ i ]+'.' );
+ }
+ t.end();
+});
+
+tape( 'the function supports an offset parameter', function test( t ) {
+ var workspace;
+ var expected;
+ var v;
+ var i;
+ var N;
+ var r;
+
+ N = 4;
+ r = new Float64Array([
+ -999.0,
+ 1.0, // 0
+ -999.0,
+ 2.0, // 1
+ -999.0,
+ 3.0, // 2
+ -999.0,
+ 4.0, // 3
+ -999.0
+ ]);
+ expected = new Float64Array([
+ -999.0,
+ 10.0, // 0
+ -999.0,
+ -2.0, // 1
+ -999.0,
+ 2.0, // 2
+ -999.0,
+ -2.0, // 3
+ -999.0
+ ]);
+
+ workspace = new Float64Array( 3 + ( ( ( 2 * N ) + 34 ) * 2 ) );
+ rffti( N, workspace, 2, 3 );
+ rfftf( N, r, 2, 1, workspace, 2, 3 );
+
+ for ( i = 0; i < r.length; i++ ) {
+ v = r[ i ];
+ t.strictEqual( isAlmostSameValue( v, expected[ i ], 1 ), true, 'within tolerance. index: '+i+'. y: '+v+'. E: '+expected[ i ]+'.' );
+ }
+ t.end();
+});