feat: add lapack/base/dlarfg

lib/node_modules/@stdlib/lapack/base/dlarfg/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.
+
+-->
+
+# dlarfg
+
+> LAPACK routine to generate a real elementary reflector `H` of order `N` such that applying `H` to a vector `[alpha; X]` zeros out `X`.
+
+<section class="intro">
+
+The `dlarfg` routine generates a **real elementary reflector** (also known as a Householder reflector) of order `N`, which can be used to zero out selected components of a vector. Specifically, it constructs a reflector matrix `H` such that:
+
+<!-- <equation class="equation" label="eq:lu_decomposition" align="center" raw="H \cdot \begin{bmatrix}\alpha \\ x \end{bmatrix} = \begin{bmatrix} \beta \\ 0 \end{bmatrix}, \quad \text{and} \quad H^T H = I" alt="Transformation of vector [α; x] into [β; 0] using orthogonal reflector H, and condition that H is orthogonal (HᵀH = I)."> -->
+
+
+```math
+H \cdot \begin{bmatrix}\alpha \\ x \end{bmatrix} = \begin{bmatrix} \beta \\ 0 \end{bmatrix}, \quad \text{and} \quad H^T H = I
+```
+
+<!-- </equation> -->
+
+
+Here:
+
+-   `α` is a scalar.
+-   `X` is a real vector of length `N-1`.
+-   `β` is a scalar value.
+-   `H` is an orthogonal matrix known as a Householder reflector.
+
+The reflector `H` is constructed in the form:
+
+<!-- <equation class="equation" label="eq:matrix_a" align="center" raw="H = I - \tau \begin{bmatrix}1 \\ v \end{bmatrix} \begin{bmatrix}1 & v^T \end{bmatrix}" alt="Definition of Householder reflector H as identity minus tau times the outer product of the vector [1; v] with its transpose."> -->
+
+```math
+H = I - \tau \begin{bmatrix}1 \\ v \end{bmatrix} \begin{bmatrix}1 & v^T \end{bmatrix}
+```
+
+<!-- </equation> -->
+
+Where:
+
+-   `τ` (`tau`) is a real scalar.
+-   `V` is a real vector of length `N-1` that defines the Householder vector.
+-   The vector `[1; V]` is the Householder direction.
+
+The values of `τ` and `V` are chosen so that applying `H` to the vector `[α; x]` results in a new vector `[β; 0]`, i.e., only the first component remains nonzero. The reflector matrix `H` is **symmetric and orthogonal**, satisfying `H^T = H` and `H^T H = I`.
+
+### Special Cases
+
+-   If all elements of `x` are zero, then `τ = 0` and `H = I`, the identity matrix.
+-   Otherwise, `τ` satisfies `1 ≤ τ ≤ 2`, ensuring numerical stability in transformations.
+
+This elementary reflector is commonly used in algorithms for QR factorization and other orthogonal transformations.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var dlarfg = require( '@stdlib/lapack/base/dlarfg' );
+```
+
+#### dlarfg( N, X, incx, out )
+
+Generates a real elementary reflector `H` of order `N` such that applying `H` to a vector `[alpha; X]` zeros out `X`.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var X = new Float64Array( [ 2.0, 3.0, 4.0 ] );
+var out = new Float64Array( [ 4.0, 0.0 ] );
+
+dlarfg( 4, X, 1, out );
+// X => <Float64Array>[ ~0.19, ~0.28, ~0.37 ]
+// out => <Float64Array>[ ~-6.7, ~1.6 ]
+```
+
+The function has the following parameters:
+
+-   **N**: number of rows/columns of the elementary reflector `H`.
+-   **X**: a [`Float64Array`][mdn-float64array] which is overwritten by the vector `V`. Should have `N - 1` indexed elements.
+-   **incx**: stride length of `X`.
+-   **out**: output [`Float64Array`][mdn-float64array]. The first element of `out` represents alpha and the second element of `out` represents `tau`.
+
+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 -->
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+// Initial arrays...
+var X0 = new Float64Array( [ 0.0, 2.0, 3.0, 4.0 ] );
+var out0 = new Float64Array( [ 0.0, 4.0, 0.0 ] );
+
+// Create offset views...
+var X1 = new Float64Array( X0.buffer, X0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var out1 = new Float64Array( out0.buffer, out0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+dlarfg( 4, X1, 1, out1 );
+// X0 => <Float64Array>[ 0.0, ~0.19, ~0.28, ~0.37 ]
+// out0 => <Float64Array>[ 0.0, ~6.7, ~1.6 ]
+```
+
+#### dlarfg.ndarray( N, X, strideX, offsetX, out, strideOut, offsetOut )
+
+Generates a real elementary reflector `H` of order `N` such that applying `H` to a vector `[alpha; X]` zeros out `X` using alternative indexing semantics.
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var X = new Float64Array( [ 2.0, 3.0, 4.0 ] );
+var out = new Float64Array( [ 4.0, 0.0 ] );
+
+dlarfg.ndarray( 4, X, 1, 0, out, 1, 0 );
+// X => <Float64Array>[ ~0.19, ~0.28, ~0.37 ]
+// out => <Float64Array>[ ~-6.7, ~1.6 ]
+```
+
+The function has the following parameters:
+
+-   **N**: number of rows/columns of the elementary reflector `H`.
+-   **X**: a [`Float64Array`][mdn-float64array] which is overwritten by the vector `V`. Should have `N - 1` indexed elements.
+-   **strideX**: stride length of `X`.
+-   **offsetX**: starting index of `X`.
+-   **out**: output [`Float64Array`][mdn-float64array]. The first element of `out` represents alpha and the second element of `out` represents `tau`.
+-   **strideOut**: stride length of `out`.
+-   **offsetOut**: starting index of `out`.
+
+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,
+
+```javascript
+var Float64Array = require( '@stdlib/array/float64' );
+
+var X = new Float64Array( [ 0.0, 2.0, 3.0, 4.0 ] );
+var out = new Float64Array( [ 0.0, 4.0, 0.0 ] );
+
+dlarfg.ndarray( 4, X, 1, 1, out, 1, 1 );
+// X => <Float64Array>[ 0.0, ~0.19, ~0.28, ~0.37 ]
+// out => <Float64Array>[ 0.0, ~6.7, ~1.6 ]
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   `dlarfg()` corresponds to the [LAPACK][lapack] routine [`dlarfg`][lapack-dlarfg].
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var uniform = require( '@stdlib/random/array/discrete-uniform' );
+var Float64Array = require( '@stdlib/array/float64' );
+var dlarfg = require( '@stdlib/lapack/base/dlarfg' );
+
+var N = 4;
+
+var X = uniform( N - 1, -10, 10, {
+    'dtype': 'float64'
+});
+console.log( 'X: ', X );
+
+var alpha = 4.0;
+
+var out = new Float64Array( [ alpha, 0.0 ] );
+
+dlarfg( N, X, 1, out );
+
+console.log( 'V: ', X );
+console.log( 'beta: ', out[ 0 ] );
+console.log( 'tau: ', out[ 1 ] );
+```
+
+</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-dlarfg]: https://netlib.org/lapack/explore-html-3.6.1/d7/da9/dlarfg_8f_aabb59655e820b3551af27781bd716143.html
+
+[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 -->