simple-squiggle

complexEigs.js (21144B)

---

 import { clone } from '../../../utils/object.js';
 export function createComplexEigs(_ref) {
   var {
     addScalar,
     subtract,
     flatten,
     multiply,
     multiplyScalar,
     divideScalar,
     sqrt,
     abs,
     bignumber,
     diag,
     inv,
     qr,
     usolve,
     usolveAll,
     equal,
     complex,
     larger,
     smaller,
     matrixFromColumns,
     dot
   } = _ref;
 
   /**
    * @param {number[][]} arr the matrix to find eigenvalues of
    * @param {number} N size of the matrix
    * @param {number|BigNumber} prec precision, anything lower will be considered zero
    * @param {'number'|'BigNumber'|'Complex'} type
    * @param {boolean} findVectors should we find eigenvectors?
    *
    * @returns {{ values: number[], vectors: number[][] }}
    */
   function complexEigs(arr, N, prec, type, findVectors) {
     if (findVectors === undefined) {
       findVectors = true;
     } // TODO check if any row/col are zero except the diagonal
     // make sure corresponding rows and columns have similar magnitude
     // important because of numerical stability
     // MODIFIES arr by side effect!
 
 
     var R = balance(arr, N, prec, type, findVectors); // R is the row transformation matrix
     // arr = A' = R A R⁻¹, A is the original matrix
     // (if findVectors is false, R is undefined)
     // (And so to return to original matrix: A = R⁻¹ arr R)
     // TODO if magnitudes of elements vary over many orders,
     // move greatest elements to the top left corner
     // using similarity transformations, reduce the matrix
     // to Hessenberg form (upper triangular plus one subdiagonal row)
     // updates the transformation matrix R with new row operationsq
     // MODIFIES arr by side effect!
 
     reduceToHessenberg(arr, N, prec, type, findVectors, R); // still true that original A = R⁻¹ arr R)
     // find eigenvalues
 
     var {
       values,
       C
     } = iterateUntilTriangular(arr, N, prec, type, findVectors); // values is the list of eigenvalues, C is the column
     // transformation matrix that transforms arr, the hessenberg
     // matrix, to upper triangular
     // (So U = C⁻¹ arr C and the relationship between current arr
     // and original A is unchanged.)
 
     var vectors;
 
     if (findVectors) {
       vectors = findEigenvectors(arr, N, C, R, values, prec, type);
       vectors = matrixFromColumns(...vectors);
     }
 
     return {
       values,
       vectors
     };
   }
   /**
    * @param {number[][]} arr
    * @param {number} N
    * @param {number} prec
    * @param {'number'|'BigNumber'|'Complex'} type
    * @returns {number[][]}
    */
 
 
   function balance(arr, N, prec, type, findVectors) {
     var big = type === 'BigNumber';
     var cplx = type === 'Complex';
     var realzero = big ? bignumber(0) : 0;
     var one = big ? bignumber(1) : cplx ? complex(1) : 1;
     var realone = big ? bignumber(1) : 1; // base of the floating-point arithmetic
 
     var radix = big ? bignumber(10) : 2;
     var radixSq = multiplyScalar(radix, radix); // the diagonal transformation matrix R
 
     var Rdiag;
 
     if (findVectors) {
       Rdiag = Array(N).fill(one);
     } // this isn't the only time we loop thru the matrix...
 
 
     var last = false;
 
     while (!last) {
       // ...haha I'm joking! unless...
       last = true;
 
       for (var i = 0; i < N; i++) {
         // compute the taxicab norm of i-th column and row
         // TODO optimize for complex numbers
         var colNorm = realzero;
         var rowNorm = realzero;
 
         for (var j = 0; j < N; j++) {
           if (i === j) continue;
           var c = abs(arr[i][j]); // should be real
 
           colNorm = addScalar(colNorm, c);
           rowNorm = addScalar(rowNorm, c);
         }
 
         if (!equal(colNorm, 0) && !equal(rowNorm, 0)) {
           // find integer power closest to balancing the matrix
           // (we want to scale only by integer powers of radix,
           // so that we don't lose any precision due to round-off)
           var f = realone;
           var _c = colNorm;
           var rowDivRadix = divideScalar(rowNorm, radix);
           var rowMulRadix = multiplyScalar(rowNorm, radix);
 
           while (smaller(_c, rowDivRadix)) {
             _c = multiplyScalar(_c, radixSq);
             f = multiplyScalar(f, radix);
           }
 
           while (larger(_c, rowMulRadix)) {
             _c = divideScalar(_c, radixSq);
             f = divideScalar(f, radix);
           } // check whether balancing is needed
           // condition = (c + rowNorm) / f < 0.95 * (colNorm + rowNorm)
 
 
           var condition = smaller(divideScalar(addScalar(_c, rowNorm), f), multiplyScalar(addScalar(colNorm, rowNorm), 0.95)); // apply balancing similarity transformation
 
           if (condition) {
             // we should loop once again to check whether
             // another rebalancing is needed
             last = false;
             var g = divideScalar(1, f);
 
             for (var _j = 0; _j < N; _j++) {
               if (i === _j) {
                 continue;
               }
 
               arr[i][_j] = multiplyScalar(arr[i][_j], f);
               arr[_j][i] = multiplyScalar(arr[_j][i], g);
             } // keep track of transformations
 
 
             if (findVectors) {
               Rdiag[i] = multiplyScalar(Rdiag[i], f);
             }
           }
         }
       }
     } // return the diagonal row transformation matrix
 
 
     return diag(Rdiag);
   }
   /**
    * @param {number[][]} arr
    * @param {number} N
    * @param {number} prec
    * @param {'number'|'BigNumber'|'Complex'} type
    * @param {boolean} findVectors
    * @param {number[][]} R the row transformation matrix that will be modified
    */
 
 
   function reduceToHessenberg(arr, N, prec, type, findVectors, R) {
     var big = type === 'BigNumber';
     var cplx = type === 'Complex';
     var zero = big ? bignumber(0) : cplx ? complex(0) : 0;
 
     if (big) {
       prec = bignumber(prec);
     }
 
     for (var i = 0; i < N - 2; i++) {
       // Find the largest subdiag element in the i-th col
       var maxIndex = 0;
       var max = zero;
 
       for (var j = i + 1; j < N; j++) {
         var el = arr[j][i];
 
         if (smaller(abs(max), abs(el))) {
           max = el;
           maxIndex = j;
         }
       } // This col is pivoted, no need to do anything
 
 
       if (smaller(abs(max), prec)) {
         continue;
       }
 
       if (maxIndex !== i + 1) {
         // Interchange maxIndex-th and (i+1)-th row
         var tmp1 = arr[maxIndex];
         arr[maxIndex] = arr[i + 1];
         arr[i + 1] = tmp1; // Interchange maxIndex-th and (i+1)-th column
 
         for (var _j2 = 0; _j2 < N; _j2++) {
           var tmp2 = arr[_j2][maxIndex];
           arr[_j2][maxIndex] = arr[_j2][i + 1];
           arr[_j2][i + 1] = tmp2;
         } // keep track of transformations
 
 
         if (findVectors) {
           var tmp3 = R[maxIndex];
           R[maxIndex] = R[i + 1];
           R[i + 1] = tmp3;
         }
       } // Reduce following rows and columns
 
 
       for (var _j3 = i + 2; _j3 < N; _j3++) {
         var n = divideScalar(arr[_j3][i], max);
 
         if (n === 0) {
           continue;
         } // from j-th row subtract n-times (i+1)th row
 
 
         for (var k = 0; k < N; k++) {
           arr[_j3][k] = subtract(arr[_j3][k], multiplyScalar(n, arr[i + 1][k]));
         } // to (i+1)th column add n-times j-th column
 
 
         for (var _k = 0; _k < N; _k++) {
           arr[_k][i + 1] = addScalar(arr[_k][i + 1], multiplyScalar(n, arr[_k][_j3]));
         } // keep track of transformations
 
 
         if (findVectors) {
           for (var _k2 = 0; _k2 < N; _k2++) {
             R[_j3][_k2] = subtract(R[_j3][_k2], multiplyScalar(n, R[i + 1][_k2]));
           }
         }
       }
     }
 
     return R;
   }
   /**
    * @returns {{values: values, C: Matrix}}
    * @see Press, Wiliams: Numerical recipes in Fortran 77
    * @see https://en.wikipedia.org/wiki/QR_algorithm
    */
 
 
   function iterateUntilTriangular(A, N, prec, type, findVectors) {
     var big = type === 'BigNumber';
     var cplx = type === 'Complex';
     var one = big ? bignumber(1) : cplx ? complex(1) : 1;
 
     if (big) {
       prec = bignumber(prec);
     } // The Francis Algorithm
     // The core idea of this algorithm is that doing successive
     // A' = Q⁺AQ transformations will eventually converge to block-
     // upper-triangular with diagonal blocks either 1x1 or 2x2.
     // The Q here is the one from the QR decomposition, A = QR.
     // Since the eigenvalues of a block-upper-triangular matrix are
     // the eigenvalues of its diagonal blocks and we know how to find
     // eigenvalues of a 2x2 matrix, we know the eigenvalues of A.
 
 
     var arr = clone(A); // the list of converged eigenvalues
 
     var lambdas = []; // size of arr, which will get smaller as eigenvalues converge
 
     var n = N; // the diagonal of the block-diagonal matrix that turns
     // converged 2x2 matrices into upper triangular matrices
 
     var Sdiag = []; // N×N matrix describing the overall transformation done during the QR algorithm
 
     var Qtotal = findVectors ? diag(Array(N).fill(one)) : undefined; // n×n matrix describing the QR transformations done since last convergence
 
     var Qpartial = findVectors ? diag(Array(n).fill(one)) : undefined; // last eigenvalue converged before this many steps
 
     var lastConvergenceBefore = 0;
 
     while (lastConvergenceBefore <= 100) {
       lastConvergenceBefore += 1; // TODO if the convergence is slow, do something clever
       // Perform the factorization
 
       var k = 0; // TODO set close to an eigenvalue
 
       for (var i = 0; i < n; i++) {
         arr[i][i] = subtract(arr[i][i], k);
       } // TODO do an implicit QR transformation
 
 
       var {
         Q,
         R
       } = qr(arr);
       arr = multiply(R, Q);
 
       for (var _i = 0; _i < n; _i++) {
         arr[_i][_i] = addScalar(arr[_i][_i], k);
       } // keep track of transformations
 
 
       if (findVectors) {
         Qpartial = multiply(Qpartial, Q);
       } // The rightmost diagonal element converged to an eigenvalue
 
 
       if (n === 1 || smaller(abs(arr[n - 1][n - 2]), prec)) {
         lastConvergenceBefore = 0;
         lambdas.push(arr[n - 1][n - 1]); // keep track of transformations
 
         if (findVectors) {
           Sdiag.unshift([[1]]);
           inflateMatrix(Qpartial, N);
           Qtotal = multiply(Qtotal, Qpartial);
 
           if (n > 1) {
             Qpartial = diag(Array(n - 1).fill(one));
           }
         } // reduce the matrix size
 
 
         n -= 1;
         arr.pop();
 
         for (var _i2 = 0; _i2 < n; _i2++) {
           arr[_i2].pop();
         } // The rightmost diagonal 2x2 block converged
 
       } else if (n === 2 || smaller(abs(arr[n - 2][n - 3]), prec)) {
         lastConvergenceBefore = 0;
         var ll = eigenvalues2x2(arr[n - 2][n - 2], arr[n - 2][n - 1], arr[n - 1][n - 2], arr[n - 1][n - 1]);
         lambdas.push(...ll); // keep track of transformations
 
         if (findVectors) {
           Sdiag.unshift(jordanBase2x2(arr[n - 2][n - 2], arr[n - 2][n - 1], arr[n - 1][n - 2], arr[n - 1][n - 1], ll[0], ll[1], prec, type));
           inflateMatrix(Qpartial, N);
           Qtotal = multiply(Qtotal, Qpartial);
 
           if (n > 2) {
             Qpartial = diag(Array(n - 2).fill(one));
           }
         } // reduce the matrix size
 
 
         n -= 2;
         arr.pop();
         arr.pop();
 
         for (var _i3 = 0; _i3 < n; _i3++) {
           arr[_i3].pop();
 
           arr[_i3].pop();
         }
       }
 
       if (n === 0) {
         break;
       }
     } // standard sorting
 
 
     lambdas.sort((a, b) => +subtract(abs(a), abs(b))); // the algorithm didn't converge
 
     if (lastConvergenceBefore > 100) {
       var err = Error('The eigenvalues failed to converge. Only found these eigenvalues: ' + lambdas.join(', '));
       err.values = lambdas;
       err.vectors = [];
       throw err;
     } // combine the overall QR transformation Qtotal with the subsequent
     // transformation S that turns the diagonal 2x2 blocks to upper triangular
 
 
     var C = findVectors ? multiply(Qtotal, blockDiag(Sdiag, N)) : undefined;
     return {
       values: lambdas,
       C
     };
   }
   /**
    * @param {Matrix} A hessenberg-form matrix
    * @param {number} N size of A
    * @param {Matrix} C column transformation matrix that turns A into upper triangular
    * @param {Matrix} R similarity that turns original matrix into A
    * @param {number[]} values array of eigenvalues of A
    * @param {'number'|'BigNumber'|'Complex'} type
    * @returns {number[][]} eigenvalues
    */
 
 
   function findEigenvectors(A, N, C, R, values, prec, type) {
     var Cinv = inv(C);
     var U = multiply(Cinv, A, C);
     var big = type === 'BigNumber';
     var cplx = type === 'Complex';
     var zero = big ? bignumber(0) : cplx ? complex(0) : 0;
     var one = big ? bignumber(1) : cplx ? complex(1) : 1; // turn values into a kind of "multiset"
     // this way it is easier to find eigenvectors
 
     var uniqueValues = [];
     var multiplicities = [];
 
     for (var λ of values) {
       var i = indexOf(uniqueValues, λ, equal);
 
       if (i === -1) {
         uniqueValues.push(λ);
         multiplicities.push(1);
       } else {
         multiplicities[i] += 1;
       }
     } // find eigenvectors by solving U − λE = 0
     // TODO replace with an iterative eigenvector algorithm
     // (this one might fail for imprecise eigenvalues)
 
 
     var vectors = [];
     var len = uniqueValues.length;
     var b = Array(N).fill(zero);
     var E = diag(Array(N).fill(one)); // eigenvalues for which usolve failed (due to numerical error)
 
     var failedLambdas = [];
 
     var _loop = function _loop(_i4) {
       var λ = uniqueValues[_i4];
       var S = subtract(U, multiply(λ, E)); // the characteristic matrix
 
       var solutions = usolveAll(S, b);
       solutions.shift(); // ignore the null vector
       // looks like we missed something, try inverse iteration
 
       while (solutions.length < multiplicities[_i4]) {
         var approxVec = inverseIterate(S, N, solutions, prec, type);
 
         if (approxVec == null) {
           // no more vectors were found
           failedLambdas.push(λ);
           break;
         }
 
         solutions.push(approxVec);
       } // Transform back into original array coordinates
 
 
       var correction = multiply(inv(R), C);
       solutions = solutions.map(v => multiply(correction, v));
       vectors.push(...solutions.map(v => flatten(v)));
     };
 
     for (var _i4 = 0; _i4 < len; _i4++) {
       _loop(_i4);
     }
 
     if (failedLambdas.length !== 0) {
       var err = new Error('Failed to find eigenvectors for the following eigenvalues: ' + failedLambdas.join(', '));
       err.values = values;
       err.vectors = vectors;
       throw err;
     }
 
     return vectors;
   }
   /**
    * Compute the eigenvalues of an 2x2 matrix
    * @return {[number,number]}
    */
 
 
   function eigenvalues2x2(a, b, c, d) {
     // λ± = ½ trA ± ½ √( tr²A - 4 detA )
     var trA = addScalar(a, d);
     var detA = subtract(multiplyScalar(a, d), multiplyScalar(b, c));
     var x = multiplyScalar(trA, 0.5);
     var y = multiplyScalar(sqrt(subtract(multiplyScalar(trA, trA), multiplyScalar(4, detA))), 0.5);
     return [addScalar(x, y), subtract(x, y)];
   }
   /**
    * For an 2x2 matrix compute the transformation matrix S,
    * so that SAS⁻¹ is an upper triangular matrix
    * @return {[[number,number],[number,number]]}
    * @see https://math.berkeley.edu/~ogus/old/Math_54-05/webfoils/jordan.pdf
    * @see http://people.math.harvard.edu/~knill/teaching/math21b2004/exhibits/2dmatrices/index.html
    */
 
 
   function jordanBase2x2(a, b, c, d, l1, l2, prec, type) {
     var big = type === 'BigNumber';
     var cplx = type === 'Complex';
     var zero = big ? bignumber(0) : cplx ? complex(0) : 0;
     var one = big ? bignumber(1) : cplx ? complex(1) : 1; // matrix is already upper triangular
     // return an identity matrix
 
     if (smaller(abs(c), prec)) {
       return [[one, zero], [zero, one]];
     } // matrix is diagonalizable
     // return its eigenvectors as columns
 
 
     if (larger(abs(subtract(l1, l2)), prec)) {
       return [[subtract(l1, d), subtract(l2, d)], [c, c]];
     } // matrix is not diagonalizable
     // compute off-diagonal elements of N = A - λI
     // N₁₂ = 0 ⇒ S = ( N⃗₁, I⃗₁ )
     // N₁₂ ≠ 0 ⇒ S = ( N⃗₂, I⃗₂ )
 
 
     var na = subtract(a, l1);
     var nb = subtract(b, l1);
     var nc = subtract(c, l1);
     var nd = subtract(d, l1);
 
     if (smaller(abs(nb), prec)) {
       return [[na, one], [nc, zero]];
     } else {
       return [[nb, zero], [nd, one]];
     }
   }
   /**
    * Enlarge the matrix from n×n to N×N, setting the new
    * elements to 1 on diagonal and 0 elsewhere
    */
 
 
   function inflateMatrix(arr, N) {
     // add columns
     for (var i = 0; i < arr.length; i++) {
       arr[i].push(...Array(N - arr[i].length).fill(0));
     } // add rows
 
 
     for (var _i5 = arr.length; _i5 < N; _i5++) {
       arr.push(Array(N).fill(0));
       arr[_i5][_i5] = 1;
     }
 
     return arr;
   }
   /**
    * Create a block-diagonal matrix with the given square matrices on the diagonal
    * @param {Matrix[] | number[][][]} arr array of matrices to be placed on the diagonal
    * @param {number} N the size of the resulting matrix
    */
 
 
   function blockDiag(arr, N) {
     var M = [];
 
     for (var i = 0; i < N; i++) {
       M[i] = Array(N).fill(0);
     }
 
     var I = 0;
 
     for (var sub of arr) {
       var n = sub.length;
 
       for (var _i6 = 0; _i6 < n; _i6++) {
         for (var j = 0; j < n; j++) {
           M[I + _i6][I + j] = sub[_i6][j];
         }
       }
 
       I += n;
     }
 
     return M;
   }
   /**
    * Finds the index of an element in an array using a custom equality function
    * @template T
    * @param {Array<T>} arr array in which to search
    * @param {T} el the element to find
    * @param {function(T, T): boolean} fn the equality function, first argument is an element of `arr`, the second is always `el`
    * @returns {number} the index of `el`, or -1 when it's not in `arr`
    */
 
 
   function indexOf(arr, el, fn) {
     for (var i = 0; i < arr.length; i++) {
       if (fn(arr[i], el)) {
         return i;
       }
     }
 
     return -1;
   }
   /**
    * Provided a near-singular upper-triangular matrix A and a list of vectors,
    * finds an eigenvector of A with the smallest eigenvalue, which is orthogonal
    * to each vector in the list
    * @template T
    * @param {T[][]} A near-singular square matrix
    * @param {number} N dimension
    * @param {T[][]} orthog list of vectors
    * @param {number} prec epsilon
    * @param {'number'|'BigNumber'|'Complex'} type
    * @return {T[] | null} eigenvector
    *
    * @see Numerical Recipes for Fortran 77 – 11.7 Eigenvalues or Eigenvectors by Inverse Iteration
    */
 
 
   function inverseIterate(A, N, orthog, prec, type) {
     var largeNum = type === 'BigNumber' ? bignumber(1000) : 1000;
     var b; // the vector
     // you better choose a random vector before I count to five
 
     var i = 0;
 
     while (true) {
       b = randomOrthogonalVector(N, orthog, type);
       b = usolve(A, b);
 
       if (larger(norm(b), largeNum)) {
         break;
       }
 
       if (++i >= 5) {
         return null;
       }
     } // you better converge before I count to ten
 
 
     i = 0;
 
     while (true) {
       var c = usolve(A, b);
 
       if (smaller(norm(orthogonalComplement(b, [c])), prec)) {
         break;
       }
 
       if (++i >= 10) {
         return null;
       }
 
       b = normalize(c);
     }
 
     return b;
   }
   /**
    * Generates a random unit vector of dimension N, orthogonal to each vector in the list
    * @template T
    * @param {number} N dimension
    * @param {T[][]} orthog list of vectors
    * @param {'number'|'BigNumber'|'Complex'} type
    * @returns {T[]} random vector
    */
 
 
   function randomOrthogonalVector(N, orthog, type) {
     var big = type === 'BigNumber';
     var cplx = type === 'Complex'; // generate random vector with the correct type
 
     var v = Array(N).fill(0).map(_ => 2 * Math.random() - 1);
 
     if (big) {
       v = v.map(n => bignumber(n));
     }
 
     if (cplx) {
       v = v.map(n => complex(n));
     } // project to orthogonal complement
 
 
     v = orthogonalComplement(v, orthog); // normalize
 
     return normalize(v, type);
   }
   /**
    * Project vector v to the orthogonal complement of an array of vectors
    */
 
 
   function orthogonalComplement(v, orthog) {
     for (var w of orthog) {
       // v := v − (w, v)/∥w∥² w
       v = subtract(v, multiply(divideScalar(dot(w, v), dot(w, w)), w));
     }
 
     return v;
   }
   /**
    * Calculate the norm of a vector.
    * We can't use math.norm because factory can't handle circular dependency.
    * Seriously, I'm really fed up with factory.
    */
 
 
   function norm(v) {
     return abs(sqrt(dot(v, v)));
   }
   /**
    * Normalize a vector
    * @template T
    * @param {T[]} v
    * @param {'number'|'BigNumber'|'Complex'} type
    * @returns {T[]} normalized vec
    */
 
 
   function normalize(v, type) {
     var big = type === 'BigNumber';
     var cplx = type === 'Complex';
     var one = big ? bignumber(1) : cplx ? complex(1) : 1;
     return multiply(divideScalar(one, norm(v)), v);
   }
 
   return complexEigs;
 }