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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.
*/
/* eslint-disable max-len, max-params, max-statements */
'use strict';
// MODULES //
var dlamch = require( '@stdlib/lapack/base/dlamch' );
var dlapy2 = require( '@stdlib/lapack/base/dlapy2' );
var int = require( '@stdlib/math/base/special/floor' );
var pow = require( '@stdlib/math/base/special/pow' );
var ln = require( '@stdlib/math/base/special/ln' );
var sign = require( '@stdlib/math/base/special/copysign' );
var max = require( '@stdlib/math/base/special/fast/max' );
var min = require( '@stdlib/math/base/special/fast/min' );
var abs = require( '@stdlib/math/base/special/fast/abs' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
// VARIABLES //
var safmin = dlamch( 'safe minimum' );
var eps = dlamch( 'precision' );
var safmin2 = pow( dlamch( 'base' ), int( ln( safmin / eps ) / ln( dlamch('base') ) / 2.0 ) );
var safmax2 = 1.0 / safmin2;
var multpl = 4.0;
// MAIN //
/**
* Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix `A` in standard form.
*
* Given a real 2×2 matrix:
*
* ```tex
* \begin{bmatrix}
* A & B \\
* C & D
* \end{bmatrix}
* ```
*
* this routine computes an orthogonal matrix:
*
* ```tex
* \begin{bmatrix}
* \text{CS} & \text{SN} \\
* -\text{SN} & \text{CS}
* \end{bmatrix}
* ```
*
* such that the matrix is reduced to Schur (quasi-triangular) form:
*
* ```tex
* \begin{bmatrix}
* \text{AA} & \text{BB} \\
* 0 & \text{DD}
* \end{bmatrix}
* ```
*
* @private
* @param {Float64Array} A - array containing the element A(1,1)
* @param {NonNegativeInteger} offsetA - index in `A` of the element A(1,1)
* @param {Float64Array} B - array containing the element A(1,2)
* @param {NonNegativeInteger} offsetB - index in `B` of the element A(1,2)
* @param {Float64Array} C - array containing the element A(2,1)
* @param {NonNegativeInteger} offsetC - index in `C` of the element A(2,1)
* @param {Float64Array} D - array containing the element A(2,2)
* @param {NonNegativeInteger} offsetD - index in `D` of the element A(2,2)
* @param {Float64Array} RT1R - output array for the real part of the first eigenvalue
* @param {NonNegativeInteger} offsetRT1R - index in `RT1R` at which to store the value
* @param {Float64Array} RT1I - output array for the imaginary part of the first eigenvalue
* @param {NonNegativeInteger} offsetRT1I - index in `RT1I` at which to store the value
* @param {Float64Array} RT2R - output array for the real part of the second eigenvalue
* @param {NonNegativeInteger} offsetRT2R - index in `RT2R` at which to store the value
* @param {Float64Array} RT2I - output array for the imaginary part of the second eigenvalue
* @param {NonNegativeInteger} offsetRT2I - index in `RT2I` at which to store the value
* @param {Float64Array} CS - output array for cosine of the rotation
* @param {NonNegativeInteger} offsetCS - index in `CS` at which to store the value
* @param {Float64Array} SN - output array for sine of the rotation
* @param {NonNegativeInteger} offsetSN - index in `SN` at which to store the value
* @returns {void}
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* var A = new Float64Array( [ 4.0 ] );
* var B = new Float64Array( [ -5.0 ] );
* var C = new Float64Array( [ 2.0 ] );
* var D = new Float64Array( [ -3.0 ] );
* var RT1R = new Float64Array( 1 );
* var RT1I = new Float64Array( 1 );
* var RT2R = new Float64Array( 1 );
* var RT2I = new Float64Array( 1 );
* var CS = new Float64Array( 1 );
* var SN = new Float64Array( 1 );
*
* dlanv2( A, 0, B, 0, C, 0, D, 0, RT1R, 0, RT1I, 0, RT2R, 0, RT2I, 0, CS, 0, SN, 0 );
* // A => <Float64Array>[ 2.0 ]
* // B => <Float64Array>[ -7.0 ]
* // C => <Float64Array>[ 0.0 ]
* // D => <Float64Array>[ -1.0 ]
* // RT1R => <Float64Array>[ 2.0 ]
* // RT1I => <Float64Array>[ 0.0 ]
* // RT2R => <Float64Array>[ -1.0 ]
* // RT2I => <Float64Array>[ 0.0 ]
* // CS => <Float64Array>[ ~0.93 ]
* // SN => <Float64Array>[ ~0.34 ]
*/
function dlanv2( A, offsetA, B, offsetB, C, offsetC, D, offsetD, RT1R, offsetRT1R, RT1I, offsetRT1I, RT2R, offsetRT2R, RT2I, offsetRT2I, CS, offsetCS, SN, offsetSN ) {
var bcmax;
var bcmis;
var sigma;
var scale;
var count;
var temp;
var tau;
var sab;
var sac;
var sn1;
var cs1;
var aa;
var bb;
var cc;
var dd;
var p;
var z;
if ( C[ offsetC ] === 0.0 ) {
CS[ offsetCS ] = 1.0;
SN[ offsetSN ] = 0.0;
} else if ( B[ offsetB ] === 0.0 ) {
CS[ offsetCS ] = 0.0;
SN[ offsetSN ] = 1.0;
temp = D[ offsetD ];
D[ offsetD ] = A[ offsetA ];
A[ offsetA ] = temp;
B[ offsetB ] = -C[ offsetC ];
C[ offsetC ] = 0.0;
} else if ( A[ offsetA ] === D[ offsetD ] && ( sign( 1.0, B[ offsetB ] ) !== sign( 1.0, C[ offsetC ] ) ) ) {
CS[ offsetCS ] = 1.0;
SN[ offsetSN ] = 0.0;
} else {
temp = A[ offsetA ] - D[ offsetD ];
p = 0.5 * temp;
bcmax = max( abs( B[ offsetB ] ), abs( C[ offsetC ] ) );
bcmis = min( abs( B[ offsetB ] ), abs( C[ offsetC ] ) ) * sign( 1.0, B[ offsetB ] ) * sign( 1.0, C[ offsetC ] );
scale = max( abs( p ), bcmax );
z = ( ( p / scale ) * p ) + ( ( bcmax / scale ) * bcmis );
if ( z > multpl * eps ) {
z = p + sign( sqrt( scale ) * sqrt( z ), p );
A[ offsetA ] = D[ offsetD ] + z;
D[ offsetD ] -= ( bcmax / z ) * bcmis;
tau = dlapy2( C[ offsetC ], z );
CS[ offsetCS ] = z / tau;
SN[ offsetSN ] = C[ offsetC ] / tau;
B[ offsetB ] -= C[ offsetC ];
C[ offsetC ] = 0.0;
} else {
count = 0;
sigma = B[ offsetB ] + C[ offsetC ];
while ( true ) {
scale = max( abs( temp ), abs( sigma ) );
if ( count <= 20 ) {
if ( scale >= safmax2 ) {
sigma *= safmin2;
temp *= safmin2;
count += 1;
continue;
}
if ( scale <= safmin2 ) {
sigma *= safmax2;
temp *= safmax2;
count += 1;
continue;
}
}
break;
}
p = 0.5 * temp;
tau = dlapy2( sigma, temp );
CS[ offsetCS ] = sqrt( 0.5 * ( 1.0 + ( abs( sigma ) / tau ) ) );
SN[ offsetSN ] = -( p / ( tau * CS[ offsetCS ] ) ) * sign( 1.0, sigma );
aa = ( A[ offsetA ] * CS[ offsetCS ] ) + ( B[ offsetB ] * SN[ offsetSN ] );
bb = -( A[ offsetA ] * SN[ offsetSN ] ) + ( B[ offsetB ] * CS[ offsetCS ] );
cc = ( C[ offsetC ] * CS[ offsetCS ] ) + ( D[ offsetD ] * SN[ offsetSN ] );
dd = -( C[ offsetC ] * SN[ offsetSN ] ) + ( D[ offsetD ] * CS[ offsetCS ] );
A[ offsetA ] = ( aa * CS[ offsetCS ] ) + ( cc * SN[ offsetSN ] );
B[ offsetB ] = ( bb * CS[ offsetCS ] ) + ( dd * SN[ offsetSN ] );
C[ offsetC ] = -( aa * SN[ offsetSN ] ) + ( cc * CS[ offsetCS ] );
D[ offsetD ] = -( bb * SN[ offsetSN ] ) + ( dd * CS[ offsetCS ] );
temp = 0.5 * ( A[ offsetA ] + D[ offsetD ] );
A[ offsetA ] = temp;
D[ offsetD ] = temp;
if ( C[ offsetC ] !== 0.0 ) {
if ( B[ offsetB ] === 0.0 ) {
B[ offsetB ] -= C[ offsetC ];
C[ offsetC ] = 0.0;
temp = CS[ offsetCS ];
CS[ offsetCS ] = -SN[ offsetSN ];
SN[ offsetSN ] = temp;
} else if ( sign( 1.0, B[ offsetB ] ) === sign( 1.0, C[ offsetC ] ) ) {
sab = sqrt( abs( B[ offsetB ] ) );
sac = sqrt( abs( C[ offsetC ] ) );
p = sign( sab*sac, C[ offsetC ] );
tau = 1.0 / sqrt( abs( B[ offsetB ]+C[ offsetC ] ) );
A[ offsetA ] = temp + p;
D[ offsetD ] = temp - p;
B[ offsetB ] -= C[ offsetC ];
C[ offsetC ] = 0.0;
cs1 = sab*tau;
sn1 = sac*tau;
temp = ( CS[ offsetCS ] * cs1 ) - ( SN[ offsetSN ] * sn1 );
SN[ offsetSN ] = ( CS[ offsetCS ] * sn1 ) + ( SN[ offsetSN ] * cs1 );
CS[ offsetCS ] = temp;
}
}
}
}
RT1R[ offsetRT1R ] = A[ offsetA ];
RT2R[ offsetRT2R ] = D[ offsetD ];
if ( C[ offsetC ] === 0.0 ) {
RT1I[ offsetRT1I ] = 0.0;
RT2I[ offsetRT2I ] = 0.0;
} else {
RT1I[ offsetRT1I ] = sqrt( abs( B[ offsetB ] * C[ offsetC ] ) );
RT2I[ offsetRT2I ] = -RT1I[ offsetRT1I ];
}
}
// EXPORTS //
module.exports = dlanv2;
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