simple-squiggle

expm.js (4546B)

---

 import { isSparseMatrix } from '../../utils/is.js';
 import { format } from '../../utils/string.js';
 import { factory } from '../../utils/factory.js';
 var name = 'expm';
 var dependencies = ['typed', 'abs', 'add', 'identity', 'inv', 'multiply'];
 export var createExpm = /* #__PURE__ */factory(name, dependencies, _ref => {
   var {
     typed,
     abs,
     add,
     identity,
     inv,
     multiply
   } = _ref;
 
   /**
    * 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
    */
   return typed(name, {
     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 isSparseMatrix(A) ? A.createSparseMatrix(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);
   }
 });