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node_modules/fraction.js/examples/hesse-convergence.js

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+/*
+Fraction.js v5.0.0 10/1/2024
+https://raw.org/article/rational-numbers-in-javascript/
+
+Copyright (c) 2024, Robert Eisele (https://raw.org/)
+Licensed under the MIT license.
+*/
+const Fraction = require('fraction.js');
+
+/*
+We have the polynom f(x) = 1/3x_1^2 + x_2^2 + x_1 * x_2 + 3
+
+The gradient of f(x):
+
+grad(x) = | x_1^2+x_2 |
+          | 2x_2+x_1  |
+
+And thus the Hesse-Matrix H:
+| 2x_1  1 |
+|  1    2 |
+
+The inverse Hesse-Matrix H^-1 is
+| -2 / (1-4x_1)    1 / (1 - 4x_1)     |
+| 1 / (1 - 4x_1)   -2x_1 / (1 - 4x_1) |
+
+We now want to find lim ->oo x[n], with the starting element of (3 2)^T
+
+*/
+
+// Get the Hesse Matrix
+function H(x) {
+
+  var z = Fraction(1).sub(Fraction(4).mul(x[0]));
+
+  return [
+    Fraction(-2).div(z),
+    Fraction(1).div(z),
+    Fraction(1).div(z),
+    Fraction(-2).mul(x[0]).div(z),
+  ];
+}
+
+// Get the gradient of f(x)
+function grad(x) {
+
+  return [
+    Fraction(x[0]).mul(x[0]).add(x[1]),
+    Fraction(2).mul(x[1]).add(x[0])
+  ];
+}
+
+// A simple matrix multiplication helper
+function matrMult(m, v) {
+
+  return [
+    Fraction(m[0]).mul(v[0]).add(Fraction(m[1]).mul(v[1])),
+    Fraction(m[2]).mul(v[0]).add(Fraction(m[3]).mul(v[1]))
+  ];
+}
+
+// A simple vector subtraction helper
+function vecSub(a, b) {
+
+  return [
+    Fraction(a[0]).sub(b[0]),
+    Fraction(a[1]).sub(b[1])
+  ];
+}
+
+// Main function, gets a vector and the actual index
+function run(V, j) {
+
+  var t = H(V);
+  //console.log("H(X)");
+  for (var i in t) {
+
+    //	console.log(t[i].toFraction());
+  }
+
+  var s = grad(V);
+  //console.log("vf(X)");
+  for (var i in s) {
+
+    //	console.log(s[i].toFraction());
+  }
+
+  //console.log("multiplication");
+  var r = matrMult(t, s);
+  for (var i in r) {
+
+    //	console.log(r[i].toFraction());
+  }
+
+  var R = (vecSub(V, r));
+
+  console.log("X" + j);
+  console.log(R[0].toFraction(), "= " + R[0].valueOf());
+  console.log(R[1].toFraction(), "= " + R[1].valueOf());
+  console.log("\n");
+
+  return R;
+}
+
+
+// Set the starting vector
+var v = [3, 2];
+
+for (var i = 0; i < 15; i++) {
+
+  v = run(v, i);
+}