mathjs/test/function/algebra/decomposition/lup.test.js

Math.js is an extensive math library for JavaScript and Node.js. It features a flexible expression parser and offers an integrated solution to work with numbers, big numbers, complex numbers, units, and matrices.

// test lup
var assert = require('assert'),
    approx = require('../../../../tools/approx'),
    math = require('../../../../index');

describe('lup', function () {

  it('should decompose matrix, n x n, no permutations, array', function () {

    var m = [[2, 1], [1, 4]];

    var r = math.lup(m);
    // L
    assert.deepEqual(r.L.valueOf(), [[1, 0], [0.5, 1]]);
    // U
    assert.deepEqual(r.U.valueOf(), [[2, 1], [0, 3.5]]);
    // P
    assert.deepEqual(r.p, [0, 1]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, n x n, no permutations, sparse', function () {

    var m = math.matrix([[2, 1], [1, 4]], 'sparse');

    var r = math.lup(m);
    // L
    assert.deepEqual(r.L.valueOf(), [[1, 0], [0.5, 1]]);
    // U
    assert.deepEqual(r.U.valueOf(), [[2, 1], [0, 3.5]]);
    // P
    assert.deepEqual(r.p, [0, 1]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, n x n, no permutations, dense format', function () {

    var m = math.matrix([[2, 1], [1, 4]], 'dense');

    var r = math.lup(m);
    // L
    assert.deepEqual(r.L.valueOf(), [[1, 0], [0.5, 1]]);
    // U
    assert.deepEqual(r.U.valueOf(), [[2, 1], [0, 3.5]]);
    // P
    assert.deepEqual(r.p, [0, 1]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });
  
  it('should decompose matrix, m x n, m < n, no permutations, dense format', function () {
    var m = math.matrix(
      [
        [2, 1, 1],
        [1, 4, 5]
      ]
    );

    var r = math.lup(m);
    // L
    assert.deepEqual(
      r.L,
      math.matrix(
        [
          [1, 0],
          [0.5, 1]
        ]
      ));
    // U
    assert.deepEqual(
      r.U,
      math.matrix(
        [
          [2, 1, 1],
          [0, 3.5, 4.5]
        ]
      ));
    // P
    assert.deepEqual(r.p, [0, 1]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, m x n, m > n, no permutations, dense format', function () {
    var m = math.matrix(
      [
        [8, 2],
        [6, 4],
        [4, 1]
      ]
    );

    var r = math.lup(m);
    // L
    assert.deepEqual(
      r.L,
      math.matrix(
        [
          [1, 0],
          [0.75, 1],
          [0.5, 0]
        ]
      ));
    // U
    assert.deepEqual(
      r.U,
      math.matrix(
        [
          [8, 2],
          [0, 2.5]
        ]
      ));
    // P
    assert.deepEqual(r.p, [0, 1, 2]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, n x n, dense format', function () {
    var m = math.matrix(
      [
        [16, -120, 240, -140],
        [-120, 1200, -2700, 1680],
        [240, -2700, 6480, -4200],
        [-140, 1680, -4200, 2800]
      ]
    );

    var r = math.lup(m);
    // L
    approx.deepEqual(
      r.L.valueOf(),
      [
        [1, 0, 0, 0],  
        [-0.5, 1, 0, 0],
        [-0.5833333333333334, -0.7, 1, 0],
        [0.06666666666666667, -0.4, -0.5714285714285776, 1]
      ]);
    // U
    approx.deepEqual(
      r.U.valueOf(),
      [
        [240, -2700, 6480, -4200],
        [0, -150, 540, -420], 
        [0, 0, -42, 56],
        [0, 0, 0, 4]
      ]);
    // P
    assert.deepEqual(r.p, [3, 1, 0, 2]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, 3 x 3, zero pivote value, dense format', function () {
    var m = math.matrix(
      [
        [1, 2, 3], 
        [2, 4, 6], 
        [4, 8, 9]
      ]);

    var r = math.lup(m);
    // L
    approx.deepEqual(
      r.L.valueOf(),
      [
        [1, 0, 0],  
        [0.5, 1, 0],
        [0.25, 0, 1.0]
      ]);
    // U
    approx.deepEqual(
      r.U.valueOf(),
      [
        [4, 8, 9],
        [0, 0, 1.5], 
        [0, 0, 0.75]
      ]);
    // P
    assert.deepEqual(r.p, [2, 1, 0]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, 3 x 2, complex numbers, dense format', function () {
    var m = math.matrix(
      [
        [math.complex(0, 3), 10],
        [math.complex(0, 1), 1],
        [math.complex(0, 1), 1]
      ]);

    var r = math.lup(m);
    // L
    approx.deepEqual(
      r.L.valueOf(),
      [
        [1, 0],
        [math.complex(0.3333333, 0), 1],
        [math.complex(0.3333333, 0), 1]
      ]);
    // U
    approx.deepEqual(
      r.U.valueOf(),
      [
        [math.complex(0, 3), 10],
        [0, math.complex(-2.3333333333, 0)]
      ]);
    // P
    assert.deepEqual(r.p, [0, 1, 2]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });
  
  it('should decompose matrix, m x n, m < n, no permutations, sparse', function () {
    var m = math.matrix(
      [
        [2, 1, 1],
        [1, 4, 5]
      ], 
      'sparse');

    var r = math.lup(m);
    // L
    assert.deepEqual(
      r.L.valueOf(),
      [
        [1, 0],
        [0.5, 1]
      ]);
    // U
    assert.deepEqual(
      r.U.valueOf(),
      [
        [2, 1, 1],
        [0, 3.5, 4.5]
      ]);
    // P
    assert.deepEqual(r.p, [0, 1]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, m x n, m > n, no permutations, sparse', function () {
    var m = math.matrix(
      [
        [8, 2],
        [6, 4],
        [4, 1]
      ],
      'sparse');

    var r = math.lup(m);
    // L
    assert.deepEqual(
      r.L.valueOf(),
      [
        [1, 0],
        [0.75, 1],
        [0.5, 0]
      ]);
    // U
    assert.deepEqual(
      r.U.valueOf(),
      [
        [8, 2],
        [0, 2.5]
      ]);
    // P
    assert.deepEqual(r.p, [0, 1, 2]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, n x n, sparse', function () {
    var m = math.matrix(
      [
        [16, -120, 240, -140],
        [-120, 1200, -2700, 1680],
        [240, -2700, 6480, -4200],
        [-140, 1680, -4200, 2800]
      ],
      'sparse');

    var r = math.lup(m);
    // L
    approx.deepEqual(
      r.L.valueOf(),
      [
        [1, 0, 0, 0],  
        [-0.5, 1, 0, 0],
        [-0.5833333333333334, -0.7, 1, 0],
        [0.06666666666666667, -0.4, -0.5714285714285776, 1]
      ]);
    // U
    approx.deepEqual(
      r.U.valueOf(),
      [
        [240, -2700, 6480, -4200],
        [0, -150, 540, -420], 
        [0, 0, -42, 56],
        [0, 0, 0, 4]
      ]);
    // P
    assert.deepEqual(r.p, [3, 1, 0, 2]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, 3 x 3, zero pivote value, sparse', function () {
    var m = math.matrix(
      [
        [1, 2, 3], 
        [2, 4, 6], 
        [4, 8, 9]
      ],
      'sparse');

    var r = math.lup(m);
    // L
    approx.deepEqual(
      r.L.valueOf(),
      [
        [1, 0, 0],  
        [0.5, 1, 0],
        [0.25, 0, 1.0]
      ]);
    // U
    approx.deepEqual(
      r.U.valueOf(),
      [
        [4, 8, 9],
        [0, 0, 1.5], 
        [0, 0, 0.75]
      ]);
    // P
    assert.deepEqual(r.p, [2, 1, 0]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });

  it('should decompose matrix, 3 x 2, complex numbers, sparse', function () {
    var m = math.matrix(
      [
        [math.complex(0, 3), 10],
        [math.complex(0, 1), 1],
        [math.complex(0, 1), 1]
      ], 'sparse');

    var r = math.lup(m);
    // L
    approx.deepEqual(
      r.L.valueOf(),
      [
        [1, 0],
        [math.complex(0.3333333, 0), 1],
        [math.complex(0.3333333, 0), 1]
      ]);
    // U
    approx.deepEqual(
      r.U.valueOf(),
      [
        [math.complex(0, 3), 10],
        [0, math.complex(-2.3333333333, 0)]
      ]);
    // P
    assert.deepEqual(r.p, [0, 1, 2]);
    // verify
    approx.deepEqual(math.multiply(_p(r.p), m).valueOf(), math.multiply(r.L, r.U).valueOf());
  });
  
  /**
   * Creates a Matrix out of a row permutation vector
   */
  function _p(p) {
    // identity matrix
    var identity = math.eye(p.length);
    // array
    var data = [];
    // loop rows
    for (var i = 0, l = p.length; i < l; i++) {
      // swap row
      data[p[i]] = identity._data[i];
    }
    return data;
  }
});