feat: Add Gram Schmidt Orthogonalisation (TheAlgorithms#1339)

* Add Gram Schmidt Orthogonalisation * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp After removing '&' while passing the parameters. The changes in array B are not reflected. * Update gram_schmidt.cpp * Update gram_schmidt.cpp All the changes have been done and changes in B are also reflected. * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Fix spelling of brief * updating DIRECTORY.md * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update gram_schmidt.cpp * Update linear_algebra/gram_schmidt.cpp * Update linear_algebra/gram_schmidt.cpp * Update linear_algebra/gram_schmidt.cpp Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com> Co-authored-by: Ayaan Khan <[email protected]>
langbin2014 · Oct 31, 2020 · fb5e00f · fb5e00f
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diff --git a/DIRECTORY.md b/DIRECTORY.md
@@ -115,6 +115,9 @@
   * [Linear Probing Hash Table](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/hashing/linear_probing_hash_table.cpp)
   * [Quadratic Probing Hash Table](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/hashing/quadratic_probing_hash_table.cpp)
 
+## Linear Algebra
+  * [Gram Schmidt](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/linear_algebra/gram_schmidt.cpp)
+
 ## Machine Learning
   * [Adaline Learning](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/machine_learning/adaline_learning.cpp)
   * [Kohonen Som Topology](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/machine_learning/kohonen_som_topology.cpp)

diff --git a/linear_algebra/gram_schmidt.cpp b/linear_algebra/gram_schmidt.cpp
@@ -0,0 +1,251 @@
+/**  
+ * @file   
+ * @brief [Gram Schmidt Orthogonalisation Process](https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process)
+ *
+ * @details
+ * Takes the input of Linearly Independent Vectors,
+ * returns vectors orthogonal to each other.
+ * 
+ * ### Algorithm
+ * Take the first vector of given LI vectors as first vector of Orthogonal vectors.
+ * Take projection of second input vector on the first vector of Orthogonal vector 
+ * and subtract it from the 2nd LI vector.
+ * Take projection of third vector on the second vector of Othogonal vectors and subtract it from the 3rd LI vector.
+ * Keep repeating the above process until all the vectors in the given input array are exhausted. 
+ *
+ * For Example:
+ * In R2, 
+ * Input LI Vectors={(3,1),(2,2)}
+ * then Orthogonal Vectors= {(3, 1),(-0.4, 1.2)}
+ *
+ *  Have defined maximum dimension of vectors to be 10 and number of vectors
+ *  taken is 20.
+ *  Please do not give linearly dependent vectors
+ *
+ *
+ * @author [Akanksha Gupta](https://github.com/Akanksha-Gupta920)
+ */
+
+#include <iostream>    /// for io operations
+#include <cassert>     /// for assert
+#include <cmath>       /// for fabs
+#include <array>       /// for std::array
+
+/**
+ * @namespace linear_algebra
+ * @brief Linear Algebra algorithms
+ */
+namespace linear_algebra {
+/**
+ * @namespace gram_schmidt
+ * @brief Functions for [Gram Schmidt Orthogonalisation Process](https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process)
+ */
+namespace gram_schmidt {
+/**
+ * Dot product function.
+ * Takes 2 vectors along with their dimension as input and returns the dot product.
+ * @param x vector 1
+ * @param y vector 2
+ * @param c dimension of the vectors
+ * 
+ * @returns sum
+ */
+double dot_product(const std::array<double, 10>& x, const std::array<double, 10>& y, const int& c) {
+  double sum = 0;
+  for (int i = 0; i < c; ++i) {
+    sum += x[i] * y[i];
+  }
+  return sum;
+}
+
+/**
+ * Projection Function
+ * Takes input of 2 vectors along with their dimension and evaluates their projection in temp 
+ *
+ * @param x Vector 1
+ * @param y Vector 2
+ * @param c dimension of each vector
+ *
+ * @returns factor
+ */
+double projection(const std::array<double, 10>& x,const std::array<double, 10>& y,
+                 const int& c) {
+  double dot = dot_product(x, y, c); ///The dot product of two vectors is taken
+  double anorm = dot_product(y, y, c); ///The norm of the second vector is taken.
+  double factor = dot / anorm; ///multiply that factor with every element in a 3rd vector, whose initial values are same as the 2nd vector.
+  return factor;
+}
+
+/**
+ * Function to print the orthogonalised vector
+ *
+ * @param r number of vectors
+ * @param c dimenaion of vectors
+ * @param B stores orthogonalised vectors
+ *
+ * @returns void
+ */
+void display(const int& r,const int& c,const std::array<std::array<double, 10>, 20>& B) {
+  for (int i = 0; i < r; ++i) {
+    std::cout << "Vector " << i + 1 << ": ";
+    for (int j = 0; j < c; ++j) {
+      std::cout << B[i][j] << " ";
+    }
+    std::cout << '\n';
+  }
+}
+
+/**
+ * Function for the process of Gram Schimdt Process
+ * @param r number of vectors
+ * @param c dimension of vectors
+ * @param A stores input of given LI vectors
+ * @param B stores orthogonalised vectors
+ * 
+ * @returns void
+ */
+void gram_schmidt(int r,const int& c,const std::array<std::array<double, 10>, 20>& A,
+                  std::array<std::array<double, 10>, 20> B) {
+  if (c < r) {   /// we check whether appropriate dimensions are given or not.
+    std::cout
+        << "Dimension of vector is less than number of vector, hence \n first "
+        << c << " vectors are orthogonalised\n";
+    r = c;
+  }
+
+  int k = 1;
+
+  while (k <= r) {
+    if (k == 1) {
+      for (int j = 0; j < c; j++) B[0][j] = A[0][j]; ///First vector is copied as it is.
+    }
+
+    else {
+      std::array<double, 10> all_projection{};  ///array to store projections
+      for (int i = 0; i < c; ++i) {
+        all_projection[i] = 0;  ///First initialised to zero
+      }
+
+      int l = 1;
+      while (l < k) {
+        std::array<double, 10> temp{}; ///to store previous projected array
+        double factor; ///to store the factor by which the previous array will change 
+        factor = projection(A[k - 1], B[l - 1], c);
+        for(int i = 0; i < c; ++i)
+         temp[i] = B[l - 1][i] * factor;  ///projected array created
+        for (int j = 0; j < c; ++j) {
+          all_projection[j] = all_projection[j] + temp[j];   ///we take the projection with all the previous vector and add them.
+        }
+        l++;
+      }
+      for (int i = 0; i < c; ++i) {
+        B[k - 1][i] = A[k - 1][i] - all_projection[i];  ///subtract total projection vector from the input vector
+      }
+    }
+    k++;
+  }
+  display(r, c, B); //for displaying orthogoanlised vectors
+}
+}  // namespace gram_schmidt
+}  // namespace linear_algebra
+/**
+ * Test Function. Process has been tested for 3 Sample Inputs
+ * @returns void
+ */
+static void test() {
+  std::array<std::array<double, 10>, 20> a1 = {
+      {{1, 0, 1, 0}, {1, 1, 1, 1}, {0, 1, 2, 1}}};
+  std::array<std::array<double, 10>, 20> b1 = {{0}};
+       double dot1 = 0;
+ linear_algebra::gram_schmidt::gram_schmidt(3, 4, a1, b1);
+  int flag = 1;
+  for (int i = 0; i < 2; ++i)
+    for (int j = i + 1; j < 3; ++j) {
+      dot1 = fabs(linear_algebra::gram_schmidt::dot_product(b1[i], b1[j], 4));
+      if (dot1 > 0.1) {
+        flag = 0;
+        break;
+      }
+    }
+  if (flag == 0) std::cout << "Vectors are linearly dependent\n";
+  assert(flag == 1);
+  std::cout << "Passed Test Case 1\n ";
+
+  std::array<std::array<double, 10>, 20> a2 = {{{3, 1}, {2, 2}}};
+  std::array<std::array<double, 10>, 20> b2 = {{0}};
+  double dot2 = 0;
+  linear_algebra::gram_schmidt::gram_schmidt(2, 2, a2, b2);
+  flag = 1;
+  for (int i = 0; i < 1; ++i)
+    for (int j = i + 1; j < 2; ++j) {
+      dot2 = fabs(linear_algebra::gram_schmidt::dot_product(b2[i], b2[j], 2));
+      if (dot2 > 0.1) {
+        flag = 0;
+        break;
+      }
+    }
+  if (flag == 0) std::cout << "Vectors are linearly dependent\n";
+  assert(flag == 1);
+  std::cout << "Passed Test Case 2\n";
+
+  std::array<std::array<double, 10>, 20> a3 = {{{1, 2, 2}, {-4, 3, 2}}};
+  std::array<std::array<double, 10>, 20> b3 = {{0}};
+  double dot3 = 0;
+  linear_algebra::gram_schmidt::gram_schmidt(2, 3, a3, b3);
+  flag = 1;
+  for (int i = 0; i < 1; ++i)
+    for (int j = i + 1; j < 2; ++j) {
+      dot3 = fabs(linear_algebra::gram_schmidt::dot_product(b3[i], b3[j], 3));
+      if (dot3 > 0.1) {
+        flag = 0;
+        break;
+      }
+    }
+  if (flag == 0) std::cout << "Vectors are linearly dependent\n" ;
+  assert(flag == 1);
+  std::cout << "Passed Test Case 3\n";
+}
+
+/**
+ * @brief Main Function
+ * @return 0 on exit
+ */
+int main() {
+  int r=0, c=0;
+  test(); // perform self tests
+  std::cout << "Enter the dimension of your vectors\n";
+  std::cin >> c;
+  std::cout << "Enter the number of vectors you will enter\n";
+  std::cin >> r;
+
+  std::array<std::array<double, 10>, 20>
+      A{};  ///a 2-D array for storing all vectors
+  std::array<std::array<double, 10>, 20> B = {
+      {0}};  /// a 2-D array for storing orthogonalised vectors
+  /// storing vectors in array A
+  for (int i = 0; i < r; ++i) {
+    std::cout << "Enter vector " << i + 1 <<'\n';  ///Input of vectors is taken
+    for (int j = 0; j < c; ++j) {
+      std::cout << "Value " << j + 1 << "th of vector: ";
+      std::cin >> A[i][j];
+    }
+    std::cout <<'\n';
+  }
+
+  linear_algebra::gram_schmidt::gram_schmidt(r, c, A, B);
+
+  double dot = 0;
+  int flag = 1; ///To check whether vectors are orthogonal or  not
+  for (int i = 0; i < r - 1; ++i) {
+    for (int j = i + 1; j < r; ++j) {
+      dot = fabs(linear_algebra::gram_schmidt::dot_product(B[i], B[j], c));
+      if (dot > 0.1) /// take make the process numerically stable, upper bound for the dot product take 0.1
+      {
+        flag = 0;
+        break;
+      }
+    }
+  }
+  if (flag == 0) std::cout << "Vectors are linearly dependent\n"; 
+  return 0;
+}