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 dlacpy = require( '@stdlib/lapack/base/dlacpy' ).ndarray;
var floor = require( '@stdlib/math/base/special/floor' );
var dtrmm = require( './dtrmm.js' );
// MAIN //
/**
* Forms the triangular factor T of a real block reflector H of order N, which is defined as a product of K elementary reflectors.
*
* ## Notes
*
* - If `direct` = 'forward', `H = H(1) H(2) . . . H(k)` and `T` is upper triangular.
* - If `direct` = 'backward', `H = H(k) . . . H(2) H(1)` and `T` is lower triangular.
* - If `storev` = 'columnwise', the vector which defines the elementary reflector `H(i)` is stored in the i-th column of the array `V`, and `H = I - V * T * V**T`.
* - If `storev` = 'rowwise', the vector which defines the elementary reflector `H(i)` is stored in the i-th row of the array `V`, and `H = I - V**T * T * V`.
*
* @private
* @param {string} direct - specifies the order in which the elementary reflectors are multiplied to form the block reflector `H`
* @param {string} storev - specifies how the vectors which define the elementary reflectors are stored
* @param {NonNegativeInteger} N - order of the block reflector `H`
* @param {NonNegativeInteger} K - order of the triangular factor `T` (the number of elementary reflectors)
* @param {Float64Array} V - matrix of reflector vectors
* @param {integer} strideV1 - stride of first dimension of `V`
* @param {integer} strideV2 - stride of second dimension of `V`
* @param {NonNegativeInteger} offsetV - starting index for `V`
* @param {Float64Array} TAU - array of scalar factors of the elementary reflector `H(i)`
* @param {integer} strideTAU - stride length for `TAU`
* @param {NonNegativeInteger} offsetTAU - starting index for `TAU`
* @param {Float64Array} T - output triangular matrix
* @param {integer} strideT1 - stride of first dimension of `T`
* @param {integer} strideT2 - stride of second dimension of `T`
* @param {NonNegativeInteger} offsetT - starting index for `T`
* @returns {Float64Array} `T`
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* var V = new Float64Array( [ 1.0, 0.2, 0.3, -0.4, 0.5, 0.0, 1.0, -0.6, 0.7, -0.8, 0.0, 0.0, 1.0, 0.9, 1.1 ] );
* var TAU = new Float64Array( [ 1.2, 0.7, 1.5 ] );
* var T = new Float64Array( 9 );
*
* dlarft( 'forward', 'rowwise', 5, 3, V, 5, 1, 0, TAU, 1, 0, T, 3, 1, 0 );
* // T => <Float64Array>[ ~1.2, ~0.5544, ~-0.17514, 0.0, 0.7, 0.8925, 0.0, 0.0, 1.5 ]
*/
function dlarft( direct, storev, N, K, V, strideV1, strideV2, offsetV, TAU, strideTAU, offsetTAU, T, strideT1, strideT2, offsetT ) {
var colv;
var dirf;
var lq;
var ql;
var qr;
var i;
var j;
var l;
if ( N === 0 || K === 0 ) {
return T;
}
if ( N === 1 || K === 1 ) {
T[ offsetT ] = TAU[ offsetTAU ];
return T;
}
l = floor( K / 2 );
if ( direct === 'forward' ) {
dirf = true;
}
else {
dirf = false;
}
if ( storev === 'columnwise' ) {
colv = true;
}
else {
colv = false;
}
// QR happens when we have forward direction in column storage
qr = dirf & colv;
// LQ happens when we have forward direction in row storage
lq = dirf & ( !colv );
// QL happens when we have backward direction in column storage
ql = ( !dirf ) & colv;
// The last case is RQ. Due to how we structured this, if the above 3 are false, then RQ must be true, so we never store this RQ happens when we have backward direction in row storage `RQ = ( !dirf ) & ( !colv )`
if ( qr ) {
/*
Break V apart into 6 components
V = |---------------|
|V_{1,1} 0 |
|V_{2,1} V_{2,2}|
|V_{3,1} V_{3,2}|
|---------------|
V_{1,1}\in\R^{l,l} unit lower triangular
V_{2,1}\in\R^{k-l,l} rectangular
V_{3,1}\in\R^{n-k,l} rectangular
V_{2,2}\in\R^{k-l,k-l} unit lower triangular
V_{3,2}\in\R^{n-k,k-l} rectangular
We will construct the T matrix
T = |---------------|
|T_{1,1} T_{1,2}|
|0 T_{2,2}|
|---------------|
T is the triangular factor obtained from block reflectors.
To motivate the structure, assume we have already computed T_{1,1}
and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
T_{1,1}\in\R^{l, l} upper triangular
T_{2,2}\in\R^{k-l, k-l} upper triangular
T_{1,2}\in\R^{l, k-l} rectangular
Where l = floor(k/2)
Then, consider the product:
(I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
= I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
Define T_{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
Then, we can define the matrix V as
V = |-------|
|V_1 V_2|
|-------|
So, our product is equivalent to the matrix product
I - V*T*V'
This means, we can compute T_{1,1} and T_{2,2}, then use this information
to compute T_{1,2}
*/
// Compute T_{1,1} recursively
dlarft( direct, storev, N, l, V, strideV1, strideV2, offsetV, TAU, strideTAU, offsetTAU, T, strideT1, strideT2, offsetT );
// Compute T_{2,2} recursively
dlarft( direct, storev, N - l, K - l, V, strideV1, strideV2, offsetV + ( ( strideV1 + strideV2 ) * l ), TAU, strideTAU, offsetTAU + ( strideTAU * l ), T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * l ) );
// Compute T_{1,2} = V_{2,1}
for ( j = 0; j < l; j++ ) {
for ( i = 0; i < K - l; i++ ) {
T[ offsetT + ( strideT1 * j ) + ( strideT2 * ( l + i ) ) ] = V[ offsetV + ( strideV1 * ( l + i ) ) + ( strideV2 * j ) ];
}
}
// T_{1,2} = T_{1,2}*V_{2,2}
dtrmm( 'right', 'lower', 'no-transpose', 'unit', l, K - l, 1, V, strideV1, strideV2, offsetV + ( ( strideV1 + strideV2 ) * l ), T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
// T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
// Note: We assume K <= N, and GEMM will do nothing if N=K
dgemm( 'transpose', 'no-transpose', l, K - l, N - K, 1, V, strideV1, strideV2, offsetV + ( strideV1 * K ), V, strideV1, strideV2, offsetV + ( strideV1 * K ) + ( strideV2 * l ), 1, T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
/*
At this point, we have that T_{1,2} = V_1'*V_2
All that is left is to pre and post multiply by -T_{1,1} and T_{2,2} respectively.
T_{1,2} = -T_{1,1}*T_{1,2}
*/
dtrmm( 'left', 'upper', 'no-transpose', 'non-unit', l, K - l, -1, T, strideT1, strideT2, offsetT, T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
// T_{1,2} = T_{1,2}*T_{2,2}
dtrmm( 'right', 'upper', 'no-transpose', 'non-unit', l, K - l, 1, T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * l ), T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
} else if ( lq ) {
/*
Break V apart into 6 components
V = |----------------------|
|V_{1,1} V_{1,2} V{1,3}|
|0 V_{2,2} V{2,3}|
|----------------------|
V_{1,1}\in\R^{l,l} unit upper triangular
V_{1,2}\in\R^{l,k-l} rectangular
V_{1,3}\in\R^{l,n-k} rectangular
V_{2,2}\in\R^{k-l,k-l} unit upper triangular
V_{2,3}\in\R^{k-l,n-k} rectangular
Where l = floor(k/2)
We will construct the T matrix
T = |---------------|
|T_{1,1} T_{1,2}|
|0 T_{2,2}|
|---------------|
T is the triangular factor obtained from block reflectors.
To motivate the structure, assume we have already computed T_{1,1}
and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
T_{1,1}\in\R^{l, l} upper triangular
T_{2,2}\in\R^{k-l, k-l} upper triangular
T_{1,2}\in\R^{l, k-l} rectangular
Then, consider the product:
(I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
= I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
Then, we can define the matrix V as
V = |---|
|V_1|
|V_2|
|---|
So, our product is equivalent to the matrix product
I - V'*T*V
This means, we can compute T_{1,1} and T_{2,2}, then use this information
to compute T_{1,2}
*/
// Compute T_{1,1} recursively
dlarft( direct, storev, N, l, V, strideV1, strideV2, offsetV, TAU, strideTAU, offsetTAU, T, strideT1, strideT2, offsetT );
// Compute T_{2,2} recursively
dlarft( direct, storev, N - l, K - l, V, strideV1, strideV2, offsetV + ( ( strideV1 + strideV2 ) * l ), TAU, strideTAU, offsetTAU + ( strideTAU * l ), T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * l ) );
// Compute T_{1,2} = V_{1,2}
dlacpy( 'all', l, K - l, V, strideV1, strideV2, offsetV + ( strideV2 * l ), T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
// T_{1,2} = T_{1,2}*V_{2,2}
dtrmm( 'right', 'upper', 'transpose', 'unit', l, K - l, 1, V, strideV1, strideV2, offsetV + ( ( strideV1 + strideV2 ) * l ), T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
// T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
// Note: We assume K <= N, and GEMM will do nothing if N=K
dgemm( 'no-transpose', 'transpose', l, K - l, N - K, 1, V, strideV1, strideV2, offsetV + ( strideV2 * K ), V, strideV1, strideV2, offsetV + ( strideV1 * l ) + ( strideV2 * K ), 1, T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
/*
At this point, we have that T_{1,2} = V_1*V_2'
All that is left is to pre and post multiply by -T_{1,1} and T_{2,2} respectively.
T_{1,2} = -T_{1,1}*T_{1,2}
*/
dtrmm( 'left', 'upper', 'no-transpose', 'non-unit', l, K - l, -1, T, strideT1, strideT2, offsetT, T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
// T_{1,2} = T_{1,2}*T_{2,2}
dtrmm( 'right', 'upper', 'no-transpose', 'non-unit', l, K - l, 1, T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * l ), T, strideT1, strideT2, offsetT + ( strideT2 * l ) );
} else if ( ql ) {
/*
Break V apart into 6 components
V = |---------------|
|V_{1,1} V_{1,2}|
|V_{2,1} V_{2,2}|
|0 V_{3,2}|
|---------------|
V_{1,1}\in\R^{n-k,k-l} rectangular
V_{2,1}\in\R^{k-l,k-l} unit upper triangular
V_{1,2}\in\R^{n-k,l} rectangular
V_{2,2}\in\R^{k-l,l} rectangular
V_{3,2}\in\R^{l,l} unit upper triangular
We will construct the T matrix
T = |---------------|
|T_{1,1} 0 |
|T_{2,1} T_{2,2}|
|---------------|
T is the triangular factor obtained from block reflectors.
To motivate the structure, assume we have already computed T_{1,1}
and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
T_{2,2}\in\R^{l, l} non-unit lower triangular
T_{2,1}\in\R^{k-l, l} rectangular
Where l = floor(k/2)
Then, consider the product:
(I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
= I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
Then, we can define the matrix V as
V = |-------|
|V_1 V_2|
|-------|
So, our product is equivalent to the matrix product
I - V*T*V'
This means, we can compute T_{1,1} and T_{2,2}, then use this information
to compute T_{2,1}
*/
// Compute T_{1,1} recursively
dlarft( direct, storev, N - l, K - l, V, strideV1, strideV2, offsetV, TAU, strideTAU, offsetTAU, T, strideT1, strideT2, offsetT );
// Compute T_{2,2} recursively
dlarft( direct, storev, N - l, K - l, V, strideV1, strideV2, offsetV + ( ( strideV1 + strideV2 ) * l ), TAU, strideTAU, offsetTAU + ( strideTAU * l ), T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * l ) );
// Compute T_{2,1} = V_{2,2}
for ( j = 0; j < K - l; j++ ) {
for ( i = 0; i < l; i++ ) {
T[ offsetT + ( strideT1 * ( K - l + i ) ) + ( strideT2 * j ) ] = V[ offsetV + ( strideV1 * ( N - K + j ) ) + ( strideV2 * ( K - l + i ) ) ];
}
}
// T_{2,1} = T_{2,1}*V_{2,1}
dtrmm( 'right', 'upper', 'no-transpose', 'unit', l, K - l, 1, V, strideV1, strideV2, offsetV + ( strideV1 * ( N - K ) ), T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
// T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
// Note: We assume K <= N, and GEMM will do nothing if N=K
dgemm( 'transpose', 'no-transpose', l, K - l, N - K, 1, V, strideV1, strideV2, offsetV + ( strideV2 * ( K - l ) ), V, strideV1, strideV2, offsetV, 1, T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
/*
At this point, we have that T_{2,1} = V_2'*V_1
All that is left is to pre and post multiply by -T_{2,2} and T_{1,1} respectively.
T_{2,1} = -T_{2,2}*T_{2,1}
*/
dtrmm( 'left', 'lower', 'no-transpose', 'non-unit', l, K - l, -1, T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * ( K - l ) ), T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
// T_{2,1} = T_{2,1}*T_{1,1}
dtrmm( 'right', 'lower', 'no-transpose', 'non-unit', l, K - l, 1, T, strideT1, strideT2, offsetT, T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
} else {
/*
Else means RQ case
Break V apart into 6 components
V = |-----------------------|
|V_{1,1} V_{1,2} 0 |
|V_{2,1} V_{2,2} V_{2,3}|
|-----------------------|
V_{1,1}\in\R^{k-l,n-k} rectangular
V_{1,2}\in\R^{k-l,k-l} unit lower triangular
V_{2,1}\in\R^{l,n-k} rectangular
V_{2,2}\in\R^{l,k-l} rectangular
V_{2,3}\in\R^{l,l} unit lower triangular
We will construct the T matrix
T = |---------------|
|T_{1,1} 0 |
|T_{2,1} T_{2,2}|
|---------------|
T is the triangular factor obtained from block reflectors.
To motivate the structure, assume we have already computed T_{1,1}
and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
T_{2,2}\in\R^{l, l} non-unit lower triangular
T_{2,1}\in\R^{k-l, l} rectangular
Where l = floor(k/2)
Then, consider the product:
(I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
= I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
Then, we can define the matrix V as
V = |---|
|V_1|
|V_2|
|---|
So, our product is equivalent to the matrix product
I - V'*T*V
This means, we can compute T_{1,1} and T_{2,2}, then use this information
to compute T_{2,1}
*/
// Compute T_{1,1} recursively
dlarft( direct, storev, N - l, K - l, V, strideV1, strideV2, offsetV, TAU, strideTAU, offsetTAU, T, strideT1, strideT2, offsetT );
// Compute T_{2,2} recursively
dlarft( direct, storev, N, l, V, strideV1, strideV2, offsetV + ( strideV1 * ( K - l ) ), TAU, strideTAU, offsetTAU + ( strideTAU * ( K - l ) ), T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * ( K - l ) ) );
// Compute T_{2,1} = V_{2,2}
dlacpy( 'all', l, K - l, V, strideV1, strideV2, offsetV + ( strideV1 * ( K - l ) ) + ( strideV2 * ( N - K ) ), T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
// T_{2,1} = T_{2,1}*V_{1,2}
dtrmm( 'right', 'lower', 'transpose', 'unit', l, K - l, 1, V, strideV1, strideV2, offsetV + ( strideV2 * ( N - K ) ), T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
// T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1}
// Note: We assume K <= N, and GEMM will do nothing if N=K
dgemm('no-transpose', 'transpose', l, K - l, N - K, 1, V, strideV1, strideV2, offsetV + ( strideV1 * ( K - l ) ), V, strideV1, strideV2, offsetV, 1, T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
/*
At this point, we have that T_{2,1} = V_2*V_1'
All that is left is to pre and post multiply by -T_{2,2} and T_{1,1} respectively.
T_{2,1} = -T_{2,2}*T_{2,1}
*/
dtrmm( 'left', 'lower', 'no-transpose', 'non-unit', l, K - l, -1, T, strideT1, strideT2, offsetT + ( ( strideT1 + strideT2 ) * ( K - l ) ), T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
// T_{2,1} = T_{2,1}*T_{1,1}
dtrmm( 'right', 'lower', 'no-transpose', 'non-unit', l, K - l, 1, T, strideT1, strideT2, offsetT, T, strideT1, strideT2, offsetT + ( strideT1 * ( K - l ) ) );
}
return T;
}
// EXPORTS //
module.exports = dlarft;
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