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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x 20x 20x 20x 2x 2x 2x 2x 2x | /** * @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` usinig alternative indexing semantics. * * `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} 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 ) { base( N, X, strideX, offsetX, out, strideOut, offsetOut ); } // EXPORTS // module.exports = dlarfg; |