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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.
*/
'use strict';
// MODULES //
var base = require( './base.js' );
// MAIN //
/**
* Generates a real elementary reflector `H` of order `N` such that applying `H` to a vector `[alpha; X]` zeros out `X`.
*
* `H` is a Householder matrix with the form:
*
* ```tex
* H \cdot \begin{bmatrix} \alpha \\ x \end{bmatrix} = \begin{bmatrix} \beta \\ 0 \end{bmatrix}, \quad \text{and} \quad H^T H = I
* ```
*
* where:
*
* - `tau` is a scalar
* - `X` is a vector of length `N-1`
* - `beta` is a scalar value
* - `H` is an orthogonal matrix known as a Householder reflector.
*
* The reflector `H` is constructed in the form:
*
* ```tex
* H = I - \tau \begin{bmatrix}1 \\ v \end{bmatrix} \begin{bmatrix}1 & v^T \end{bmatrix}
* ```
*
* where:
*
* - `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 `tau` and `V` are chosen so that applying `H` to the vector `[alpha; X]` results in a new vector `[beta; 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 `tau = 0` and `H = I`, the identity matrix.
* - Otherwise, `tau` satisfies `1 ≤ tau ≤ 2`, ensuring numerical stability in transformations.
*
* ## Notes
*
* - `X` should have `N-1` indexed elements
* - The output array contains the following two elements: `alpha` and `tau`
*
* @param {NonNegativeInteger} N - number of rows/columns of the elementary reflector `H`
* @param {Float64Array} X - input vector
* @param {integer} incx - stride length for `X`
* @param {Float64Array} out - output array
* @returns {void}
*
* @example
* 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 ]
*/
function dlarfg( N, X, incx, out ) {
base( N, X, incx, 0, out, 1, 0 );
}
// EXPORTS //
module.exports = dlarfg;
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