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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 */
'use strict';
// FUNCTIONS //
/**
* Solve AT * X = B using the LU factorization of A computed by `dgttrf`, overwriting B with the solution.
*
* @private
* @param {NonNegativeInteger} N - order of the matrix A
* @param {NonNegativeInteger} nrhs - number of right-hand sides, i.e., the number of columns of the matrix B
* @param {Float64Array} DL - multipliers that define the matrix L
* @param {integer} sdl - stride length for DL
* @param {NonNegativeInteger} odl - starting index of DL
* @param {Float64Array} D - N diagonal elements of the upper triangular matrix U
* @param {integer} sd - stride length for D
* @param {NonNegativeInteger} od - starting index of D
* @param {Float64Array} DU - elements of the first super-diagonal of U
* @param {integer} sdu - stride length for DU
* @param {NonNegativeInteger} odu - starting index of DU
* @param {Float64Array} DU2 - elements of the second super-diagonal of U
* @param {integer} sdu2 - stride length for DU2
* @param {NonNegativeInteger} odu2 - starting index of DU2
* @param {Int32Array} IPIV - vector of pivot indices
* @param {integer} si - stride length for IPIV
* @param {NonNegativeInteger} oi - starting index for IPIV
* @param {Float64Array} B - right-hand side matrix B, overwritten by the solution matrix X
* @param {integer} sb1 - stride of the first dimension of B
* @param {integer} sb2 - stride of the second dimension of B
* @param {NonNegativeInteger} ob - starting index of B
* @returns {Float64Array} the solution matrix X
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
* var Int32Array = require( '@stdlib/array/int32' );
*
* var DL = new Float64Array( [ 0.25, 0.26666667 ] );
* var D = new Float64Array( [ 4.0, 3.75, 3.73333333 ] );
* var DU = new Float64Array( [ 1.0, 0.73333333 ] );
* var DU2 = new Float64Array( [ 0.0 ] );
* var IPIV = new Int32Array( [ 0, 1, 2 ] );
* var B = new Float64Array( [ 7.0, 8.0, 7.0 ] );
*
* var out = transpose( 3, 1, DL, 1, 0, D, 1, 0, DU, 1, 0, DU2, 1, 0, IPIV, 1, 0, B, 1, 1, 0 );
* // out => <Float64Array>[ ~1.44, ~1.25, ~1.55 ]
*/
function transpose( N, nrhs, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi, B, sb1, sb2, ob ) {
var idu2;
var temp;
var ipi;
var idu;
var idl;
var id;
var ib;
var ip;
var i;
var j;
ip = oi;
ib = ob;
idl = odl;
id = od;
idu = odu;
idu2 = odu2;
if ( nrhs <= 1 ) {
for ( j = 0; j < nrhs; j++) {
B[ ib + (sb2*j) ] = B[ ib + (sb2*j) ] / D[ id ];
if ( N > 1 ) {
B[ ib + sb1 + (sb2*j) ] = ( B[ ib + sb1 + (sb2*j) ] - ( DU[ idu ] * B[ ib + (sb2*j) ] ) ) / D[ id + sd ];
}
for ( i = 2; i < N; i++ ) {
B[ ib + (sb1*i) + (sb2*j) ] = ( B[ ib + (sb1*i) + (sb2*j) ] - ( DU[ idu + (sdu*(i-1)) ] * B[ ib + (sb1*(i-1)) + (sb2*j) ] ) - ( DU2[ idu2 + (sdu2*(i-2)) ] * B[ ib + (sb1*(i-2)) + (sb2*j) ] ) ) / D[ id + (sd*i) ];
}
for (i = N - 2; i >= 0; i--) {
ipi = IPIV[ ip + (i*si) ];
temp = B[ ib + (sb1*i) + (sb2*j) ] - ( DL[ idl + (sdl*i) ] * B[ ib + (sb1*(i+1)) + (sb2*j) ] );
B[ ib + (sb1*i) + (sb2*j) ] = B[ ib + (sb1*ipi) + (sb2*j) ];
B[ ib + (sb1*ipi) + (sb2*j) ] = temp;
}
}
} else {
for ( j = 0; j < nrhs; j++ ) {
B[ ib + (sb2*j) ] /= D[ id ];
if ( N > 1 ) {
B[ ib + sb1 + (sb2*j) ] = ( B[ ib + sb1 + (sb2*j) ] - ( DU[idu] * B[ ib + (sb2*j) ] ) ) / D[ id + sd ];
}
for ( i = 2; i < N; i++ ) {
B[ ib + (sb1*i) + (sb2*j) ] = ( B[ ib + (sb1*i) + (sb2*j) ] - ( DU[ idu + (sdu*(i-1)) ] * B[ ib + (sb1*(i-1)) + (sb2*j) ] ) - ( DU2[ idu2 + (sdu2*(i-2)) ] * B[ ib + (sb1*(i-2)) + (sb2*j) ] ) ) / D[ id + (sd*i) ];
}
for ( i = N - 2; i >= 0; i-- ) {
if ( IPIV[ ip + (i * si) ] === i ) {
B[ ib + (sb1*i) + (sb2*j) ] -= DL[ idl + (sdl*i) ] * B[ ib + (sb1*(i+1)) + (sb2*j) ];
} else {
temp = B[ ib + (sb1*(i+1)) + (sb2*j) ];
B[ ib + (sb1*(i+1)) + (sb2*j) ] = B[ ib + (sb1*i) + (sb2*j) ] - ( DL[ idl + (sdl*i) ] * temp );
B[ ib + (sb1*i) + (sb2*j) ] = temp;
}
}
}
}
return B;
}
/**
* Solve A * X = B using the LU factorization of A computed by `dgttrf`, overwriting B with the solution.
*
* @private
* @param {NonNegativeInteger} N - order of the matrix A
* @param {NonNegativeInteger} nrhs - number of right-hand sides, i.e., the number of columns of the matrix B
* @param {Float64Array} DL - multipliers that define the matrix L
* @param {integer} sdl - stride length for DL
* @param {NonNegativeInteger} odl - starting index of DL
* @param {Float64Array} D - N diagonal elements of the upper triangular matrix U
* @param {integer} sd - stride length for D
* @param {NonNegativeInteger} od - starting index of D
* @param {Float64Array} DU - elements of the first super-diagonal of U
* @param {integer} sdu - stride length for DU
* @param {NonNegativeInteger} odu - starting index of DU
* @param {Float64Array} DU2 - elements of the second super-diagonal of U
* @param {integer} sdu2 - stride length for DU2
* @param {NonNegativeInteger} odu2 - starting index of DU2
* @param {Int32Array} IPIV - vector of pivot indices
* @param {integer} si - stride length for IPIV
* @param {NonNegativeInteger} oi - starting index for IPIV
* @param {Float64Array} B - right-hand side matrix B, overwritten by the solution matrix X
* @param {integer} sb1 - stride of the first dimension of B
* @param {integer} sb2 - stride of the second dimension of B
* @param {NonNegativeInteger} ob - starting index of B
* @returns {Float64Array} the solution matrix X
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
* var Int32Array = require( '@stdlib/array/int32' );
*
* var DL = new Float64Array( [ 0.25, 0.26666667 ] );
* var D = new Float64Array( [ 4.0, 3.75, 3.73333333 ] );
* var DU = new Float64Array( [ 1.0, 0.73333333 ] );
* var DU2 = new Float64Array( [ 0.0 ] );
* var IPIV = new Int32Array( [ 0, 1, 2 ] );
* var B = new Float64Array( [ 7.0, 8.0, 7.0 ] );
*
* var out = noTranspose( 3, 1, DL, 1, 0, D, 1, 0, DU, 1, 0, DU2, 1, 0, IPIV, 1, 0, B, 1, 1, 0 );
* // out => <Float64Array>[ ~1.40, ~1.39, ~1.43 ]
*/
function noTranspose( N, nrhs, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi, B, sb1, sb2, ob ) {
var temp;
var idu2;
var ipi;
var idl;
var idu;
var ip;
var ib;
var id;
var j;
var i;
ip = oi;
ib = ob;
idl = odl;
id = od;
idu = odu;
idu2 = odu2;
if ( nrhs <= 1 ) {
for ( j = 0; j < nrhs; j++ ) {
for ( i = 0; i < N - 1; i++ ) {
ipi = IPIV[ ip+(si*i) ];
temp = B[ ib + (sb1*(i+1-ipi+i)) + (sb2*j) ] - ( DL[ idl + (i*sdl) ] * B[ ib + (sb1*ipi) + (sb2*j) ] );
B[ ib + (sb1*i) + (sb2*j) ] = B[ ib + (sb1*ipi) + (sb2*j) ];
B[ ib + (sb1*(i+1)) + (sb2*j) ] = temp;
}
B[ ib + (sb1*(N-1)) + (sb2*j) ] /= D[ id + (sd*(N-1)) ];
if ( N > 1 ) {
B[ ib + (sb1*(N-2)) + (sb2*j) ] = ( B[ ib + (sb1*(N-2)) + (sb2*j) ] - ( DU[ idu + (sdu*(N-2)) ] * B[ ib + (sb1*(N-1)) + (sb2*j) ] ) ) / D[ id + (sd*(N-2)) ];
}
for ( i = N - 3; i >= 0; i-- ) {
B[ ib + (sb1*i) + (sb2*j) ] = ( B[ ib + (sb1*i) + (sb2*j) ] - ( DU[ idu + (sdu*i) ] * B[ ib + (sb1*(i+1)) + (sb2*j) ] ) - ( DU2[ idu2 + (sdu2*i) ] * B[ ib + (sb1*(i+2)) + (sb2*j) ] ) ) / D[ id + (sd*i) ];
}
}
} else {
for ( j = 0; j < nrhs; j++ ) {
for ( i = 0; i < N - 1; i++ ) {
if ( IPIV[ ip + (si*i) ] === i ) {
B[ ib + (sb1*(i+1)) + (sb2*j) ] -= DL[ idl + (i*sdl) ] * B[ ib + (sb1*i) + (sb2*j) ];
} else {
temp = B[ ib + (sb1*i) + (sb2*j) ];
B[ ib + (sb1*i) + (sb2*j) ] = B[ ib + (sb1*(i+1)) + (sb2*j) ];
B[ ib + (sb1*(i+1)) + (sb2*j) ] = temp - ( DL[ idl + (i*sdl) ] * B[ ib + (sb1*i) + (sb2*j) ] );
}
}
B[ ib + (sb1*(N-1)) + (sb2*j) ] /= D[ id + (sd*(N-1)) ];
if ( N > 1 ) {
B[ ib + (sb1*(N-2)) + (sb2 * j) ] = ( B[ ib + (sb1*(N-2)) + (sb2*j) ] - ( DU[ idu + (sdu*(N-2)) ] * B[ ib + (sb1*(N-1)) + (sb2*j) ] ) ) / D[ id + (sd*(N-2)) ];
}
for ( i = N - 3; i >= 0; i-- ) {
B[ ib + (sb1*i) + (sb2*j) ] = ( B[ ib + (sb1*i) + (sb2*j) ] - ( DU[ idu + (sdu*i) ] * B[ ib + (sb1*(i+1)) + (sb2*j) ] ) - ( DU2[ idu2 + (sdu2*i) ] * B[ ib + (sb1*(i+2)) + (sb2*j) ] ) ) / D[ id + (sd*i) ];
}
}
}
return B;
}
// MAIN //
/**
* Solves a system of linear equations with a tri diagonal matrix using the LU factorization computed by `dgttrf`.
*
* ## Notes
*
* - To solve A * X = B (no transpose), use itrans = 0.
* - To solve AT * X = B (transpose), use itrans = 1.
* - To solve AT * X = B (conjugate transpose = transpose), use itrans = 2.
*
* @param {integer} itrans - specifies the form of the system of equations
* @param {NonNegativeInteger} N - order of the matrix A
* @param {NonNegativeInteger} nrhs - number of right-hand sides, i.e., the number of columns of the matrix B
* @param {Float64Array} DL - multipliers that define the matrix L
* @param {integer} sdl - stride length for DL
* @param {NonNegativeInteger} odl - starting index of DL
* @param {Float64Array} D - N diagonal elements of the upper triangular matrix U
* @param {integer} sd - stride length for D
* @param {NonNegativeInteger} od - starting index of D
* @param {Float64Array} DU - elements of the first super-diagonal of U
* @param {integer} sdu - stride length for DU
* @param {NonNegativeInteger} odu - starting index of DU
* @param {Float64Array} DU2 - elements of the second super-diagonal of U
* @param {integer} sdu2 - stride length for DU2
* @param {NonNegativeInteger} odu2 - starting index of DU2
* @param {Int32Array} IPIV - vector of pivot indices
* @param {integer} si - stride length for IPIV
* @param {NonNegativeInteger} oi - starting index for IPIV
* @param {Float64Array} B - right-hand side matrix B, overwritten by the solution matrix X
* @param {integer} sb1 - stride of the first dimension of B
* @param {integer} sb2 - stride of the second dimension of B
* @param {NonNegativeInteger} ob - starting index of B
* @returns {Float64Array} the solution matrix X
*
* @example
* var Float64Array = require( '@stdlib/array/float64' );
* var Int32Array = require( '@stdlib/array/int32' );
*
* var DL = new Float64Array( [ 0.25, 0.26666667 ] );
* var D = new Float64Array( [ 4.0, 3.75, 3.73333333 ] );
* var DU = new Float64Array( [ 1.0, 0.73333333 ] );
* var DU2 = new Float64Array( [ 0.0 ] );
* var IPIV = new Int32Array( [ 0, 1, 2 ] );
* var B = new Float64Array( [ 7.0, 8.0, 7.0 ] );
*
* var out = dgtts2( 1, 3, 1, DL, 1, 0, D, 1, 0, DU, 1, 0, DU2, 1, 0, IPIV, 1, 0, B, 1, 1, 0 );
* // out => <Float64Array>[ ~1.44, ~1.25, ~1.55 ]
*/
function dgtts2( itrans, N, nrhs, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi, B, sb1, sb2, ob ) {
if ( N === 0 || nrhs === 0 ) {
return B;
}
if ( itrans === 0 ) {
// Solve A * X = B using the LU factorization of A, overwriting B with the solution
return noTranspose( N, nrhs, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi, B, sb1, sb2, ob );
}
// Solve Solve A**T * X = B using the LU factorization of A, overwriting B with the solution.
return transpose( N, nrhs, DL, sdl, odl, D, sd, od, DU, sdu, odu, DU2, sdu2, odu2, IPIV, si, oi, B, sb1, sb2, ob );
}
// EXPORTS //
module.exports = dgtts2;
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