Press n or j to go to the next uncovered block, b, p or k for the previous block.
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* @license Apache-2.0
*
* Copyright (c) 2026 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 */
'use strict';
// MODULES //
var dgemm = require( '@stdlib/blas/base/dgemm' ).ndarray;
var min = require( '@stdlib/math/base/special/min' );
var max = require( '@stdlib/math/base/special/max' );
var floor = require( '@stdlib/math/base/special/floor' );
var dgebd2 = require( './dgebd2.js' );
var dlabrd = require( './dlabrd.js' );
// MAIN //
/**
* Reduces a general real `M-by-N` matrix `A` to upper or lower bi-diagonal form `B` by an orthogonal transformation `Q**T * A * P = B`.
*
* ## Notes
*
* - If `M >= N`,
*
* - `B` is upper bi-diagonal form.
* - The diagonal and the first superdiagonal are overwritten with the upper bi-diagonal matrix `B`.
* - Elements below the diagonal, with the array `TAUQ`, represent the orthogonal matrix `Q` as a product of elementary reflectors.
* - Elements above the first super-diagonal, with the array `TAUP`, represent the orthogonal matrix `P` as a product of elementary reflectors.
*
* - If `M < N`,
*
* - `B` is lower bi-diagonal form.
* - The diagonal and the first sub-diagonal are overwritten with the lower bi-diagonal matrix `B`.
* - Elements below the first sub-diagonal, with the array `TAUQ`, represent the orthogonal matrix `Q` as a product of elementary reflectors.
* - Elements on and above the diagonal, with the array `TAUP`, represent the orthogonal matrix `P` as a product of elementary reflectors.
*
* @private
* @param {NonNegativeInteger} M - number of rows in `A`
* @param {NonNegativeInteger} N - number of columns in `A`
* @param {Float64Array} A - input matrix
* @param {integer} strideA1 - stride of the first dimension of `A`
* @param {integer} strideA2 - stride of the second dimension of `A`
* @param {NonNegativeInteger} offsetA - starting index for `A`
* @param {Float64Array} D - diagonal elements of the bi-diagonal matrix `B` (length `min(M,N)`)
* @param {integer} strideD - stride length for `D`
* @param {NonNegativeInteger} offsetD - starting index for `D`
* @param {Float64Array} E - off-diagonal elements of the bi-diagonal matrix `B` (length `min(M,N)-1`)
* @param {integer} strideE - stride length for `E`
* @param {NonNegativeInteger} offsetE - starting index for `E`
* @param {Float64Array} TAUQ - scalars factors of the elementary reflectors that represent the orthogonal matrix `Q` (length `NB`)
* @param {integer} strideTAUQ - stride length for `TAUQ`
* @param {NonNegativeInteger} offsetTAUQ - starting index for `TAUQ`
* @param {Float64Array} TAUP - scalars factors of the elementary reflectors that represent the orthogonal matrix `P` (length `NB`)
* @param {integer} strideTAUP - stride length for `TAUP`
* @param {NonNegativeInteger} offsetTAUP - starting index for `TAUP`
* @param {Float64Array} WORK - workspace array (length >= `max(1,LWORK)`)
* @param {integer} strideWORK - stride length of `WORK`
* @param {NonNegativeInteger} offsetWORK - starting index for `WORK`
* @param {integer} LWORK - length of WORK array
* @returns {integer} status code
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* var A = new Float64Array( [ 4.0, -1.0, 2.5, 7.0, 8.5, -2.0 ] );
* var D = new Float64Array( [ 0.0, 0.0 ] );
* var E = new Float64Array( [ 0.0 ] );
* var TAUQ = new Float64Array( [ 0.0, 0.0 ] );
* var TAUP = new Float64Array( [ 0.0, 0.0 ] );
* var WORK = new Float64Array( 100 );
*
* dgebrd( 3, 2, A, 1, 3, 0, D, 1, 0, E, 1, 0, TAUQ, 1, 0, TAUP, 1, 0, WORK, 1, 0, 100 );
* // A => <Float64Array>[ ~-4.822, ~-0.113, ~0.283, ~-3.007, ~-10.78, ~-0.237 ]
* // D => <Float64Array>[ ~-4.822, ~-10.78 ]
* // E => <Float64Array>[ ~-3.007 ]
* // TAUQ => <Float64Array>[ ~1.83, ~1.894 ]
* // TAUP => <Float64Array>[ 0.0, 0.0 ]
* // WORK[ 0 ] => 3.0
*/
function dgebrd( M, N, A, strideA1, strideA2, offsetA, D, strideD, offsetD, E, strideE, offsetE, TAUQ, strideTAUQ, offsetTAUQ, TAUP, strideTAUP, offsetTAUP, WORK, strideWORK, offsetWORK, LWORK ) { // eslint-disable-line stdlib/jsdoc-doctest-decimal-point
var ldwrkx;
var ldwrky;
var lwkmin;
var lwkopt;
var lquery;
var minmn;
var nbmin;
var nb;
var nx;
var ws;
var i;
var j;
minmn = min( M, N );
if ( minmn === 0 ) {
lwkmin = 1;
lwkopt = 1;
} else {
lwkmin = max( M, N );
nb = 32; // The optimal blocksize derived from the Fortran LAPACK call `ilaenv( 1, 'DGEBRD', ' ', m, n, -1, -1 )`
lwkopt = ( M + N ) * nb;
}
WORK[ offsetWORK ] = lwkopt;
lquery = ( LWORK === -1 );
if ( ( LWORK < lwkmin ) && !lquery ) {
return -10;
}
if ( lquery ) {
return 0;
}
// Quick return if possible
if ( minmn === 0 ) {
WORK[ offsetWORK ] = 1;
return 0;
}
// Determine the block size
ldwrkx = M;
ldwrky = N;
ws = max( M, N );
nx = minmn;
if ( nb > 1 && nb < minmn ) {
// Set the crossover point NX
nx = max( nb, 128 ); // The crossover point derived from the Fortran LAPACK call `ilaenv( 3, 'DGEBRD', ' ', m, n, -1, -1 )`
// Determine when to switch from blocked to unblocked code
if ( nx < minmn ) {
ws = lwkopt;
if ( LWORK < ws ) {
// Not enough workspace for optimal NB,consider using a smaller block size
nbmin = 2; // The minimum block size derived from the Fortran LAPACK call `ilaenv( 2, 'DGEBRD', ' ', m, n, -1, -1 )`
if ( LWORK >= ( M + N ) * nbmin ) {
nb = floor( LWORK / ( M + N ) );
} else {
nb = 1;
nx = minmn;
}
}
}
} else {
nx = minmn;
}
for ( i = 0; i < minmn - nx; i += nb ) {
// Reduce rows and columns i:i+nb-1 to bi-diagonal form and return the matrices X and Y which are needed to update the unreduced part of the matrix.
dlabrd( M - i, N - i, nb, A, strideA1, strideA2, offsetA + ( i * strideA1 ) + ( i * strideA2 ), D, strideD, offsetD + ( i * strideD ), E, strideE, offsetE + ( i * strideE ), TAUQ, strideTAUQ, offsetTAUQ + ( i * strideTAUQ ), TAUP, strideTAUP, offsetTAUP + ( i * strideTAUP ), WORK, strideWORK, ldwrkx, offsetWORK, WORK, strideWORK, ldwrky, offsetWORK + ( ldwrkx*nb ) );
// Update the trailing submatrix A(i+nb:m,i+nb:n), using an update of the form A := A - V*Y**T - X*U**T
dgemm('no-transpose', 'transpose', M - i - nb, N - i - nb, nb, -1.0, A, strideA1, strideA2, offsetA + ( ( i + nb ) * strideA1 ) + ( i * strideA2 ), WORK, strideWORK, ldwrky, offsetWORK + ( ldwrkx * nb ) + nb, 1.0, A, strideA1, strideA2, offsetA + ( ( i + nb ) * strideA1 ) + ( ( i + nb ) * strideA2 ) );
dgemm('no-transpose', 'no-transpose', M - i - nb, N - i - nb, nb, -1.0, WORK, strideWORK, ldwrkx, offsetWORK + nb, A, strideA1, strideA2, offsetA + ( i * strideA1 ) + (( i + nb ) * strideA2), 1.0, A, strideA1, strideA2, offsetA + ( ( i + nb ) * strideA1 ) + ( ( i + nb ) * strideA2 ) );
// Copy diagonal and off-diagonal elements of B back into A
if ( M >= N ) {
for ( j = i; j < i + nb; j++ ) {
A[ offsetA + ( j * strideA1 ) + ( j * strideA2 ) ] = D[ offsetD + ( j * strideD ) ];
A[ offsetA + ( j * strideA1 ) + ( ( j + 1 ) * strideA2 ) ] = E[ offsetE + ( j * strideE ) ];
}
} else {
for ( j = i; j < i + nb; j++ ) {
A[ offsetA + ( j * strideA1 ) + ( j * strideA2 ) ] = D[ offsetD + ( j * strideD ) ];
A[ offsetA + ( ( j + 1 ) * strideA1 ) + ( j * strideA2 ) ] = E[ offsetE + ( j * strideE ) ];
}
}
}
// Use unblocked code to reduce the remainder of the matrix
dgebd2( M - i, N - i, A, strideA1, strideA2, offsetA + ( i * strideA1 ) + ( i * strideA2 ), D, strideD, offsetD + ( i * strideD ), E, strideE, offsetE + ( i * strideE ), TAUQ, strideTAUQ, offsetTAUQ + ( i * strideTAUQ ), TAUP, strideTAUP, offsetTAUP + ( i * strideTAUP ), WORK, strideWORK, offsetWORK );
WORK[ offsetWORK ] = ws;
return 0;
}
// EXPORTS //
module.exports = dgebrd;
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