freeCodeCamp/curriculum/challenges/italian/10-coding-interview-prep/rosetta-code/cramers-rule.md

id, title, challengeType, forumTopicId, dashedName

id	title	challengeType	forumTopicId	dashedName
59713da0a428c1a62d7db430	Regola di Cramer	5	302239	cramers-rule

--description--

In <a href="https://it.wikipedia.org/wiki/Algebra_lineare"">algebra lineare, la regola di Cramer è una formula esplicita per la risoluzione di un sistema di equazioni lineari con tante soluzioni quante sono le variabili, valida ogni volta che il sistema ha una soluzione unica. Esprime la soluzione in termine di determinanti della matrice quadrata dei coefficienti e delle matrici ottenute da essa sostituendo una delle colonne con il vettore dei termini a destra dell'uguale nelle equazioni.

Dati

$\left\{\begin{matrix}a_1x + b_1y + c_1z&= {\color{red}d_1}\\a_2x + b_2y + c_2z&= {\color{red}d_2}\\a_3x + b_3y + c_3z&= {\color{red}d_3}\end{matrix}\right.$

che in forma matriciale è

\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}=\\begin{bmatrix} {\\color{red}d_1} \\\\ {\\color{red}d_2} \\\\ {\\color{red}d_3} \\end{bmatrix}.

Allora i valodi di x, y e z possono essere trovati come segue:

x = \\frac{\\begin{vmatrix} {\\color{red}d_1} & b_1 & c_1 \\\\ {\\color{red}d_2} & b_2 & c_2 \\\\ {\\color{red}d_3} & b_3 & c_3 \\end{vmatrix} } { \\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}}, \\quad y = \\frac {\\begin{vmatrix} a_1 & {\\color{red}d_1} & c_1 \\\\ a_2 & {\\color{red}d_2} & c_2 \\\\ a_3 & {\\color{red}d_3} & c_3 \\end{vmatrix}} {\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}}, \\text{ and }z = \\frac { \\begin{vmatrix} a_1 & b_1 & {\\color{red}d_1} \\\\ a_2 & b_2 & {\\color{red}d_2} \\\\ a_3 & b_3 & {\\color{red}d_3} \\end{vmatrix}} {\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix} }.

--instructions--

Dato il seguente sistema di equazioni:

\\begin{cases} 2w-x+5y+z=-3 \\\\ 3w+2x+2y-6z=-32 \\\\ w+3x+3y-z=-47 \\\\ 5w-2x-3y+3z=49 \\\\ \\end{cases}

risolvi per w, x, y e z, usando la regola di Cramer.

--hints--

cramersRule dovrebbe essere una funzione.

assert(typeof cramersRule === 'function');

cramersRule([[2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3]], [-3, -32, -47, 49]) dovrebbe restituire [2, -12, -4, 1].

assert.deepEqual(cramersRule(matrices[0], freeTerms[0]), answers[0]);

cramersRule([[3, 1, 1], [2, 2, 5], [1, -3, -4]], [3, -1, 2]) dovrebbe restituire [1, 1, -1].

assert.deepEqual(cramersRule(matrices[1], freeTerms[1]), answers[1]);

--seed--

--after-user-code--

const matrices = [
  [
    [2, -1, 5, 1],
    [3, 2, 2, -6],
    [1, 3, 3, -1],
    [5, -2, -3, 3]
  ],
  [
    [3, 1, 1],
    [2, 2, 5],
    [1, -3, -4]
  ]
];
const freeTerms = [[-3, -32, -47, 49], [3, -1, 2]];

const answers = [[2, -12, -4, 1], [1, 1, -1]];

--seed-contents--

function cramersRule(matrix, freeTerms) {

  return true;
}

--solutions--

/**
 * Compute Cramer's Rule
 * @param  {array} matrix    x,y,z, etc. terms
 * @param  {array} freeTerms
 * @return {array}           solution for x,y,z, etc.
 */
function cramersRule(matrix, freeTerms) {
  const det = detr(matrix);
  const returnArray = [];
  let i;

  for (i = 0; i < matrix[0].length; i++) {
    const tmpMatrix = insertInTerms(matrix, freeTerms, i);
    returnArray.push(detr(tmpMatrix) / det);
  }
  return returnArray;
}

/**
 * Inserts single dimensional array into
 * @param  {array} matrix multidimensional array to have ins inserted into
 * @param  {array} ins single dimensional array to be inserted vertically into matrix
 * @param  {array} at  zero based offset for ins to be inserted into matrix
 * @return {array}     New multidimensional array with ins replacing the at column in matrix
 */
function insertInTerms(matrix, ins, at) {
  const tmpMatrix = clone(matrix);
  let i;
  for (i = 0; i < matrix.length; i++) {
    tmpMatrix[i][at] = ins[i];
  }
  return tmpMatrix;
}
/**
 * Compute the determinate of a matrix.  No protection, assumes square matrix
 * function borrowed, and adapted from MIT Licensed numericjs library (www.numericjs.com)
 * @param  {array} m Input Matrix (multidimensional array)
 * @return {number}   result rounded to 2 decimal
 */
function detr(m) {
  let ret = 1;
  let j;
  let k;
  const A = clone(m);
  const n = m[0].length;
  let alpha;

  for (j = 0; j < n - 1; j++) {
    k = j;
    for (let i = j + 1; i < n; i++) { if (Math.abs(A[i][j]) > Math.abs(A[k][j])) { k = i; } }
    if (k !== j) {
      const temp = A[k]; A[k] = A[j]; A[j] = temp;
      ret *= -1;
    }
    const Aj = A[j];
    for (let i = j + 1; i < n; i++) {
      const Ai = A[i];
      alpha = Ai[j] / Aj[j];
      for (k = j + 1; k < n - 1; k += 2) {
        const k1 = k + 1;
        Ai[k] -= Aj[k] * alpha;
        Ai[k1] -= Aj[k1] * alpha;
      }
      if (k !== n) { Ai[k] -= Aj[k] * alpha; }
    }
    if (Aj[j] === 0) { return 0; }
    ret *= Aj[j];
  }
  return Math.round(ret * A[j][j] * 100) / 100;
}

/**
 * Clone two dimensional Array using ECMAScript 5 map function and EcmaScript 3 slice
 * @param  {array} m Input matrix (multidimensional array) to clone
 * @return {array}   New matrix copy
 */
function clone(m) {
  return m.map(a => a.slice());
}