Gram - Schmidt

#include <iostream>
#include <stdlib.h>
#include <math.h>
#include <string>
#include <vector>
#include <algorithm>

using namespace std;

// Auxiliary Functions
// 1. Display the vector
void displayVector(vector <double> v){
    cout << "{";
    for(unsigned int i=0; i<v.size(); i++){
        if(i != v.size()-1){
            cout << v[i] << ", ";
        }
        else{
            cout << v[i] << "}";
        }
    }
    cout << endl;
}

// 2. Calculate the norm of a vector
double norm(vector <double> v){
    double sum = 0;
    for(unsigned int i=0; i<v.size(); i++){
        sum += v[i] * v[i];
    }
    return sqrt(sum);
}

// 3. Add 2 vectors
vector <double> add(vector <double> v1, vector <double> v2){
    vector <double> result = vector <double> ();
    if(v1.size() == v2.size()){
        for(unsigned int i=0; i<v1.size(); i++){
            result.push_back(v1[i] + v2[i]);
        }
        return result;
    }
    else{
        cout << "Cannot add 2 vectors with different size/dimension." << endl;
    }
}

// 4. Subtract 2 vectors
vector <double> subtract(vector <double> v1, vector <double> v2){
    vector <double> result = vector <double> ();
    if(v1.size() == v2.size()){
        for(unsigned int i=0; i<v1.size(); i++){
            result.push_back(v1[i] - v2[i]);
        }
        return result;
    }
    else{
        cout << "Cannot add 2 vectors with different size/dimension." << endl;
    }
}

// 5. Multiply 2 vectors
double multiply(vector <double> v1, vector <double> v2){
    double sum = 0;
    if(v1.size() == v2.size()){
        for(unsigned int i=0; i<v1.size(); i++){
            sum += v1[i] * v2[i];
        }
        return sum;
    }
    else{
        cout << "Cannot multiply 2 vectors with different size/dimension." << endl;
    }
}

// 6. Multiply all the elements of a vector by a number
vector <double> multiplyVectorByNumber(vector <double> v, double number){
    for(unsigned int i=0; i<v.size(); i++){
        v[i] *= number;
    }
    return v;
}

// 7. Divide all the elements of a vector by a number
vector <double> divideVectorByNumber(vector <double> v, double number){
    for(unsigned int i=0; i<v.size(); i++){
        v[i] /= number;
    }
    return v;
}

// 8. Gram - Schmidt
vector < vector <double> > gramSchmidt(vector < vector <double> > sVector){
    // Create the basic variable
    vector < vector <double> > bases = vector < vector <double> > ();
    // Starting the session
    vector <double> g1 = sVector[0];
    vector <double> base1 = divideVectorByNumber(g1, norm(g1));
    bases.push_back(base1);
    // For-loop
    for(int i=1; i<sVector.size(); i++){
        vector <double> gi = sVector[i];
        for(int j=0; j<i; j++){
            gi = subtract( gi , multiplyVectorByNumber(bases[j], multiply(bases[j], gi)) );
        }
        bases.push_back(divideVectorByNumber(gi, norm(gi)));
    }
    return bases;
}




int main()
{
    // 1. Display the problem, ask the user for data
    cout << "******************" << endl;
    cout << "* GRAM - SCHMIDT *" << endl;
    cout << "******************" << endl << endl;
    cout << "First, give me how many vectors you have: ";
    int n;
    cin >> n;
    while(n < 2){
        cout << "First, give me how many vectors you have: ";
        cin >> n;
    }
    cout << endl;
    cout << "Now, I want you give me these vectors' dimension (how many coordinates each vector has): ";
    int m;
    cin >> m;
    while(m < 2){
        cout << "Now, I want you give me these vectors' dimension (how many coordinates each vector has): ";
        cin >> m;
    }
    cout << endl << endl;

    // 2. Create the "n" vectors
    vector < vector <double> > sVector = vector < vector <double> > ();
    for(int i=0; i<n; i++){
        cout << "Vector " << (i+1) << ": " << endl;
        vector <double> si = vector <double> ();
        for(int j=0; j<m; j++){
            cout << "Coordinate " << (j+1) << ": ";
            double coordinate;
            cin >> coordinate;
            si.push_back(coordinate);
        }
        sVector.push_back(si);
        cout << endl;
    }

    // 3. Show the user the "n" vector he has created
    for(unsigned int i=0; i<sVector.size(); i++){
        cout << "s" << (i+1) << " = ";
        displayVector(sVector[i]);
    }
    cout << endl;
    /*
    // Checking the functions
    cout << "Norm of s1 = ||s1|| = " << norm(sVector[0]) << endl;
    cout << "Norm of s2 = ||s2|| = " << norm(sVector[1]) << endl;
    vector <double> sumS1S2 = add(sVector[0], sVector[1]);
    cout << "s1 + s2 = ";
    displayVector(sumS1S2);
    vector <double> subtractS1S2 = subtract(sVector[0], sVector[1]);
    cout << "s1 - s2 = ";
    displayVector(subtractS1S2);
    double productS1S2 = multiply(sVector[0], sVector[1]);
    cout << "s1 * s2 = " << productS1S2;
    vector <double> prod = multiplyVectorByNumber(sVector[0], 8);
    cout << "\n8 * s1 = ";
    displayVector(prod);
    vector <double> normalizationOfS1 = divideVectorByNumber(sVector[0], norm(sVector[0]));
    cout << "Normalizing s1 ----> (s1/ ||s1||) = ";
    displayVector(normalizationOfS1);
    */
    cout << "~~~~ Gram - Schmidt having the vectors ";
    for(int i=0; i<sVector.size(); i++){
        if(i != sVector.size()-1){
            cout << "s" << (i+1) << ", ";
        }
        else{
            cout << "s" << (i+1) << " ~~~~" << endl;
        }
    }
    vector < vector <double> > bases = gramSchmidt(sVector);
    for(unsigned int i=0; i<sVector.size(); i++){
        cout << "base" << (i+1) << " = ";
        displayVector(bases[i]);
    }
    cout << endl << endl;
    cout << "~~~~ Confirmation ~~~~" << endl;
    for(unsigned int i=0; i<sVector.size(); i++){
        cout << "s" << (i+1) << " = ";
        for(unsigned int j=0; j<bases.size(); j++){
            if(j != bases.size()-1){
                cout << multiply(sVector[i], bases[j]) << "*";
                displayVector(bases[j]);
                cout << " + ";
            }
            else{
                cout << multiply(sVector[i], bases[j]) << "*";
                displayVector(bases[j]);
            }
        }
    }

    return 0;
}