feat: add lapack/base/dlarf1f

lib/node_modules/@stdlib/lapack/base/dlarf1f/README.md

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2025 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.
+
+-->
+
+# dlarf1f
+
+> Apply a real elementary reflector `H = I - tau * v * v ^ T` to a real M by N matrix `C`.
+
+<section class="intro">
+
+In linear algebra, a `Householder transformation` (or an `elementary reflector`) is a linear transformation that describes a reflection about a plane or a hyperplane containing the origin.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dlarf1f = require( '@stdlib/lapack/base/dlarf1f' );
+```
+
+#### dlarf1f( order, side, M, N, V, incv, tau, C, LDC, work )
+
+Applies a real elementary reflector `H = I - tau * v * v ^ T` to a real M by N matrix `C`.
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
+var work = new Float64Array( 3 );
+
+var out = dlarf1f( 'row-major', 'left', 4, 3, V, 1, 1.0, C, 3, work );
+// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
+```
+
+The function has the following parameters:
+
+-   **order**: storage layout.
+-   **side**: specifies the side of multiplication with `C`. Use `left` to form `H * C` and `right` to from `C * H`.
+-   **M**: number of rows in `C`.
+-   **N**: number of columns in `C`.
+-   **V**: the vector `v` in the representation of `H` as a [`Float64Array`][mdn-float64array].
+-   **incv**: stride length for `V`. If `incv` is negative, the elements of `V` are accessed in reverse order.
+-   **tau**: the value of `tau` in representation of `H`.
+-   **C**: input matrix stored in linear memory as a [`Float64Array`][mdn-float64array].
+-   **LDC**: stride of the first dimension of `C` (a.k.a., leading dimension of the matrix `C`).
+-   **work**: workspace [`Float64Array`][mdn-float64array].
+
+Based on the `side` of multiplication, the expected storage of the array changes:
+
+-   `work` should have `N` indexed elements if `side` = `left` and `M` indexed elements if `side` = `right`.
+-   `V` should have `1 + (M-1) * abs(incv)` indexed elements if `side` = `left` and `1 + (N-1) * abs(incv)` indexed elements if `side` = `right`.
+
+The sign of the increment parameter `incv` determines the order in which elements of `V` are accessed. For example, to access elements in reverse order,
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
+var work = new Float64Array( 3 );
+
+var out = dlarf1f( 'row-major', 'left', 4, 3, V, -1, 1.0, C, 3, work );
+// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
+```
+
+To perform strided access over `V`, provide an `abs(incv)` value greater than one. For example, to access every other element in `V`,
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+var V = new Float64Array( [ 0.5, 999, 0.5, 999, 0.5, 999, 0.5 ] );
+var work = new Float64Array( 3 );
+
+var out = dlarf1f( 'row-major', 'left', 4, 3, V, 2, 1.0, C, 3, work );
+// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
+```
+
+Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
+
+<!-- eslint-disable stdlib/capitalized-comments, max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+// Initial arrays...
+var C0 = new Float64Array( [ 0.0, 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+var V0 = new Float64Array( [ 0.0, 0.5, 0.5, 0.5, 0.5 ] );
+var work0 = new Float64Array( [ 0.0, 0.0, 0.0, 0.0 ] );
+
+// Create offset views...
+var C1 = new Float64Array( C0.buffer, C0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var V1 = new Float64Array( V0.buffer, V0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var work1 = new Float64Array( work0.buffer, work0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+var our = dlarf1f( 'row-major', 'left', 4, 3, V1, 1, 1.0, C1, 3, work1 );
+// C0 => <Float64Array>[ 0.0, -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
+```
+
+#### dlarf1f.ndarray( side, M, N, V, sv, ov, tau, C, sc1, sc2, oc, work, sw, ow )
+
+Applies a real elementary reflector `H = I - tau * v * v ^ T` to a real M by N matrix `C` using alternative indexing semantics.
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
+var work = new Float64Array( 3 );
+
+var out = dlarf1f.ndarray( 'left', 4, 3, V, 1, 0, 1.0, C, 3, 1, 0, work, 1, 0 );
+// returns <Float64Array>[ -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
+```
+
+The function has the following additional parameters:
+
+-   **side**: specifies the side of multiplication with `C`. Use `left` to form `H * C` and `right` to from `C * H`.
+-   **M**: number of rows in `C`.
+-   **N**: number of columns in `C`.
+-   **V**: the vector `v` in the representation of `H` as a [`Float64Array`][mdn-float64array].
+-   **sv**: stride length for `V`.
+-   **ov**: starting index for `V`.
+-   **tau**: the value of `tau` in representation of `H`.
+-   **C**: input matrix as a [`Float64Array`][mdn-float64array].
+-   **sc1**: stride of the first dimension of `C`.
+-   **sc2**: stride of the second dimension of `C`.
+-   **oc**: starting index for `C`.
+-   **work**: workspace array as a [`Float64Array`][mdn-float64array].
+-   **sw**: stride length for `work`.
+-   **ow**: starting index for `work`.
+
+Based on the `side` of multiplication, the expected storage of the array changes:
+
+-   `work` should have `N` indexed elements if `side` = `left` and `M` indexed elements if `side` = `right`.
+-   `V` should have `1 + (M-1) * abs(incv)` indexed elements if `side` = `left` and `1 + (N-1) * abs(incv)` indexed elements if `side` = `right`.
+
+While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example,
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var C = new Float64Array( [ 0.0, 0.0, 0.0, 0.0, 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+var V = new Float64Array( [ 0.0, 0.0, 0.5, 0.5, 0.5, 0.5 ] );
+var work = new Float64Array( [ 0.0, 0.0, 0.0, 0.0 ] );
+
+var out = dlarf1f.ndarray( 'left', 4, 3, V, 1, 2, 1.0, C, 3, 1, 4, work, 1, 0 );
+// C => <Float64Array>[ 0.0, 0.0, 0.0, 0.0, -4.5, -10.5, -16.5, -0.75, -1.75, -2.75, 0.25, -0.75, -1.75, 1.25,  0.25, -0.75 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   `C` is overwritten by `H * C` if side = `left` and `C * H` if side = `right`.
+-   `dlarf1f()` corresponds to the [LAPACK][LAPACK] function [`dlarf1f`][lapack-dlarf1f].
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+<!-- eslint-disable max-len -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+var ndarray2array = require( '@stdlib/ndarray/base/to-array' );
+var shape2strides = require( '@stdlib/ndarray/base/shape2strides' );
+var dlarf1f = require( '@stdlib/lapack/base/dlarf1f' );
+
+// Specify matrix meta data:
+var shape = [ 4, 3 ];
+var order = 'row-major';
+var strides = shape2strides( shape, order );
+
+// Create a matrix stored in linear memory:
+var C = new Float64Array( [ 1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0 ] );
+console.log( ndarray2array( C, shape, strides, 0, order ) );
+
+// Define the vector `v` and workspace array:
+var V = new Float64Array( [ 0.5, 0.5, 0.5, 0.5 ] );
+var work = new Float64Array( 3 );
+
+// Apply the elementary reflector:
+dlarf1f( order, 'left', shape[ 0 ], shape[ 1 ], V, 1, 1.0, C, strides[ 0 ], work );
+console.log( ndarray2array( C, shape, strides, 0, order ) );
+```
+
+</section>
+
+<!-- /.examples -->
+
+<!-- C interface documentation. -->
+
+* * *
+
+<section class="c">
+
+## C APIs
+
+<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
+
+<section class="intro">
+
+</section>
+
+<!-- /.intro -->
+
+<!-- C usage documentation. -->
+
+<section class="usage">
+
+### Usage
+
+```c
+TODO
+```
+
+#### TODO
+
+TODO.
+
+```c
+TODO
+```
+
+TODO
+
+```c
+TODO
+```
+
+</section>
+
+<!-- /.usage -->
+
+<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="notes">
+
+</section>
+
+<!-- /.notes -->
+
+<!-- C API usage examples. -->
+
+<section class="examples">
+
+### Examples
+
+```c
+TODO
+```
+
+</section>
+
+<!-- /.examples -->
+
+</section>
+
+<!-- /.c -->
+
+<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
+
+<section class="related">
+
+</section>
+
+<!-- /.related -->
+
+<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="links">
+
+[lapack]: https://www.netlib.org/lapack/explore-html/
+
+[lapack-dlarf1f]: https://netlib.org/lapack/explore-html/d2/d97/group__larf_ga3b4608752c4f72758c2d297e1e7cf3f0.html#ga3b4608752c4f72758c2d297e1e7cf3f0
+
+[mdn-float64array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Float64Array
+
+[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
+
+</section>
+
+<!-- /.links -->