All files / lapack/base/dlaqr5/lib dlaqr1.js

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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 abs = require( '@stdlib/math/base/special/fast/abs' );
 
 
// MAIN //
 
/**
* Given a 2-by-2 or a 3-by-3 matrix `H`, this function sets `V` to a scalar multiple of the first column of `K` where `K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)`.
*
* ## Notes
*
* -   It is expected that either `sr1 = sr2` and `si1 + si2 = 0` or `si1 = si2 = 0` (i.e., they represent complex conjugate values).
* -   This is useful for starting double implicit shift bulges in the QR algorithm.
* -   `V` should have at least `N` indexed elements.
*
* @private
* @param {PositiveInteger} N - number of row/columns in `H`
* @param {Float64Array} H - input matrix
* @param {integer} strideH1 - stride of the first dimension of `H`
* @param {integer} strideH2 - stride of the second dimension of `H`
* @param {NonNegativeInteger} offsetH - index offset for `H`
* @param {number} sr1 - real part of the first conjugate complex shift
* @param {number} si1 - imaginary part of the first conjugate complex shift
* @param {number} sr2 - real part of the second conjugate complex shift
* @param {number} si2 - imaginary part of the second conjugate complex shift
* @param {Float64Array} V - output array
* @param {integer} strideV - stride length for `V`
* @param {NonNegativeInteger} offsetV - index offset for `V`
* @returns {Float64Array} `V`
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* var H = new Float64Array( [ 1.0, 3.0, 2.0, 2.0, 4.0, 6.0, 0.0, 5.0, 7.0 ] ); // => [ [ 1.0, 3.0, 2.0 ], [ 2.0, 4.0, 6.0 ], [ 0.0, 5.0, 7.0 ] ]
* var V = new Float64Array( 3 );
*
* var out = dlaqr1( 3, H, 3, 1, 0, 1.5, 0.0, 2.5, 0.0, V, 1, 0 );
* // returns <Float64Array>[ ~1.93, ~0.57, ~2.86 ]
*/
function dlaqr1( N, H, strideH1, strideH2, offsetH, sr1, si1, sr2, si2, V, strideV, offsetV ) { // eslint-disable-line max-params, max-len
	var h21s;
	var h31s;
	var h11;
	var h12;
	var h13;
	var h21;
	var h22;
	var h23;
	var h31;
	var h32;
	var h33;
	var iv;
	var s;
	var i;

	h11 = offsetH;
	h12 = offsetH + strideH2;
	h21 = offsetH + strideH1;
	h22 = h21 + strideH2;

	if ( N === 2 ) {
		s = abs( H[ h11 ] - sr2 ) + abs( si2 ) + abs( H[ h21 ] );
		if ( s === 0.0 ) {
			V[ offsetV ] = 0.0;
			V[ strideV + offsetV ] = 0.0;
			return V;
		}
		h21s = H[ h21 ] / s;
		V[ offsetV ] = ( h21s * H[ h12 ] ) + ( ( H[ h11 ]-sr1 ) * ( ( H[ h11 ]-sr2 ) / s ) ) - ( si1*( si2 / s ) ); // eslint-disable-line max-len
		V[ offsetV + strideV ] = h21s*( H[ h11 ]+H[ h22 ]-sr1-sr2 );
		return V;
	}

	h13 = h12 + strideH2;
	h31 = h21 + strideH1;
	h33 = h22 + strideH1 + strideH2;
	h23 = h22 + strideH2;
	h32 = h22 + strideH1;

	s = abs( H[ h11 ]-sr2 ) + abs( si2 ) + abs( H[ h21 ] ) + abs( H[ h31 ] );
	if ( s === 0.0 ) {
		iv = offsetV;
		for ( i = 0; i < 3; i++ ) {
			V[ iv ] = 0.0;
			iv += strideV;
		}
		return V;
	}
	h21s = H[ h21 ] / s;
	h31s = H[ h31 ] / s;
	iv = offsetV;
	V[ iv ] = (( H[ h11 ]-sr1 )*( ( H[ h11 ]-sr2 ) / s )) - (si1*( si2 / s )) + ((H[ h12 ]*h21s) + (H[ h13 ]*h31s)); // eslint-disable-line max-len
	iv += strideV;
	V[ iv ] = (h21s*( H[ h11 ]+H[ h22 ]-sr1-sr2 )) + (H[ h23 ]*h31s);
	iv += strideV;
	V[ iv ] = (h31s*( H[ h11 ]+H[ h33 ]-sr1-sr2 )) + (h21s*H[ h32 ]);
	return V;
}
 
 
// EXPORTS //
 
module.exports = dlaqr1;