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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 dnrm2 = require( '@stdlib/blas/base/dnrm2' ).ndarray;
var sign = require( '@stdlib/math/base/special/copysign' );
var dlamch = require( '@stdlib/lapack/base/dlamch' );
var abs = require( '@stdlib/math/base/special/abs' );
var dscal = require( '@stdlib/blas/base/dscal' ).ndarray;
var dlapy2 = require( '@stdlib/lapack/base/dlapy2' );
// 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`
*
* @private
* @param {NonNegativeInteger} N - number of rows/columns of the elementary reflector `H`
* @param {Float64Array} X - input vector
* @param {integer} strideX - stride length for `X`
* @param {NonNegativeInteger} offsetX - starting index of `X`
* @param {Float64Array} out - output array
* @param {integer} strideOut - stride length for `out`
* @param {NonNegativeInteger} offsetOut - starting index of `out`
* @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, 0, out, 1, 0 );
* // X => <Float64Array>[ ~0.19, ~0.28, ~0.37 ]
* // out => <Float64Array>[ ~-6.7, ~1.6 ]
*/
function dlarfg( N, X, strideX, offsetX, out, strideOut, offsetOut ) {
var safemin;
var rsafmin;
var xnorm;
var alpha;
var beta;
var tau;
var knt;
var i;
if ( N <= 1 ) {
out[ offsetOut + strideOut ] = 0.0;
return;
}
xnorm = dnrm2( N - 1, X, strideX, offsetX );
alpha = out[ offsetOut ];
if ( xnorm === 0.0 ) {
out[ strideOut + offsetOut ] = 0.0;
} else {
beta = -1.0 * sign( dlapy2( alpha, xnorm ), alpha );
safemin = dlamch( 'safemin' ) / dlamch( 'epsilon' );
knt = 0;
if ( abs( beta ) < safemin ) {
rsafmin = 1.0 / safemin;
while ( abs( beta ) < safemin && knt < 20 ) {
knt += 1;
dscal( N-1, rsafmin, X, strideX, offsetX );
beta *= rsafmin;
alpha *= rsafmin;
}
xnorm = dnrm2( N - 1, X, strideX, offsetX );
beta = -1.0 * sign( dlapy2( alpha, xnorm ), alpha );
}
tau = ( beta - alpha ) / beta;
dscal( N-1, 1.0 / ( alpha - beta ), X, strideX, offsetX );
for ( i = 0; i < knt; i++ ) {
beta *= safemin;
}
out[ offsetOut ] = beta;
out[ strideOut + offsetOut ] = tau;
}
}
// EXPORTS //
module.exports = dlarfg;
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