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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, max-lines-per-function */
'use strict';
// MODULES //
var dswap = require( '@stdlib/blas/base/dswap' ).ndarray;
var dlamch = require( '@stdlib/lapack/base/dlamch' );
var dnrm2 = require( '@stdlib/blas/base/dnrm2' ).ndarray;
var idamax = require( '@stdlib/blas/base/idamax' ).ndarray;
var abs = require( '@stdlib/math/base/special/abs' );
var isnan = require( '@stdlib/assert/is-nan' );
var max = require( '@stdlib/math/base/special/maxn' );
var min = require( '@stdlib/math/base/special/minn' );
var dscal = require( '@stdlib/blas/base/dscal' ).ndarray;
// VARIABLES //
var sclfac = 2.0;
var factor = 0.95;
// MAIN //
/**
* Balances a general real matrix `A`.
*
* ## Notes
*
* The job parameter can be one of the following:
*
* - 'none': none, return immediately
* - 'permute': permute only
* - 'scale': scale only
* - 'both': both permute and scale
*
* The matrix `A` is overwritten by the balanced matrix. Scale factors are stored in the `scale` array. If `P[ j ]` is the index of the row and column interchanged with row and column j (zero-based) and `D[ j ]` is the scaling factor applied to row and column j, then:
*
* - `scale[ j ]` = `P[ j ]` for j = `0`,...,`out[ 0 ]-1`
* - `scale[ j ]` = `D[ j ]` for j = `out[ 0 ]`,...,`out[ 1 ]`
* - `scale[ j ]` = `P[ j ]` for j = `out[ 1 ]+1`,...,`N-1`.
*
* The order in which the interchanges are made is `N-1` to `out[ 1 ]+1`, then `0` to `out[ 0 ]-1`.
*
* @private
* @param {string} job - indicates the operations to be performed
* @param {NonNegativeInteger} N - number of rows/columns in matrix `A`
* @param {Float64Array} A - input matrix to be balanced
* @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 {Int32Array} out - stores the first and last row/column of the balanced submatrix
* @param {integer} strideOut - stride of `out`
* @param {NonNegativeInteger} offsetOut - starting index for `out`
* @param {Float64Array} scale - array containing permutation and scaling information
* @param {integer} strideScale - stride of `scale`
* @param {NonNegativeInteger} offsetScale - starting index for `scale`
* @throws {RangeError} should not return NaN
* @returns {integer} status code
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
* var Int32Array = require( '@stdlib/array/int32' );
*
* var A = new Float64Array( [ 1.0, 100.0, 0.0, 2.0, 200.0, 0.0, 0.0, 0.0, 3.0 ] );
* var out = new Int32Array( 2 );
* var scale = new Float64Array( 3 );
*
* dgebal( 'both', 3, A, 3, 1, 0, out, 1, 0, scale, 1, 0 );
* // A => <Float64Array>[ 1, 12.5, 0, 16, 200, 0, 0, 0, 3 ]
* // out => <Int32Array>[ 0, 1 ]
* // scale => <Float64Array>[ 8, 1, 2 ]
*/
function dgebal( job, N, A, strideA1, strideA2, offsetA, out, strideOut, offsetOut, scale, strideScale, offsetScale ) {
var canSwap;
var noconv;
var sfmin1;
var sfmin2;
var sfmax1;
var sfmax2;
var ica;
var ira;
var ia1;
var ia2;
var ia3;
var ia4;
var ca;
var ra;
var is;
var c;
var r;
var k;
var l;
var i;
var j;
var g;
var f;
var s;
// Quick return if possible
if ( N === 0 ) {
out[ offsetOut ] = 0.0; // ilo
out[ offsetOut + strideOut ] = -1.0; // ihi (invalid)
return 0;
}
if ( job === 'none' ) {
is = offsetScale;
for ( i = 0; i < N; i++ ) {
scale[ is ] = 1.0;
is += strideScale;
}
out[ offsetOut ] = 0.0; // ilo
out[ offsetOut + strideOut ] = N - 1; // ihi
return 0;
}
// Permutation to isolate eigenvalues if possible
k = 0;
l = N - 1;
if ( job !== 'scale' ) {
// Row and column exchange
noconv = true;
while ( noconv ) {
// Search for rows isolating an eigenvalue and push them down
noconv = false;
is = offsetScale + ( l * strideScale ); // Follows scale
ia1 = offsetA + ( l * strideA2 ); // Follows `i`th column of A
ia2 = offsetA + ( l * strideA2 ); // Follows `l`th column of A
ia3 = offsetA + (l*strideA1) + (k*strideA2); // Follows `i`th row of A
ia4 = offsetA + (l*strideA1) + (k*strideA2); // Follows `l`th row of A
for ( i = l; i >= 0; i-- ) {
canSwap = true;
for ( j = 0; j <= l; j++ ) {
if ( i !== j && A[ offsetA + (i*strideA1) + (j*strideA2) ] !== 0.0 ) {
canSwap = false;
break;
}
}
if ( canSwap ) {
scale[ is ] = i;
if ( i !== l ) {
dswap( l+1, A, strideA1, ia1, A, strideA1, ia2 );
dswap( N - k, A, strideA2, ia3, A, strideA2, ia4 );
}
noconv = true;
// Check if remaining submatrix is empty and return
if ( l === 0.0 ) {
out[ offsetOut ] = 0.0; // ilo
out[ offsetOut + strideOut ] = 0.0; // ihi
return 0;
}
l -= 1;
is -= strideScale;
ia2 -= strideA2;
ia4 -= strideA1;
}
ia1 -= strideA2;
ia3 -= strideA1;
}
}
noconv = true;
while ( noconv ) {
// Search for columns isolating an eigenvalue and push them left
noconv = false;
is = offsetScale + ( k * strideScale ); // Follows scale
ia1 = offsetA + ( k * strideA2 ); // Follows `j`th column of A
ia2 = offsetA + ( k * strideA2 ); // Follows `k`th column of A
ia3 = offsetA + ( k * strideA1 ); // Follows `j`th row of A
ia4 = offsetA + ( k * strideA1 ); // Follows `k`th row of A
for ( j = k; j <= l; j++ ) {
canSwap = true;
for ( i = k; i <= l; i++ ) {
if ( i !== j && A[ offsetA + (i*strideA1) + (j*strideA2) ] !== 0.0 ) {
canSwap = false;
break;
}
}
if ( canSwap ) {
scale[ is ] = j;
if ( j !== k ) {
dswap( l+1, A, strideA1, ia1, A, strideA1, ia2 );
dswap( N-k, A, strideA2, ia3, A, strideA2, ia4 );
}
noconv = true;
k += 1;
is += strideScale;
ia2 += strideA2;
ia4 += strideA1;
}
ia1 += strideA2;
ia3 += strideA1;
}
}
}
// Initialize scale for non-permuted submatrix
is = offsetScale + ( k * strideScale );
for ( i = k; i <= l; i++ ) {
scale[ is ] = 1.0;
is += strideScale;
}
if ( job === 'permute' ) {
out[ offsetOut ] = k; // ilo
out[ offsetOut + strideOut ] = l; // ihi
return 0;
}
// Balance the submatrix in rows K to L, iterative loop for norm reduction (job = 'B')
sfmin1 = dlamch( 'S' ) / dlamch( 'P' );
sfmax1 = 1.0 / sfmin1;
sfmin2 = sfmin1 * sclfac;
sfmax2 = 1.0 / sfmin2;
noconv = true;
while ( noconv ) {
is = offsetScale + ( k * strideScale ); // Follows scale
ia1 = offsetA + ( k * strideA1 ) + ( k * strideA2 ); // Follows A[ k, i ]
ia2 = offsetA + ( k * strideA1 ) + ( k * strideA2 ); // Follows A[ i, k ]
ia3 = offsetA + ( k * strideA2 ); // follows `i`th column of A
noconv = false;
for ( i = k; i <= l; i++ ) {
c = dnrm2( l-k+1, A, strideA1, ia1 );
r = dnrm2( l-k+1, A, strideA2, ia2 );
ica = idamax( l+1, A, strideA1, ia3 );
ca = abs( A[ offsetA + (ica*strideA1) + (i*strideA2) ] );
ira = idamax( N-k+1, A, strideA2, ia2 );
ra = abs( A[ offsetA + (i*strideA1) + ((ira+k)*strideA2) ] );
if ( c === 0.0 || r === 0.0 ) {
ia1 += strideA2;
ia2 += strideA1;
is += strideScale;
ia3 += strideA2;
continue;
}
if ( isnan( c ) || isnan( r ) || isnan( ca ) || isnan( ra ) ) {
throw new RangeError( 'should not return NaN' );
}
g = r / sclfac;
f = 1.0;
s = c + r;
while ( c < g && max( f, c, ca ) < sfmax2 && min( r, g, ra ) > sfmin2 ) {
f *= sclfac;
c *= sclfac;
ca *= sclfac;
r /= sclfac;
g /= sclfac;
ra /= sclfac;
}
g = c / sclfac;
while ( g >= r && max( r, ra ) < sfmax2 && min( f, c, g, ca ) > sfmin2 ) {
f /= sclfac;
c /= sclfac;
g /= sclfac;
ca /= sclfac;
r *= sclfac;
ra *= sclfac;
}
// Now balance
if ( ( c + r ) >= factor * s ) {
ia1 += strideA2;
ia2 += strideA1;
is += strideScale;
ia3 += strideA2;
continue;
}
if ( f < 1.0 && scale[ is ] < 1.0 ) {
if ( f * scale[ is ] <= sfmin1 ) {
ia1 += strideA2;
ia2 += strideA1;
is += strideScale;
ia3 += strideA2;
continue;
}
}
if ( f > 1.0 && scale[ is ] > 1.0 ) {
if ( scale[ is ] >= sfmax1 / f ) {
ia1 += strideA2;
ia2 += strideA1;
is += strideScale;
ia3 += strideA2;
continue;
}
}
g = 1.0 / f;
scale[ is ] *= f;
noconv = true;
dscal( N-k, g, A, strideA2, ia2 );
dscal( l+1, f, A, strideA1, ia3 );
ia1 += strideA2;
ia2 += strideA1;
is += strideScale;
ia3 += strideA2;
}
}
out[ offsetOut ] = k; // ilo
out[ offsetOut + strideOut ] = l; // ihi
return 0;
}
// EXPORTS //
module.exports = dgebal;
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