feat: add row echelon matrix algorithm (#1454)

* feat: add row echelon matrix algorithm

* test: add self-tests for row echelon algorithm

* fix: replace rounding with float tolerance

* chore: use correct style

* fix: use error tolerance and segregate testcases

* chore: add necessary explaining comments

Maths/RowEchelon.js

@@ -0,0 +1,150 @@
+/**
+ * Given a two dimensional matrix, find its row echelon form.
+ *
+ * For more info: https://en.wikipedia.org/wiki/Row_echelon_form
+ *
+ * @param {number[[]]} matrix - Two dimensional array of rational numbers.
+ * @returns {number[[]]} - Two dimensional array of rational numbers (row echelon form).
+ *
+ * @example
+ * const matrix = [
+ *                          [2,3,4,5,7],
+ *                          [9,8,4,0,9],
+ *                          [5,7,4,3,9],
+ *                          [3,4,0,2,1]
+ *                ]
+ *
+ * const result = rowEchelon(matrix)
+ *
+ * // The function returns the corresponding row echelon form:
+ * // result:
+ * //             [
+ * //                       [1,  1.5,  2,        2.5,      3.5],
+ * //                       [0,  1,    2.54545,  4.09091,  4.09091],
+ * //                       [0,  0,    1,        1.57692,  1.36539],
+ * //                       [0,  0,    0,        1,        -0.25]
+ * //             ]
+ */
+
+// Set a tolerance value for floating-point comparisons
+const tolerance = 0.000001
+
+// Check if all the rows have same length of elements
+const isMatrixValid = (matrix) => {
+  let numRows = matrix.length
+  let numCols = matrix[0].length
+  for (let i = 0; i < numRows; i++) {
+    if (numCols !== matrix[i].length) {
+      return false
+    }
+  }
+
+  // Check for input other than a 2D matrix
+  if (
+    !Array.isArray(matrix) ||
+    matrix.length === 0 ||
+    !Array.isArray(matrix[0])
+  ) {
+    return false
+  }
+  return true
+}
+
+const checkNonZero = (currentRow, currentCol, matrix) => {
+  let numRows = matrix.length
+  for (let i = currentRow; i < numRows; i++) {
+    // Checks if the current element is not very near to zero.
+    if (!isTolerant(0, matrix[i][currentCol], tolerance)) {
+      return true
+    }
+  }
+  return false
+}
+
+const swapRows = (currentRow, withRow, matrix) => {
+  let numCols = matrix[0].length
+  let tempValue = 0
+  for (let j = 0; j < numCols; j++) {
+    tempValue = matrix[currentRow][j]
+    matrix[currentRow][j] = matrix[withRow][j]
+    matrix[withRow][j] = tempValue
+  }
+}
+
+// Select a pivot element in the current column to facilitate row operations.
+// Pivot element is the first non-zero element found from the current row
+// down to the last row.
+const selectPivot = (currentRow, currentCol, matrix) => {
+  let numRows = matrix.length
+  for (let i = currentRow; i < numRows; i++) {
+    if (matrix[i][currentCol] !== 0) {
+      swapRows(currentRow, i, matrix)
+      return
+    }
+  }
+}
+
+// Multiply each element of the given row with a factor.
+const scalarMultiplication = (currentRow, factor, matrix) => {
+  let numCols = matrix[0].length
+  for (let j = 0; j < numCols; j++) {
+    matrix[currentRow][j] *= factor
+  }
+}
+
+// Subtract one row from another row
+const subtractRow = (currentRow, fromRow, matrix) => {
+  let numCols = matrix[0].length
+  for (let j = 0; j < numCols; j++) {
+    matrix[fromRow][j] -= matrix[currentRow][j]
+  }
+}
+
+// Check if two numbers are equal within a given tolerance
+const isTolerant = (a, b, tolerance) => {
+  const absoluteDifference = Math.abs(a - b)
+  return absoluteDifference <= tolerance
+}
+
+const rowEchelon = (matrix) => {
+  // Check if the input matrix is valid; if not, throw an error.
+  if (!isMatrixValid(matrix)) {
+    throw new Error('Input is not a valid 2D matrix.')
+  }
+
+  let numRows = matrix.length
+  let numCols = matrix[0].length
+  let result = matrix
+
+  // Iterate through the rows (i) and columns (j) of the matrix.
+  for (let i = 0, j = 0; i < numRows && j < numCols; ) {
+    // If the current column has all zero elements below the current row,
+    // move to the next column.
+    if (!checkNonZero(i, j, result)) {
+      j++
+      continue
+    }
+
+    // Select a pivot element and normalize the current row.
+    selectPivot(i, j, result)
+    let factor = 1 / result[i][j]
+    scalarMultiplication(i, factor, result)
+
+    // Make elements below the pivot element zero by performing
+    // row operations on subsequent rows.
+    for (let x = i + 1; x < numRows; x++) {
+      factor = result[x][j]
+      if (isTolerant(0, factor, tolerance)) {
+        continue
+      }
+      scalarMultiplication(i, factor, result)
+      subtractRow(i, x, result)
+      factor = 1 / factor
+      scalarMultiplication(i, factor, result)
+    }
+    i++
+  }
+  return result
+}
+
+export { rowEchelon }

Maths/test/RowEchelon.test.js

@@ -0,0 +1,89 @@
+import { rowEchelon } from '../RowEchelon'
+describe('Determinant', () => {
+  const tolerance = 0.000001
+  test.each([
+    [
+      [
+        [8, 1, 3, 5],
+        [4, 6, 8, 2],
+        [3, 5, 6, 8]
+      ],
+      [
+        [1, 0.125, 0.375, 0.625],
+        [0, 1, 1.18182, -0.09091],
+        [0, 0, 1, -11.0769]
+      ]
+    ],
+    [
+      [
+        [6, 8, 1, 3, 5],
+        [1, 4, 6, 8, 2],
+        [0, 3, 5, 6, 8],
+        [2, 5, 9, 7, 8],
+        [5, 5, 7, 0, 1]
+      ],
+      [
+        [1, 1.33333, 0.16667, 0.5, 0.83333],
+        [0, 1, 2.1875, 2.8125, 0.4375],
+        [0, 0, 1, 1.56, -4.28003],
+        [0, 0, 0, 1, -3.3595],
+        [0, 0, 0, 0, 1]
+      ]
+    ],
+    [
+      [
+        [1, 3, 5],
+        [6, 8, 2],
+        [5, 6, 8],
+        [7, 9, 9],
+        [5, 0, 6]
+      ],
+      [
+        [1, 3, 5],
+        [0, 1, 2.8],
+        [0, 0, 1],
+        [0, 0, 0],
+        [0, 0, 0]
+      ]
+    ],
+    [
+      [
+        [0, 7, 8, 1, 3, 5],
+        [0, 6, 4, 6, 8, 2],
+        [0, 7, 3, 5, 6, 8],
+        [6, 8, 1, 0, 0, 4],
+        [3, 3, 5, 7, 3, 1],
+        [1, 2, 1, 0, 9, 7],
+        [8, 8, 0, 2, 3, 1]
+      ],
+      [
+        [1, 1.33333, 0.16667, 0, 0, 0.66667],
+        [0, 1, 0.66667, 1, 1.33333, 0.33333],
+        [0, 0, 1, 1.2, 1.99999, -3.4],
+        [0, 0, 0, 1, 1.3, -1.4],
+        [0, 0, 0, 0, 1, -2.32854],
+        [0, 0, 0, 0, 0, 1],
+        [0, 0, 0, 0, 0, 0]
+      ]
+    ]
+  ])('Should return the matrix in row echelon form.', (matrix, expected) => {
+    for (let i = 0; i < matrix.length; i++) {
+      for (let j = 0; j < matrix[i].length; j++) {
+        expect(rowEchelon(matrix)[i][j]).toBeCloseTo(expected[i][j], tolerance)
+      }
+    }
+  })
+
+  test.each([
+    [
+      [
+        [8, 1, 3, 5],
+        [4, 6, 8, 2, 7],
+        [3, 5, 6, 8]
+      ],
+      'Input is not a valid 2D matrix.'
+    ]
+  ])('Should return the error message.', (matrix, expected) => {
+    expect(() => rowEchelon(matrix)).toThrowError(expected)
+  })
+})