ysjj-system/node_modules/mathjs/lib/esm/function/algebra/decomposition/qr.js

import _extends from "@babel/runtime/helpers/extends";
import { factory } from '../../../utils/factory.js';
var name = 'qr';
var dependencies = ['typed', 'matrix', 'zeros', 'identity', 'isZero', 'equal', 'sign', 'sqrt', 'conj', 'unaryMinus', 'addScalar', 'divideScalar', 'multiplyScalar', 'subtract', 'complex'];
export var createQr = /* #__PURE__ */factory(name, dependencies, _ref => {
  var {
    typed,
    matrix,
    zeros,
    identity,
    isZero,
    equal,
    sign,
    sqrt,
    conj,
    unaryMinus,
    addScalar,
    divideScalar,
    multiplyScalar,
    subtract,
    complex
  } = _ref;
  /**
   * Calculate the Matrix QR decomposition. Matrix `A` is decomposed in
   * two matrices (`Q`, `R`) where `Q` is an
   * orthogonal matrix and `R` is an upper triangular matrix.
   *
   * Syntax:
   *
   *    math.qr(A)
   *
   * Example:
   *
   *    const m = [
   *      [1, -1,  4],
   *      [1,  4, -2],
   *      [1,  4,  2],
   *      [1,  -1, 0]
   *    ]
   *    const result = math.qr(m)
   *    // r = {
   *    //   Q: [
   *    //     [0.5, -0.5,   0.5],
   *    //     [0.5,  0.5,  -0.5],
   *    //     [0.5,  0.5,   0.5],
   *    //     [0.5, -0.5,  -0.5],
   *    //   ],
   *    //   R: [
   *    //     [2, 3,  2],
   *    //     [0, 5, -2],
   *    //     [0, 0,  4],
   *    //     [0, 0,  0]
   *    //   ]
   *    // }
   *
   * See also:
   *
   *    lup, lusolve
   *
   * @param {Matrix | Array} A    A two dimensional matrix or array
   * for which to get the QR decomposition.
   *
   * @return {{Q: Array | Matrix, R: Array | Matrix}} Q: the orthogonal
   * matrix and R: the upper triangular matrix
   */
  return _extends(typed(name, {
    DenseMatrix: function DenseMatrix(m) {
      return _denseQR(m);
    },
    SparseMatrix: function SparseMatrix(m) {
      return _sparseQR(m);
    },
    Array: function Array(a) {
      // create dense matrix from array
      var m = matrix(a);
      // lup, use matrix implementation
      var r = _denseQR(m);
      // result
      return {
        Q: r.Q.valueOf(),
        R: r.R.valueOf()
      };
    }
  }), {
    _denseQRimpl
  });
  function _denseQRimpl(m) {
    // rows & columns (m x n)
    var rows = m._size[0]; // m
    var cols = m._size[1]; // n

    var Q = identity([rows], 'dense');
    var Qdata = Q._data;
    var R = m.clone();
    var Rdata = R._data;

    // vars
    var i, j, k;
    var w = zeros([rows], '');
    for (k = 0; k < Math.min(cols, rows); ++k) {
      /*
       * **k-th Household matrix**
       *
       * The matrix I - 2*v*transpose(v)
       * x     = first column of A
       * x1    = first element of x
       * alpha = x1 / |x1| * |x|
       * e1    = tranpose([1, 0, 0, ...])
       * u     = x - alpha * e1
       * v     = u / |u|
       *
       * Household matrix = I - 2 * v * tranpose(v)
       *
       *  * Initially Q = I and R = A.
       *  * Household matrix is a reflection in a plane normal to v which
       *    will zero out all but the top right element in R.
       *  * Appplying reflection to both Q and R will not change product.
       *  * Repeat this process on the (1,1) minor to get R as an upper
       *    triangular matrix.
       *  * Reflections leave the magnitude of the columns of Q unchanged
       *    so Q remains othoganal.
       *
       */

      var pivot = Rdata[k][k];
      var sgn = unaryMinus(equal(pivot, 0) ? 1 : sign(pivot));
      var conjSgn = conj(sgn);
      var alphaSquared = 0;
      for (i = k; i < rows; i++) {
        alphaSquared = addScalar(alphaSquared, multiplyScalar(Rdata[i][k], conj(Rdata[i][k])));
      }
      var alpha = multiplyScalar(sgn, sqrt(alphaSquared));
      if (!isZero(alpha)) {
        // first element in vector u
        var u1 = subtract(pivot, alpha);

        // w = v * u1 / |u|    (only elements k to (rows-1) are used)
        w[k] = 1;
        for (i = k + 1; i < rows; i++) {
          w[i] = divideScalar(Rdata[i][k], u1);
        }

        // tau = - conj(u1 / alpha)
        var tau = unaryMinus(conj(divideScalar(u1, alpha)));
        var s = void 0;

        /*
         * tau and w have been choosen so that
         *
         * 2 * v * tranpose(v) = tau * w * tranpose(w)
         */

        /*
         * -- calculate R = R - tau * w * tranpose(w) * R --
         * Only do calculation with rows k to (rows-1)
         * Additionally columns 0 to (k-1) will not be changed by this
         *   multiplication so do not bother recalculating them
         */
        for (j = k; j < cols; j++) {
          s = 0.0;

          // calculate jth element of [tranpose(w) * R]
          for (i = k; i < rows; i++) {
            s = addScalar(s, multiplyScalar(conj(w[i]), Rdata[i][j]));
          }

          // calculate the jth element of [tau * transpose(w) * R]
          s = multiplyScalar(s, tau);
          for (i = k; i < rows; i++) {
            Rdata[i][j] = multiplyScalar(subtract(Rdata[i][j], multiplyScalar(w[i], s)), conjSgn);
          }
        }
        /*
         * -- calculate Q = Q - tau * Q * w * transpose(w) --
         * Q is a square matrix (rows x rows)
         * Only do calculation with columns k to (rows-1)
         * Additionally rows 0 to (k-1) will not be changed by this
         *   multiplication so do not bother recalculating them
         */
        for (i = 0; i < rows; i++) {
          s = 0.0;

          // calculate ith element of [Q * w]
          for (j = k; j < rows; j++) {
            s = addScalar(s, multiplyScalar(Qdata[i][j], w[j]));
          }

          // calculate the ith element of [tau * Q * w]
          s = multiplyScalar(s, tau);
          for (j = k; j < rows; ++j) {
            Qdata[i][j] = divideScalar(subtract(Qdata[i][j], multiplyScalar(s, conj(w[j]))), conjSgn);
          }
        }
      }
    }

    // return matrices
    return {
      Q,
      R,
      toString: function toString() {
        return 'Q: ' + this.Q.toString() + '\nR: ' + this.R.toString();
      }
    };
  }
  function _denseQR(m) {
    var ret = _denseQRimpl(m);
    var Rdata = ret.R._data;
    if (m._data.length > 0) {
      var zero = Rdata[0][0].type === 'Complex' ? complex(0) : 0;
      for (var i = 0; i < Rdata.length; ++i) {
        for (var j = 0; j < i && j < (Rdata[0] || []).length; ++j) {
          Rdata[i][j] = zero;
        }
      }
    }
    return ret;
  }
  function _sparseQR(m) {
    throw new Error('qr not implemented for sparse matrices yet');
  }
});