mathjs/lib/function/matrix/det.js

'use strict';

var util = require('../../utils/index');
var object = util.object;
var string = util.string;

function factory (type, config, load, typed) {
  var matrix = load(require('../../type/matrix/function/matrix'));
  var add = load(require('../arithmetic/add'));
  var subtract = load(require('../arithmetic/subtract'));
  var multiply = load(require('../arithmetic/multiply'));
  var unaryMinus = load(require('../arithmetic/unaryMinus'));

  /**
   * Calculate the determinant of a matrix.
   *
   * Syntax:
   *
   *    math.det(x)
   *
   * Examples:
   *
   *    math.det([[1, 2], [3, 4]]); // returns -2
   *
   *    var A = [
   *      [-2, 2, 3],
   *      [-1, 1, 3],
   *      [2, 0, -1]
   *    ]
   *    math.det(A); // returns 6
   *
   * See also:
   *
   *    inv
   *
   * @param {Array | Matrix} x  A matrix
   * @return {number} The determinant of `x`
   */
  var det = typed('det', {
    'any': function (x) {
      return object.clone(x);
    },

    'Array | Matrix': function det (x) {
      var size;
      if (x && x.isMatrix === true) {
        size = x.size();
      }
      else if (Array.isArray(x)) {
        x = matrix(x);
        size = x.size();
      }
      else {
        // a scalar
        size = [];
      }

      switch (size.length) {
        case 0:
          // scalar
          return object.clone(x);

        case 1:
          // vector
          if (size[0] == 1) {
            return object.clone(x.valueOf()[0]);
          }
          else {
            throw new RangeError('Matrix must be square ' +
            '(size: ' + string.format(size) + ')');
          }

        case 2:
          // two dimensional array
          var rows = size[0];
          var cols = size[1];
          if (rows == cols) {
            return _det(x.clone().valueOf(), rows, cols);
          }
          else {
            throw new RangeError('Matrix must be square ' +
            '(size: ' + string.format(size) + ')');
          }

        default:
          // multi dimensional array
          throw new RangeError('Matrix must be two dimensional ' +
          '(size: ' + string.format(size) + ')');
      }
    }
  });

  det.toTex = '\\det\\left(${args[0]}\\right)';

  return det;

  /**
   * Calculate the determinant of a matrix
   * @param {Array[]} matrix  A square, two dimensional matrix
   * @param {number} rows     Number of rows of the matrix (zero-based)
   * @param {number} cols     Number of columns of the matrix (zero-based)
   * @returns {number} det
   * @private
   */
  function _det (matrix, rows, cols) {
    if (rows == 1) {
      // this is a 1 x 1 matrix
      return object.clone(matrix[0][0]);
    }
    else if (rows == 2) {
      // this is a 2 x 2 matrix
      // the determinant of [a11,a12;a21,a22] is det = a11*a22-a21*a12
      return subtract(
          multiply(matrix[0][0], matrix[1][1]),
          multiply(matrix[1][0], matrix[0][1])
      );
    }
    else {
      // this is an n x n matrix
      var compute_mu = function (matrix) {
        var i, j;

        // Compute the matrix with zero lower triangle, same upper triangle,
        // and diagonals given by the negated sum of the below diagonal
        // elements.
        var mu = new Array(matrix.length);
        var sum = 0;
        for (i = 1; i < matrix.length; i++) {
          sum = add(sum, matrix[i][i]);
        }

        for (i = 0; i < matrix.length; i++) {
          mu[i] = new Array(matrix.length);
          mu[i][i] = unaryMinus(sum);

          for (j = 0; j < i; j++) {
            mu[i][j] = 0; // TODO: make bignumber 0 in case of bignumber computation
          }

          for (j = i + 1; j < matrix.length; j++) {
            mu[i][j] = matrix[i][j];
          }

          if (i+1 < matrix.length) {
            sum = subtract(sum, matrix[i + 1][i + 1]);
          }
        }

        return mu;
      };

      var fa = matrix;
      for (var i = 0; i < rows - 1; i++) {
        fa = multiply(compute_mu(fa), matrix);
      }

      if (rows % 2 == 0) {
        return unaryMinus(fa[0][0]);
      } else {
        return fa[0][0];
      }
    }
  }
}

exports.name = 'det';
exports.factory = factory;