gram_schmidt.cpp File Reference

Gram Schmidt Orthogonalisation Process More...

#include <iostream>
#include <cassert>
#include <cmath>
#include <array>

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Namespaces
	linear_algebra
	Linear Algebra algorithms.
	gram_schmidt
	Functions for Gram Schmidt Orthogonalisation Process

Functions
double	linear_algebra::gram_schmidt::dot_product (const std::array< double, 10 > &x, const std::array< double, 10 > &y, const int &c)
double	linear_algebra::gram_schmidt::projection (const std::array< double, 10 > &x, const std::array< double, 10 > &y, const int &c)
void	linear_algebra::gram_schmidt::display (const int &r, const int &c, const std::array< std::array< double, 10 >, 20 > &B)
void	linear_algebra::gram_schmidt::gram_schmidt (int r, const int &c, const std::array< std::array< double, 10 >, 20 > &A, std::array< std::array< double, 10 >, 20 > B)
static void	test ()
int	main ()
	Main Function. More...

Detailed Description

Gram Schmidt Orthogonalisation Process

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

Function Documentation

display()

void linear_algebra::gram_schmidt::display	(	const int &	r,
		const int &	c,
		const std::array< std::array< double, 10 >, 20 > &	B
	)

Function to print the orthogonalised vector

Parameters

r	number of vectors
c	dimenaion of vectors
B	stores orthogonalised vectors

Returns

void

                                                                                    {
  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';
  }
}

dot_product()

double linear_algebra::gram_schmidt::dot_product	(	const std::array< double, 10 > &	x,
		const std::array< double, 10 > &	y,
		const int &	c
	)

Dot product function. Takes 2 vectors along with their dimension as input and returns the dot product.

Parameters

x	vector 1
y	vector 2
c	dimension of the vectors

Returns

sum

                                                                                               {
  double sum = 0;
  for (int i = 0; i < c; ++i) {
    sum += x[i] * y[i];
  }
  return sum;
}

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gram_schmidt()

void linear_algebra::gram_schmidt::gram_schmidt	(	int	r,
		const int &	c,
		const std::array< std::array< double, 10 >, 20 > &	A,
		std::array< std::array< double, 10 >, 20 >	B
	)

Function for the process of Gram Schimdt Process

Parameters

r	number of vectors
c	dimension of vectors
A	stores input of given LI vectors
B	stores orthogonalised vectors

Returns

void

we check whether appropriate dimensions are given or not.

First vector is copied as it is.

array to store projections

First initialised to zero

to store previous projected array

to store the factor by which the previous array will change

projected array created

we take the projection with all the previous vector and add them.

subtract total projection vector from the input vector

                                                        {
  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
}

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main()

int main

(

void

)

Main Function.

Returns

0 on exit

a 2-D array for storing all vectors

a 2-D array for storing orthogonalised vectors

storing vectors in array A

Input of vectors is taken

To check whether vectors are orthogonal or not

take make the process numerically stable, upper bound for the dot product take 0.1

           {
  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;
}

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projection()

double linear_algebra::gram_schmidt::projection	(	const std::array< double, 10 > &	x,
		const std::array< double, 10 > &	y,
		const int &	c
	)

Projection Function Takes input of 2 vectors along with their dimension and evaluates their projection in temp

Parameters

x	Vector 1
y	Vector 2
c	dimension of each vector

Returns

factor

The dot product of two vectors is taken

The norm of the second vector is taken.

multiply that factor with every element in a 3rd vector, whose initial values are same as the 2nd vector.

                               {
  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;
}

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test()

static void test ( )

static

Test Function. Process has been tested for 3 Sample Inputs

Returns

void

                   {
  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";
}