mathjs/lib/function/matrix/expm.js

Math.js is an extensive math library for JavaScript and Node.js. It features a flexible expression parser with support for symbolic computation, comes with a large set of built-in functions and constants, and offers an integrated solution to work with dif

'use strict';

var format = require('../../utils/string').format;

function factory(type, config, load, typed) {
  var abs = load(require('../arithmetic/abs'));
  var add = load(require('../arithmetic/add'));
  var identity = load(require('./identity'));
  var inv = load(require('./inv'));
  var multiply = load(require('../arithmetic/multiply'));
  var SparseMatrix = type.SparseMatrix;
  /**
   * Compute the matrix exponential, expm(A) = e^A. The matrix must be square.
   * Not to be confused with exp(a), which performs element-wise
   * exponentiation.
   *
   * The exponential is calculated using the Padé approximant with scaling and
   * squaring; see "Nineteen Dubious Ways to Compute the Exponential of a
   * Matrix," by Moler and Van Loan.
   *
   * Syntax:
   *
   *     math.expm(x)
   *
   * Examples:
   *
   *     const A = [[0,2],[0,0]]
   *     math.expm(A)        // returns [[1,2],[0,1]]
   *
   * See also:
   *
   *     exp
   *
   * @param {Matrix} x  A square Matrix
   * @return {Matrix}   The exponential of x
   */

  var expm = typed('expm', {
    'Matrix': function Matrix(A) {
      // Check matrix size
      var size = A.size();

      if (size.length !== 2 || size[0] !== size[1]) {
        throw new RangeError('Matrix must be square ' + '(size: ' + format(size) + ')');
      }

      var n = size[0]; // Desired accuracy of the approximant (The actual accuracy
      // will be affected by round-off error)

      var eps = 1e-15; // The Padé approximant is not so accurate when the values of A
      // are "large", so scale A by powers of two. Then compute the
      // exponential, and square the result repeatedly according to
      // the identity e^A = (e^(A/m))^m
      // Compute infinity-norm of A, ||A||, to see how "big" it is

      var infNorm = infinityNorm(A); // Find the optimal scaling factor and number of terms in the
      // Padé approximant to reach the desired accuracy

      var params = findParams(infNorm, eps);
      var q = params.q;
      var j = params.j; // The Pade approximation to e^A is:
      // Rqq(A) = Dqq(A) ^ -1 * Nqq(A)
      // where
      // Nqq(A) = sum(i=0, q, (2q-i)!p! / [ (2q)!i!(q-i)! ] A^i
      // Dqq(A) = sum(i=0, q, (2q-i)!q! / [ (2q)!i!(q-i)! ] (-A)^i
      // Scale A by 1 / 2^j

      var Apos = multiply(A, Math.pow(2, -j)); // The i=0 term is just the identity matrix

      var N = identity(n);
      var D = identity(n); // Initialization (i=0)

      var factor = 1; // Initialization (i=1)

      var AposToI = Apos; // Cloning not necessary

      var alternate = -1;

      for (var i = 1; i <= q; i++) {
        if (i > 1) {
          AposToI = multiply(AposToI, Apos);
          alternate = -alternate;
        }

        factor = factor * (q - i + 1) / ((2 * q - i + 1) * i);
        N = add(N, multiply(factor, AposToI));
        D = add(D, multiply(factor * alternate, AposToI));
      }

      var R = multiply(inv(D), N); // Square j times

      for (var _i = 0; _i < j; _i++) {
        R = multiply(R, R);
      }

      return type.isSparseMatrix(A) ? new SparseMatrix(R) : R;
    }
  });

  function infinityNorm(A) {
    var n = A.size()[0];
    var infNorm = 0;

    for (var i = 0; i < n; i++) {
      var rowSum = 0;

      for (var j = 0; j < n; j++) {
        rowSum += abs(A.get([i, j]));
      }

      infNorm = Math.max(rowSum, infNorm);
    }

    return infNorm;
  }
  /**
   * Find the best parameters for the Pade approximant given
   * the matrix norm and desired accuracy. Returns the first acceptable
   * combination in order of increasing computational load.
   */


  function findParams(infNorm, eps) {
    var maxSearchSize = 30;

    for (var k = 0; k < maxSearchSize; k++) {
      for (var q = 0; q <= k; q++) {
        var j = k - q;

        if (errorEstimate(infNorm, q, j) < eps) {
          return {
            q: q,
            j: j
          };
        }
      }
    }

    throw new Error('Could not find acceptable parameters to compute the matrix exponential (try increasing maxSearchSize in expm.js)');
  }
  /**
   * Returns the estimated error of the Pade approximant for the given
   * parameters.
   */


  function errorEstimate(infNorm, q, j) {
    var qfac = 1;

    for (var i = 2; i <= q; i++) {
      qfac *= i;
    }

    var twoqfac = qfac;

    for (var _i2 = q + 1; _i2 <= 2 * q; _i2++) {
      twoqfac *= _i2;
    }

    var twoqp1fac = twoqfac * (2 * q + 1);
    return 8.0 * Math.pow(infNorm / Math.pow(2, j), 2 * q) * qfac * qfac / (twoqfac * twoqp1fac);
  }

  expm.toTex = {
    1: "\\exp\\left(${args[0]}\\right)"
  };
  return expm;
}

exports.name = 'expm';
exports.factory = factory;