lu - linalgebra/lu.ts - Ubique documentation

function lu



lu(x: matrix): { LU: matrix; L: matrix; U: matrix; P: array; S: number;  }

LU decomposition with partial pivoting.

Performs LU decomposition of the input matrix x using the Doolittle algorithm with partial pivoting. Returns the combined LU matrix, lower triangular matrix L, upper triangular matrix U, pivot vector P, and pivot sign S.

Examples

LU decomposition of a 2x2 square matrix

import { assertEquals } from "jsr:@std/assert";

assertEquals(lu([[3, 2], [5, 2]]), {
  LU: [
    [5, 2],
    [0.6, 0.8]
  ],
  L: [
    [1, 0],
    [0.6, 1]
  ],
  U: [
    [5, 2],
    [0, 0.8]
  ],
  P: [1, 0],
  S: -1
});

LU decomposition of a 3x3 square matrix

import { assertEquals } from "jsr:@std/assert";

assertEquals(lu([[1, 1, -1], [1, -2, 3], [2, 3, 1]]), {
  LU: [
    [2, 3, 1],
    [0.5, -3.5, 2.5],
    [0.5, 0.14285714285714285, -1.8571428571428572]
  ],
  L: [
    [1, 0, 0],
    [0.5, 1, 0],
    [0.5, 0.14285714285714285, 1]
  ],
  U: [
    [2, 3, 1],
    [0, -3.5, 2.5],
    [0, 0, -1.8571428571428572]
  ],
  P: [2, 1, 0],
  S: -1
});

LU decomposition of a 2x3 rectangular matrix

import { assertEquals } from "jsr:@std/assert";

assertEquals(lu([[5, 6, 5], [7, 8, -1]]), {
  LU: [
    [7, 8, -1],
    [0.7142857142857143, 0.2857142857142856, 5.714285714285714]
  ],
  L: [
    [1, 0, 0],
    [0.7142857142857143, 1, 0]
  ],
  U: [
    [7, 8, -1],
    [0, 0.2857142857142856, 5.714285714285714]
  ],
  P: [1, 0],
  S: -1
});

LU decomposition of a singular 3x3 matrix

import { assertEquals } from "jsr:@std/assert";

assertEquals(lu([[1, 2, 3], [4, 5, 6], [7, 8, 9]]), {
  LU: [
    [7, 8, 9],
    [0.14285714285714285, 0.8571428571428572, 1.7142857142857144],
    [0.5714285714285714, 0.5000000000000002, 1.1102230246251565e-16]
  ],
  L: [
    [1, 0, 0],
    [0.14285714285714285, 1, 0],
    [0.5714285714285714, 0.5000000000000002, 1]
  ],
  U: [
    [7, 8, 9],
    [0, 0.8571428571428572, 1.7142857142857144],
    [0, 0, 1.1102230246251565e-16]
  ],
  P: [2, 0, 1],
  S: 1
});

LU decomposition of a 3x2 rectangular matrix

import { assertEquals } from "jsr:@std/assert";

assertEquals(lu([[1, 2], [3, 4], [5, 6]]), {
  LU: [
    [5, 6],
    [0.2, 0.7999999999999998],
    [0.6, 0.5000000000000006]
  ],
  L: [
    [1, 0],
    [0.2, 1],
    [0.6, 0.5000000000000006]
  ],
  U: [
    [5, 6],
    [0, 0.7999999999999998],
    [0, 0]
  ],
  P: [2, 0, 1],
  S: 1
});

Error when input is not provided

import { assertEquals } from "jsr:@std/assert";

// lu() throws: "Not enough input arguments"

Parameters



x: matrix

The input matrix.

Return Type

{ LU: matrix; L: matrix; U: matrix; P: array; S: number;  }

Object containing LU decomposition results

Throws

If no input arguments are provided.