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) 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-lines-per-function, max-statements, max-lines, max-depth */
'use strict';
// MODULES //
var floor = require( '@stdlib/math/base/special/floor' );
var abs = require( '@stdlib/math/base/special/abs' );
var dlamch = require( '@stdlib/lapack/base/dlamch' );
var dgemm = require( '@stdlib/blas/base/dgemm' ).ndarray;
var dlacpy = require( '@stdlib/lapack/base/dlacpy' ).ndarray;
var Float64Array = require( '@stdlib/array/float64' );
var dlaset = require( '@stdlib/lapack/base/dlaset' ).ndarray;
var mod = require( '@stdlib/math/base/special/fmod' );
var min = require( '@stdlib/math/base/special/fast/min' );
var max = require( '@stdlib/math/base/special/fast/max' );
var dtrmm = require( './dtrmm.js' );
var dlarfg = require( './dlarfg.js' );
var dlaqr1 = require( './dlaqr1.js' );
// VARIABLES //
var safmin = dlamch( 'safe minimum' );
var ulp = dlamch( 'precision' );
// FUNCTIONS //
/**
* Shuffle shifts into pairs of real shifts and pairs of complex conjugate shifts, assuming that complex conjugate shifts are already adjacent to one another.
*
* @private
* @param {integer} nshifts - number of simultaneous shifts, must be even and positive
* @param {Float64Array} SR - real parts of the shifts of origin that define the QR sweep
* @param {integer} strideSR - stride length of `SR`
* @param {NonNegativeInteger} offsetSR - starting index for `SR`
* @param {Float64Array} SI - imaginary parts of the shifts of origin that define the QR sweep
* @param {integer} strideSI - stride length of `SI`
* @param {NonNegativeInteger} offsetSI - starting index of `SI`
* @returns {void}
*/
function shuffleShifts( nshifts, SR, strideSR, offsetSR, SI, strideSI, offsetSI ) {
var swap;
var isi;
var isr;
var i;
isi = offsetSI;
isr = offsetSR;
for ( i = 0; i <= nshifts - 2; i += 2 ) {
if ( SI[ isi ] !== -SI[ isi + strideSI ] ) {
swap = SR[ isr ];
SR[ isr ] = SR[ isr + strideSR ];
SR[ isr + strideSR ] = SR[ isr + (2*strideSR) ];
SR[ isr + (2*strideSR) ] = swap;
swap = SI[ isi ];
SI[ isi ] = SI[ isi + strideSI ];
SI[ isi + strideSI ] = SI[ isi + (2*strideSI) ];
SI[ isi + (2*strideSI) ] = swap;
}
isi += (2*strideSI);
isr += (2*strideSR);
}
}
// MAIN //
/**
* Performs a single, small shift multiline QR sweep.
*
* @private
* @param {boolean} wantT - boolean value indicating whether the quasi triangular Schur factor is being computed
* @param {boolean} wantZ - boolean value indicating whether the orthogonal Schur factor is being computed
* @param {integer} kacc22 - integer value ranging from 0 to 2 (inclusive), specifies the computation mode for far-from-diagonal updates
* @param {integer} N - number of rows/columns in `H`
* @param {integer} KTOP - first row and column of the submatrix of `H` where the QR sweep will be applied
* @param {integer} KBOT - last row and column of the submatrix of `H` where the QR sweep will be applied
* @param {integer} nshifts - number of simultaneous shifts, must be even and positive
* @param {Float64Array} SR - real parts of the shifts of origin that define the QR sweep
* @param {integer} strideSR - stride length of `SR`
* @param {NonNegativeInteger} offsetSR - starting index for `SR`
* @param {Float64Array} SI - imaginary parts of the shifts of origin that define the QR sweep
* @param {integer} strideSI - stride length of `SI`
* @param {NonNegativeInteger} offsetSI - starting index of `SI`
* @param {Float64Array} H - input upper hessenberg matrix
* @param {integer} strideH1 - stride of the first dimension of `H`
* @param {integer} strideH2 - stride of the second dimension of `H`
* @param {NonNegativeInteger} offsetH - starting index of `H`
* @param {integer} iloZ - starting row from where the transformation must be applied if `wantZ` is true
* @param {integer} ihiZ - ending row from where the transformation must be applied if `wantZ` is true
* @param {Float64Array} Z - the QR sweep orthogonal similarity transformation is accumulated into `Z` between the rows and columns `iloZ` and `ihiZ` if `wantZ` is true, otherwise `Z` is not referenced
* @param {integer} strideZ1 - stride of the first dimension of `Z`
* @param {integer} strideZ2 - stride of the second dimension of `Z`
* @param {NonNegativeInteger} offsetZ - starting index of `Z`
* @param {Float64Array} V - householder vectors are stored column-wise, used in forming bulges for the multi shift QR algorithm
* @param {integer} strideV1 - stride of the first dimension of `V`
* @param {integer} strideV2 - stride of the second dimension of `V`
* @param {NonNegativeInteger} offsetV - starting index of `V`
* @param {Float64Array} U - used to hold the product of householder reflector that represent accumulated orthogonal transformations from the bulge-chasing process
* @param {integer} strideU1 - stride of the first dimension of `U`
* @param {integer} strideU2 - stride of the second dimension of `U`
* @param {NonNegativeInteger} offsetU - starting index of `U`
* @param {integer} NH - number of columns in `WH` available for workspace
* @param {Float64Array} WH - workspace array
* @param {integer} strideWH1 - stride of the first dimension of `WH`
* @param {integer} strideWH2 - stride of the second dimension of `WH`
* @param {NonNegativeInteger} offsetWH - starting index of `WH`
* @param {integer} NV - number of rows in `WV` available for workspace
* @param {Float64Array} WV - workspace array
* @param {integer} strideWV1 - stride of the first dimension of `WV`
* @param {integer} strideWV2 - stride of the second dimension of `WV`
* @param {NonNegativeInteger} offsetWV - starting index of `WV`
* @returns {void}
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
*
* var H = new Float64Array( [ 1.0, 1.0, 0.0, 0.0, 0.0, 2.0, 1.5, 0.0, 0.0, 0.0, 3, 2.0, 0.0, 0.0, 0.0, 4.0 ] );
* var Z = new Float64Array( [ 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0 ] );
* var V = new Float64Array( 6 );
* var U = new Float64Array( 10 );
* var WH = new Float64Array( 16 );
* var WV = new Float64Array( 16 );
* var SR = new Float64Array( [ 1.1, 2.2 ] );
* var SI = new Float64Array( [ 0.0, 0.0 ] );
*
* dlaqr5( true, true, 0, 4, 1, 4, 2, SR, 1, 0, SI, 1, 0, H, 4, 1, 0, 1, 4, Z, 4, 1, 0, V, 2, 1, 0, U, 2, 1, 0, 4, WH, 4, 1, 0, 4, WV, 4, 1, 0 );
* // H => <Float64Array>[ 1.0, 1.0, 0.0, 0.0, 0.0, 2.0, 1.5, 0.0, 0.0, 0.0, 3, 2.0, 0.0, 0.0, 0.0, 4.0 ]
* // Z => <Float64Array>[ 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0 ]
*/
function dlaqr5( wantT, wantZ, kacc22, N, KTOP, KBOT, nshifts, SR, strideSR, offsetSR, SI, strideSI, offsetSI, H, strideH1, strideH2, offsetH, iloZ, ihiZ, Z, strideZ1, strideZ2, offsetZ, V, strideV1, strideV2, offsetV, U, strideU1, strideU2, offsetU, NH, WH, strideWH1, strideWH2, offsetWH, NV, WV, strideWV1, strideWV2, offsetWV ) {
var dlarfgOut;
var block22;
var smlnum;
var refsum;
var mstart;
var incol;
var accum;
var nbmps;
var krcol;
var bmp22;
var alpha;
var start;
var ndcol;
var beta;
var jcol;
var jlen;
var jrow;
var jtop;
var jbot;
var mend;
var mtop;
var mbot;
var tst1;
var tst2;
var step;
var h11;
var h12;
var h21;
var h22;
var kdu;
var end;
var kms;
var knz;
var kzs;
var m22;
var scl;
var k1;
var nu;
var vt;
var i2;
var i4;
var j2;
var j4;
var ns;
var ih;
var k;
var m;
var j;
dlarfgOut = new Float64Array( 2 ); // Workspace array to pass `alpha` to the `dlarfg` routine
vt = new Float64Array( 3 ); // local array
// If there are no shifts, then there is nothing to do.
if ( nshifts < 2 ) {
return;
}
// If the active block is empty or 1-by-1, then there is nothing to do.
if ( KTOP >= KBOT ) {
return;
}
/*
* Shuffle shifts into pairs of real shifts and pairs of complex conjugate shifts,
* assuming that complex conjugate shifts are already adjacent to one another.
*/
shuffleShifts( nshifts, SR, strideSR, offsetSR, SI, strideSI, offsetSI );
// `nshifts` is supposed to be even, but if it is odd, then simply reduce it by one. The shuffle above ensures that the dropped shift is real and that the remaining shifts are paired.
ns = nshifts - mod( nshifts, 2.0 );
// Machine constants for deflation
smlnum = safmin * N / ulp;
// Use accumulated reflections to update far-from-diagonal entries?
accum = ( kacc22 === 1 ) || ( kacc22 === 2 );
// If so, exploit the 2-by-2 block structure?
block22 = ( ns > 2 ) && ( kacc22 === 2 );
// Clear trash
if ( KTOP + 2 <= KBOT ) {
ih = offsetH + ( (KTOP+2) * strideH1 ) + ( KTOP * strideH2 );
H[ ih ] = 0.0;
}
// `nbmps` = number of 2-shift bulges in the chain
nbmps = ns / 2;
// KDU = width of slab
kdu = ( 6 * nbmps ) - 3;
start = ( 3 * ( 1 - nbmps ) ) + KTOP - 1;
end = KBOT - 2;
step = ( 3 * nbmps ) - 2;
// Create and chase chains of `nbmps` bulges
for ( incol = start; incol <= end; incol += step ) {
ndcol = incol + kdu;
if ( accum ) {
dlaset( 'all', kdu, kdu, 0.0, 1.0, U, strideU1, strideU2, offsetU );
}
/*
* Near-the-diagonal bulge chase. The following loop performs the
* near-the-diagonal part of a small bulge multi-shift QR sweep. Each
* `6*nbmps-2` column diagonal chunk extends from column `incol` to column
* `ndcol` (including both column `incol` and column `ndcol`). The following
* loop chases a 3*`nbmps` column long chain of `nbmps` bulges `3*nbmps-2`
* columns to the right. (`incol` may be less than `KTOP` and and `ndcol`
* may be greater than `KBOT` indicating phantom columns from which to
* chase bulges before they are actually introduced or to which to
* chase bulges beyond column `KBOT`.)
*/
for ( krcol = incol; krcol < min( incol + ( 3*nbmps ) - 3, KBOT - 2 ); krcol+= 1 ) {
/*
* Bulges number `mtop` to `mbot` are active double implicit shift bulges.
* There may or may not also be small 2-by-2 bulge, if there is room.
* The inactive bulges (if any) must wait until the active bulges
* have moved down the diagonal to make room. The phantom matrix
* paradigm described above helps keep track.
*/
mtop = max( 1, floor( ( KTOP - 1 - krcol + 2 ) / 3 ) + 1 );
mbot = min( nbmps, floor( ( KBOT - krcol ) / 3 ) );
m22 = mbot + 1;
bmp22 = ( mbot < nbmps ) && ( krcol + ( 3 * ( m22 - 1 ) ) === KBOT - 2 );
/*
* Generate reflections to chase the chain right one column.
* (The minimum value of K is KTOP-1.)
*/
for ( m = mtop; m <= mbot; m++ ) {
k = krcol + ( 3 * ( m - 1 ) );
if ( k === KTOP - 1 ) {
dlaqr1( 3, H, strideH1, strideH2, offsetH + (KTOP*strideH1) + (KTOP*strideH2), SR[ offsetSR + (((2*m)-2)*strideSR) ], SI[ offsetSI + (((2*m)-2)*strideSI) ], SR[ offsetSR + (((2*m)-1)*strideSR) ], SI[ offsetSI + (((2*m)-1)*strideSI) ], vt, 1, 0 );
alpha = V[ offsetV + (m*strideV2) ];
// Prepare the `dlarfgOut` array to pass into the routine
dlarfgOut[ 0 ] = alpha;
dlarfgOut[ 1 ] = 0.0;
// Call `dlarfg` using the `dlarfgOut` array to store outputs
dlarfg( 3, V, strideV1, offsetV + (m*strideV2), dlarfgOut, 1, 0 );
// Write the outputs to their expected positions
alpha = dlarfgOut[ 0 ];
V[ offsetV + (m*strideV2) ] = dlarfgOut[ 1 ];
} else {
beta = H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ];
V[ offsetV + (m*strideV2) + strideV1 ] = H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ];
V[ offsetV + (m*strideV2) + (2*strideV1) ] = H[ offsetH + ((k+3)*strideH1) + (k*strideH2) ];
// Prepare the `dlarfgOut` array to pass into the routine
dlarfgOut[ 0 ] = beta;
dlarfgOut[ 1 ] = 0.0;
// Call `dlarfg` using the `dlarfgOut` array to store outputs
dlarfg( 3, V, strideV1, offsetV + (m*strideV2), dlarfgOut, 1, 0 );
// Write the outputs to their expected positions
beta = dlarfgOut[ 0 ];
V[ offsetV + (m*strideV2) ] = dlarfgOut[ 1 ];
/*
* A Bulge may collapse because of vigilant deflation or
* destructive underflow. In the underflow case, try the
* two-small-subdiagonals trick to try to reinflate the bulge.
*/
if ( H[ offsetH + ((k+3)*strideH1) + (k*strideH2) ] !== 0.0 || H[ offsetH + ((k+3)*strideH1) + ((k+1)*strideH2) ] !== 0.0 || H[ offsetH + ((k+3)*strideH1) + ((k+2)*strideH2) ] === 0.0 ) {
// Typical case: not collapsed (yet).
H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] = beta;
H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ] = 0.0;
H[ offsetH + ((k+3)*strideH1) + (k*strideH2) ] = 0.0;
} else {
/*
* Atypical case: collapsed. Attempt to reintroduce
* ignoring H(K+1,K) and H(K+2,K). If the fill
* resulting from the new reflector is too large,
* then abandon it. Otherwise, use the new one.
*/
dlaqr1( 3, H, strideH1, strideH2, offsetH + ((k+1)*strideH1) + ((k+1)*strideH2), SR, strideSR, offsetSR + (((2*m)-1)*strideSR), SI, strideSI, offsetSI + (((2*m)-1)*strideSI), SR, strideSR, offsetSR + (2*m*strideSR), SI, strideSI, offsetSI + (2*m*strideSI), vt );
alpha = vt[ 0 ];
// Prepare the `dlarfgOut` array to pass into the routine
dlarfgOut[ 0 ] = alpha;
dlarfgOut[ 1 ] = 0.0;
// Call `dlarfg` using the `dlarfgOut` array to store outputs
dlarfg( 3, vt, 1, 1, dlarfgOut, 1, 0 );
// Write the outputs to their expected positions
alpha = dlarfgOut[ 0 ];
vt[ 0 ] = dlarfgOut[ 1 ];
refsum = vt[ 0 ] * ( H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] + ( vt[ 1 ] * H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ] ) );
if ( abs( H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ] - ( refsum * vt[ 1 ] ) ) + abs( refsum * vt[ 2 ] ) > ulp * ( abs( H[ offsetH + (k*strideH1) + (k*strideH2) ] ) + abs( H[ offsetH + ((k+1)*strideH1) + ((k+1)*strideH2) ] ) + abs( H[ offsetH + ((k+2)*strideH1) + ((k+2)*strideH2) ] ) ) ) {
/*
* Starting a new bulge here would create
* non-negligible fill. Use the old one with
* trepidation.
*/
H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] = beta;
H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ] = 0.0;
H[ offsetH + ((k+3)*strideH1) + (k*strideH2) ] = 0.0;
} else {
/*
* Stating a new bulge here would create only
* negligible fill. Replace the old reflector
* with the new one.
*/
H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] -= refsum;
H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ] = 0.0;
H[ offsetH + ((k+3)*strideH1) + (k*strideH2) ] = 0.0;
V[ offsetV + (m*strideV2) ] = vt[ 0 ];
V[ offsetV + (m*strideV2) + strideV1 ] = vt[ 1 ];
V[ offsetV + (m*strideV2) + (2*strideV1) ] = vt[ 2 ];
}
}
}
}
// Generate a 2-by-2 reflection, if needed.
k = krcol + ( 3 * ( m22 - 1 ) );
if ( bmp22 ) {
if ( k === KTOP - 1 ) {
dlaqr1( 2, H, strideH1, strideH2, offsetH + ((k+1)*strideH1) + ((k+1)*strideH2), SR[ offsetSR + (((2*m22)-2)*strideSR) ], SI[ offsetSI + (((2*m22)-2)*strideSI) ], SR[ offsetSR + (((2*m22)-1)*strideSR) ], SI[ offsetSI + (((2*m22)-1)*strideSI) ], vt, 1, 0 );
beta = V[ offsetV + (m22*strideV2) ];
// Prepare the `dlarfgOut` array to pass into the routine
dlarfgOut[ 0 ] = beta;
dlarfgOut[ 1 ] = 0.0;
// Call `dlarfg` using the `dlarfgOut` array to store outputs
dlarfg( 2, V, strideV1, offsetV + (m22*strideV2), dlarfgOut, 1, 0 );
// Write the outputs to their expected positions
beta = dlarfgOut[ 0 ];
V[ offsetV + (m22*strideV2) ] = dlarfgOut[ 1 ];
} else {
beta = H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ];
V[ offsetV + (m22*strideV2) + strideV1 ] = H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ];
// Prepare the `dlarfgOut` array to pass into the routine
dlarfgOut[ 0 ] = beta;
dlarfgOut[ 1 ] = 0.0;
// Call `dlarfg` using the `dlarfgOut` array to store outputs
dlarfg( 2, V, strideV1, offsetV + (m22*strideV2), dlarfgOut, 1, 0 );
// Write the outputs to their expected positions
beta = dlarfgOut[ 0 ];
V[ offsetV + (m22*strideV2) ] = dlarfgOut[ 1 ];
H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] = beta;
H[ offsetH + ((k+2)*strideH1) + (k*strideH2) ] = 0.0;
}
}
// Multiply H by reflections from the left
if ( accum ) {
jbot = min( ndcol, KBOT );
} else if ( wantT ) {
jbot = N;
} else {
jbot = KBOT;
}
for ( j = max( KTOP, krcol ); j <= jbot; j++ ) {
mend = min( mbot, floor( ( j - krcol + 2 ) / 3 ) );
for ( m = mtop; m <= mend; m++ ) {
k = krcol + ( 3 * ( m - 1 ) );
refsum = V[ offsetV + (m*strideV2) ] * ( H[ offsetH + ((k+1)*strideH1) + (j*strideH2) ] + ( V[ offsetV + (m*strideV2) + strideV1 ] * H[ offsetH + ((k+2)*strideH1) + (j*strideH2) ] ) + ( V[ offsetV + (m*strideV2) + (2*strideV1) ] * H[ offsetH + ((k+3)*strideH1) + (j*strideH2) ] ) );
H[ offsetH + ((k+1)*strideH1) + (j*strideH2) ] -= refsum;
H[ offsetH + ((k+2)*strideH1) + (j*strideH2) ] -= refsum * V[ offsetV + (m*strideV2) + strideV1 ];
H[ offsetH + ((k+3)*strideH1) + (j*strideH2) ] -= refsum * V[ offsetV + (m*strideV2) + (2*strideV1) ];
}
}
if ( bmp22 ) {
k = krcol + ( 3 * ( m22 - 1 ) );
for ( j = max( k + 1, KTOP ); j <= jbot; j++ ) {
refsum = V[ offsetV + (m22*strideV2) ] * ( H[ offsetH + ((k+1)*strideH1) + (j*strideH2) ] + ( V[ offsetV + (m22*strideV2) + strideV1 ] * H[ offsetH + ((k+2)*strideH1) + (j*strideH2) ] ) );
H[ offsetH + ((k+1)*strideH1) + (j*strideH2) ] -= refsum;
H[ offsetH + ((k+2)*strideH1) + (j*strideH2) ] -= refsum * V[ offsetV + (m22*strideV2) + strideV1 ];
}
}
/*
* Multiply H by reflections from the right.
* Delay filling in the last row until the
* vigilant deflation check is complete.
*/
if ( accum ) {
jtop = max( KTOP, incol );
} else if ( wantT ) {
jtop = 1;
} else {
jtop = KTOP;
}
for ( m = mtop; m <= mbot; m++ ) {
if ( V[ offsetV + (m*strideV2) ] !== 0.0 ) {
k = krcol + ( 3 * ( m - 1 ) );
for ( j = jtop; j <= min( KBOT, k + 3 ); j++ ) {
refsum = V[ offsetV + (m*strideV2) ] * ( H[ offsetH + (j*strideH1) + ((k+1)*strideH2) ] + ( V[ offsetV + (m*strideV2) + strideV1 ] * H[ offsetH + (j*strideH1) + ((k+2)*strideH2) ] ) + ( V[ offsetV + (m*strideV2) + (2*strideV1) ] * H[ offsetH + (j*strideH1) + ((k+3)*strideH2) ] ) );
H[ offsetH + (j*strideH1) + ((k+1)*strideH2) ] -= refsum;
H[ offsetH + (j*strideH1) + ((k+2)*strideH2) ] -= refsum * V[ offsetV + (m*strideV2) + strideV1 ];
H[ offsetH + (j*strideH1) + ((k+3)*strideH2) ] -= refsum * V[ offsetV + (m*strideV2) + (2*strideV1) ];
}
if ( accum ) {
/*
* Accumulate U. (If necessary, update Z later
* with with an efficient matrix-matrix multiply.)
*/
kms = k - incol;
for ( j = max( 1, KTOP - incol ); j <= kdu; j++ ) {
refsum = V[ offsetV + (m*strideV2) ] * ( U[ offsetU + (j*strideU1) + ((kms+1)*strideU2) ] + ( V[ offsetV + (m*strideV2) + strideV1 ] * U[ offsetU + (j*strideU1) + ((kms+2)*strideU2) ] ) + ( V[ offsetV + (m*strideV2) + (2*strideV1) ] * U[ offsetU + (j*strideU1) + ((kms+3)*strideU2) ] ) );
U[ offsetU + (j*strideU1) + ((kms+1)*strideU2) ] -= refsum;
U[ offsetU + (j*strideU1) + ((kms+2)*strideU2) ] -= refsum * V[ offsetV + (m*strideV2) + strideV1 ];
U[ offsetU + (j*strideU1) + ((kms+3)*strideU2) ] -= refsum * V[ offsetV + (m*strideV2) + (2*strideV1) ];
}
} else if ( wantZ ) {
/*
* U is not accumulated, so update Z now by
* multiplying by reflections from the right.
*/
for ( j = iloZ; j <= ihiZ; j++ ) {
refsum = V[ offsetV + (m*strideV2) ] * ( Z[ offsetZ + (j*strideZ1) + ((k+1)*strideZ2) ] + ( V[ offsetV + (m*strideV2) + strideV1 ] * Z[ offsetZ + (j*strideZ1) + ((k+2)*strideZ2) ] ) + ( V[ offsetV + (m*strideV2) + (2*strideV1) ] * Z[ offsetZ + (j*strideZ1) + ((k+3)*strideZ2) ] ) );
Z[ offsetZ + (j*strideZ1) + ((k+1)*strideZ2) ] -= refsum;
Z[ offsetZ + (j*strideZ1) + ((k+2)*strideZ2) ] -= refsum * V[ offsetV + (m*strideV2) + strideV1 ];
Z[ offsetZ + (j*strideZ1) + ((k+3)*strideZ2) ] -= refsum * V[ offsetV + (m*strideV2) + (2*strideV1) ];
}
}
}
}
// Special case: 2-by-2 reflection (if needed)
k = krcol + ( 3 * ( m22 - 1 ) );
if ( bmp22 ) {
if ( V[ offsetV + (m22*strideV2) ] !== 0.0 ) {
for ( j = jtop; j <= min( KBOT, k + 3 ); j++ ) {
refsum = V[ offsetV + (m22*strideV2) ] * ( H[ offsetH + (j*strideH1) + ((k+1)*strideH2) ] + ( V[ offsetV + (m22*strideV2) + strideV1 ] * H[ offsetH + (j*strideH1) + ((k+2)*strideH2) ] ) );
H[ offsetH + (j*strideH1) + ((k+1)*strideH2) ] -= refsum;
H[ offsetH + (j*strideH1) + ((k+2)*strideH2) ] -= refsum * V[ offsetV + (m22*strideV2) + strideV1 ];
}
if ( accum ) {
kms = k - incol;
for ( j = max( 1, KTOP - incol ); j <= kdu; j++ ) {
refsum = V[ offsetV + (m22*strideV2) ] * ( U[ offsetU + (j*strideU1) + ((kms+1)*strideU2) ] + ( V[ offsetV + (m22*strideV2) + strideV1 ] * U[ offsetU + (j*strideU1) + ((kms+2)*strideU2) ] ) );
U[ offsetU + (j*strideU1) + ((kms+1)*strideU2) ] -= refsum;
U[ offsetU + (j*strideU1) + ((kms+2)*strideU2) ] -= refsum * V[ offsetV + (m22*strideV2) + strideV1 ];
}
} else if ( wantZ ) {
for ( j = iloZ; j <= ihiZ; j++ ) {
refsum = V[ offsetV + (m22*strideV2) ] * ( Z[ offsetZ + (j*strideZ1) + ((k+1)*strideZ2) ] + ( V[ offsetV + (m22*strideV2) + strideV1 ] * Z[ offsetZ + (j*strideZ1) + ((k+2)*strideZ2) ] ) );
Z[ offsetZ + (j*strideZ1) + ((k+1)*strideZ2) ] -= refsum;
Z[ offsetZ + (j*strideZ1) + ((k+2)*strideZ2) ] -= refsum * V[ offsetV + (m22*strideV2) + strideV1 ];
}
}
}
}
// Vigilant deflation check
mstart = mtop;
if ( krcol + ( 3 * ( mstart - 1 ) ) < KTOP ) {
mstart += 1;
}
mend = mbot;
if ( bmp22 ) {
mend += 1;
}
if ( krcol === KBOT - 2 ) {
mend += 1;
}
for ( m = mstart; m <= mend; m++ ) {
k = min( KBOT - 1, krcol + ( 3 * ( m - 1 ) ) );
/*
* The following convergence test requires that the tradition
* small-compared-to-nearby-diagonals criterion and the
* Ahues & Tisseur (LAWN 122, 1997) criteria both be satisfied.
* The latter improves accuracy in some examples. Falling
* back on an alternate convergence criterion when TST1 or
* TST2 is zero (as done here) is traditional but probably
* unnecessary.
*/
if ( H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] !== 0.0 ) {
tst1 = abs( H[ offsetH + (k*strideH1) + (k*strideH2) ] ) + abs( H[ offsetH + ((k+1)*strideH1) + ((k+1)*strideH2) ] );
if ( tst1 === 0.0 ) {
if ( k >= KTOP + 1 ) {
tst1 += abs( H[ offsetH + (k*strideH1) + ((k-1)*strideH2) ] );
}
if ( k >= KTOP + 2 ) {
tst1 += abs( H[ offsetH + (k*strideH1) + ((k-2)*strideH2) ] );
}
if ( k >= KTOP + 3 ) {
tst1 += abs( H[ offsetH + (k*strideH1) + ((k-3)*strideH2) ] );
}
if ( k <= KBOT - 2 ) {
tst1 += abs( H[ offsetH + ((k+2)*strideH1) + ((k+1)*strideH2) ] );
}
if ( k <= KBOT - 3 ) {
tst1 += abs( H[ offsetH + ((k+3)*strideH1) + ((k+1)*strideH2) ] );
}
if ( k <= KBOT - 4 ) {
tst1 += abs( H[ offsetH + ((k+4)*strideH1) + ((k+1)*strideH2) ] );
}
}
if ( abs( H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] ) <= max( smlnum, ulp * tst1 ) ) {
h12 = max( abs( H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] ), abs( H[ offsetH + (k*strideH1) + ((k+1)*strideH2) ] ) );
h21 = min( abs( H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] ), abs( H[ offsetH + (k*strideH1) + ((k+1)*strideH2) ] ) );
h11 = max( abs( H[ offsetH + ((k+1)*strideH1) + ((k+1)*strideH2) ] ), abs( H[ offsetH + (k*strideH1) + (k*strideH2) ] - H[ offsetH + ((k+1)*strideH1) + ((k+1)*strideH2) ] ) );
h22 = min( abs( H[ offsetH + ((k+1)*strideH1) + ((k+1)*strideH2) ] ), abs( H[ offsetH + (k*strideH1) + (k*strideH2) ] - H[ offsetH + ((k+1)*strideH1) + ((k+1)*strideH2) ] ) );
scl = h11 + h12;
tst2 = h22 * ( h11 / scl );
if ( tst2 === 0.0 || h21 * ( h12 / scl ) <= max( smlnum, ulp * tst2 ) ) {
H[ offsetH + ((k+1)*strideH1) + (k*strideH2) ] = 0.0;
}
}
}
}
// Fill in the last row of each bulge.
mend = min( nbmps, floor( ( KBOT - krcol - 1 ) / 3 ) );
for ( m = mtop; m <= mend; m++ ) {
k = krcol + ( 3 * ( m - 1 ) );
refsum = V[ offsetV + (m*strideV2) ] * V[ offsetV + (m*strideV2) + (2*strideV1) ] * H[ offsetH + ((k+4)*strideH1) + ((k+3)*strideH2) ];
H[ offsetH + ((k+4)*strideH1) + ((k+1)*strideH2) ] = -refsum;
H[ offsetH + ((k+4)*strideH1) + ((k+2)*strideH2) ] = -refsum * V[ offsetV + (m*strideV2) + strideV1 ];
H[ offsetH + ((k+4)*strideH1) + ((k+3)*strideH2) ] -= refsum * V[ offsetV + (m*strideV2) + (2*strideV1) ];
}
} // End of near-the-diagonal bulge chase.
// Use U (if accumulated) to update far-from-diagonal entries in H. If required, use U to update Z as well.
if ( accum ) {
if ( wantT ) {
jtop = 1;
jbot = N;
} else {
jtop = KTOP;
jbot = KBOT;
}
/*
* Updates not exploiting the 2-by-2 block structure of U. K1 and NU
* keep track of the location and size of U in the special cases of
* introducing bulges and chasing bulges off the bottom. In these
* special cases and in case the number of shifts is NS = 2, there
* is no 2-by-2 block structure to exploit.
*/
if ( !block22 || incol < KTOP || ndcol > KBOT || ns <= 2 ) {
k1 = max( 1, KTOP - incol );
nu = kdu - max( 0, ndcol - KBOT ) - k1 + 1;
// Horizontal Multiply
for ( jcol = min( ndcol, KBOT ) + 1; jcol <= jbot; jcol += NH ) {
jlen = min( NH, jbot - jcol + 1 );
dgemm( 'conjugate-transpose', 'no-transpose', nu, jlen, nu, 1.0, U, strideU1, strideU2, offsetU + (k1*strideU1) + (k1*strideU2), H, strideH1, strideH2, offsetH + ((incol+k1)*strideH1) + (jcol*strideH2), 0.0, WH, strideWH1, strideWH2, offsetWH );
dlacpy( 'all', nu, jlen, WH, strideWH1, strideWH2, offsetWH, H, strideH1, strideH2, offsetH + ((incol+k1)*strideH1) + (jcol*strideH2) );
}
// Vertical multiply
for ( jrow = jtop; jrow <= max( KTOP, incol ) - 1; jrow += NV ) {
jlen = min( NV, max( KTOP, incol ) - jrow );
dgemm( 'no-transpose', 'no-transpose', jlen, nu, nu, 1.0, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+k1)*strideH2), U, strideU1, strideU2, offsetU + (k1*strideU1) + (k1*strideU2), 0.0, WV, strideWV1, strideWV2, offsetWV );
dlacpy( 'all', jlen, nu, WV, strideWV1, strideWV2, offsetWV, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+k1)*strideH2) );
}
// Z multiply (also vertical)
if ( wantZ ) {
for ( jrow = iloZ; jrow <= ihiZ; jrow += NV ) {
jlen = min( NV, ihiZ - jrow + 1 );
dgemm( 'no-transpose', 'no-transpose', jlen, nu, nu, 1.0, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+k1)*strideZ2), U, strideU1, strideU2, offsetU + (k1*strideU1) + (k1*strideU2), 0.0, WV, strideWV1, strideWV2, offsetWV );
dlacpy( 'all', jlen, nu, WV, strideWV1, strideWV2, offsetWV, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+k1)*strideZ2) );
}
}
} else {
/*
* Updates exploiting U's 2-by-2 block structure.
* (I2, I4, J2, J4 are the last rows and columns of the blocks.)
*/
i2 = floor( ( kdu + 1 ) / 2 );
i4 = kdu;
j2 = i4 - i2;
j4 = kdu;
/*
* KZS and KNZ deal with the band of zeros along the diagonal
* of one of the triangular blocks.
*/
kzs = j4 - j2 - ( ns + 1 );
knz = ns + 1;
// Horizontal multiply
for ( jcol = min( ndcol, KBOT ) + 1; jcol <= jbot; jcol += NH ) {
jlen = min( NH, jbot - jcol + 1 );
/*
* Copy bottom of H to top+KZS of scratch
* (The first KZS rows get multiplied by zero.)
*/
dlacpy( 'all', knz, jlen, H, strideH1, strideH2, offsetH + ((incol+1+j2)*strideH1) + (jcol*strideH2), WH, strideWH1, strideWH2, offsetWH + ((kzs+1)*strideWH1) );
// Multiply by U21**T
dlaset( 'all', kzs, jlen, 0.0, 0.0, WH, strideWH1, strideWH2, offsetWH );
dtrmm( 'left', 'upper', 'conjugate-transpose', 'non-unit', knz, jlen, 1.0, U, strideU1, strideU2, offsetU + ((j2+1)*strideU1) + ((1+kzs)*strideU2), WH, strideWH1, strideWH2, offsetWH + ((kzs+1)*strideWH1) );
// Multiply top of H by U11**T
dgemm( 'conjugate-transpose', 'no-transpose', i2, jlen, j2, 1.0, U, strideU1, strideU2, offsetU, H, strideH1, strideH2, offsetH + ((incol+1)*strideH1) + (jcol*strideH2), 1.0, WH, strideWH1, strideWH2, offsetWH );
// Copy top of H to bottom of WH
dlacpy( 'all', j2, jlen, H, strideH1, strideH2, offsetH + ((incol+1)*strideH1) + (jcol*strideH2), WH, strideWH1, strideWH2, offsetWH + ((i2+1)*strideWH1) );
// Multiply by U21**T
dtrmm( 'left', 'lower', 'conjugate-transpose', 'non-unit', j2, jlen, 1.0, U, strideU1, strideU2, offsetU + (1*strideU1) + ((i2+1)*strideU2), WH, strideWH1, strideWH2, offsetWH + ((i2+1)*strideWH1) );
// Multiply by U22
dgemm( 'conjugate-transpose', 'no-transpose', i4 - i2, jlen, j4 - j2, 1.0, U, strideU1, strideU2, offsetU + ((j2+1)*strideU1) + ((i2+1)*strideU2), H, strideH1, strideH2, offsetH + ((incol+1+j2)*strideH1) + (jcol*strideH2), 1.0, WH, strideWH1, strideWH2, offsetWH + ((i2+1)*strideWH1) );
// Copy it back
dlacpy( 'all', kdu, jlen, WH, strideWH1, strideWH2, offsetWH, H, strideH1, strideH2, offsetH + ((incol+1)*strideH1) + (jcol*strideH2) );
}
// Vertical multiply
for ( jrow = jtop; jrow <= max( incol, KTOP ) - 1; jrow += NV ) {
jlen = min( NV, max( incol, KTOP ) - jrow );
/*
* Copy right of H to scratch (the first KZS columns get multiplied by zero)
*/
dlacpy( 'all', jlen, knz, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+1+j2)*strideH2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+kzs)*strideWV2) );
// Multiply by U21
dlaset( 'all', jlen, kzs, 0.0, 0.0, WV, strideWV1, strideWV2, offsetWV );
dtrmm( 'right', 'upper', 'no-transpose', 'non-unit', jlen, knz, 1.0, U, strideU1, strideU2, offsetU + ((j2+1)*strideU1) + ((1+kzs)*strideU2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+kzs)*strideWV2) );
// Multiply by U11
dgemm( 'no-transpose', 'no-transpose', jlen, i2, j2, 1.0, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+1)*strideH2), U, strideU1, strideU2, offsetU, 1.0, WV, strideWV1, strideWV2, offsetWV );
// Copy left of H to right of scratch
dlacpy( 'all', jlen, j2, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+1)*strideH2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+i2)*strideWV2) );
// Multiply by U21
dtrmm( 'right', 'lower', 'no-transpose', 'non-unit', jlen, i4 - i2, 1.0, U, strideU1, strideU2, offsetU + (1*strideU1) + ((i2+1)*strideU2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+i2)*strideWV2) );
// Multiply by U22
dgemm( 'no-transpose', 'no-transpose', jlen, i4 - i2, j4 - j2, 1.0, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+1+j2)*strideH2), U, strideU1, strideU2, offsetU + ((j2+1)*strideU1) + ((i2+1)*strideU2), 1.0, WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+i2)*strideWV2) );
// Copy it back
dlacpy( 'all', jlen, kdu, WV, strideWV1, strideWV2, offsetWV, H, strideH1, strideH2, offsetH + (jrow*strideH1) + ((incol+1)*strideH2) );
}
// Multiply Z (also vertical)
if ( wantZ ) {
for ( jrow = iloZ; jrow <= ihiZ; jrow += NV ) {
jlen = min( NV, ihiZ - jrow + 1 );
/*
* Copy right of Z to left of scratch (first KZS columns get multiplied by zero)
*/
dlacpy( 'all', jlen, knz, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+1+j2)*strideZ2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+kzs)*strideWV2) );
// Multiply by U12
dlaset( 'all', jlen, kzs, 0.0, 0.0, WV, strideWV1, strideWV2, offsetWV );
dtrmm( 'right', 'upper', 'no-transpose', 'non-unit', jlen, knz, 1.0, U, strideU1, strideU2, offsetU + ((j2+1)*strideU1) + ((1+kzs)*strideU2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+kzs)*strideWV2) );
// Multiply by U11
dgemm( 'no-transpose', 'no-transpose', jlen, i2, j2, 1.0, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+1)*strideZ2), U, strideU1, strideU2, offsetU, 1.0, WV, strideWV1, strideWV2, offsetWV );
// Copy left of Z to right of scratch
dlacpy( 'all', jlen, j2, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+1)*strideZ2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+i2)*strideWV2) );
// Multiply by U21
dtrmm( 'right', 'lower', 'no-transpose', 'non-unit', jlen, i4 - i2, 1.0, U, strideU1, strideU2, offsetU + (1*strideU1) + ((i2+1)*strideU2), WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+i2)*strideWV2) );
// Multiply by U22
dgemm( 'no-transpose', 'no-transpose', jlen, i4 - i2, j4 - j2, 1.0, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+1+j2)*strideZ2), U, strideU1, strideU2, offsetU + ((j2+1)*strideU1) + ((i2+1)*strideU2), 1.0, WV, strideWV1, strideWV2, offsetWV + (1*strideWV1) + ((1+i2)*strideWV2) );
// Copy the result back to Z
dlacpy( 'all', jlen, kdu, WV, strideWV1, strideWV2, offsetWV, Z, strideZ1, strideZ2, offsetZ + (jrow*strideZ1) + ((incol+1)*strideZ2) );
}
}
}
}
}
}
// EXPORTS //
module.exports = dlaqr5;
|