# Recitation 2

## Topics Today
* Writing Matrices in C
* Manipulating Matrices in C

## Writing Matrices in C

### Matrix Declaration
#### Initialize a matrix a two-dimensional array statically

int matrix1[3][3] = {
        {1, 2, 3},
        {4, 5, 6},
        {7, 8, 9}
    };

Or you can just use one array to store the whole matrix in a row major style:

#define N 3
#define M 3
#define AT(i,j) (i*N+j)
double *matrix = (double*)malloc(N*M*sizeof(double));
int cnt = 0;
for(int i = 0; i < N; ++i){
    for(int j = 0; j < M; ++j){
         matrix[AT(i,j)]=cnt++; 
    }
}

#### Initialize a matrix using double pointer(wicked way)
* Declare a pointer to a pointer (double pointer) for the matrix

    int **matrix;

* Allocate memory for the matrix dynamically

    matrix = (int **)malloc(N * sizeof(int *));

* Check it

    if (matrix == NULL) {
        fprintf(stderr, "Memory allocation failed.\n");
    }

    for (int i = 0; i < N; i++) {
        matrix[i] = (int *)malloc(M * sizeof(int));
        if (matrix[i] == NULL) {
            fprintf(stderr, "Memory allocation failed.\n");
        }
    }

* Populate the matrix with values

    int value = 1;
    for (int i = 0; i < rows; i++) {
        for (int j = 0; j < cols; j++) {
            matrix[i][j] = value;
            value++;
        }
    }

* After all the calculation, free dynamically allocated memory

    for (int i = 0; i < rows; i++) {
        free(matrix[i]);
    }
    free(matrix);

## Manipulating Matrices in C
To incoporate all the information needed for a matrix, we can create a struct Matrix:

typedef struct Matrix{
    int rows;
    int cols;
    double ** data;
} Mat;

Define a function to initialize a Matrix to 0 given rows and cols:

Mat * initializeMatrix(int rows, int cols){
    Mat * res = (Mat*) malloc(sizeof(Mat));
    res->rows = rows;
    res->cols = cols;
    res->data = (double**)malloc(sizeof(double*)*res->rows);
    for(int i = 0; i < res->rows; ++i){
        res->data[i] = (double*)malloc(sizeof(double)*res->cols);
        for(int j = 0; j < res->cols; ++j){
            res->data[i][j] = 0.0;
        }
    }
    return res;
}

Now we can write the add function:

Mat * add(Mat*mat1, Mat*mat2){
    // we need to check if the size of the two matrices are the same
    if(mat1->rows != mat2->rows || mat1->cols != mat2->cols){
        printf("The size of two matrices are not the same");
        exit(1);
    }
    Mat * res = initializeMatrix(mat1->rows, mat2->cols);
    for(int i = 0; i < res->rows; ++i){
        for(int j = 0; j < res->cols; ++j){
            res->data[i][j] = mat1->data[i][j] + mat2->data[i][j];
        }
    }
    return res;
}

We can also define the multiplication between a matrix and a scalar:

Mat* multiplyScalar(Mat* mat1, double coef){
    Mat * res = initializeMatrix(mat1->rows, mat1->cols);
    for(int i = 0; i < res->rows; ++i){
        for(int j = 0; j < res->cols; ++j){
            res->data[i][j] = mat1->data[i][j] * coef;
        }
    }
    return res;
}

Now we have subtract function. We can either do element-wise subtraction or combine `add` function and `multiplyScalar` function:

Mat* subtract(Mat*mat1, Mat*mat2){
    // we need to check if the size of the two matrices are the same
    if(mat1->rows != mat2->rows || mat1->cols != mat2->cols){
        printf("The size of two matrices are not the same");
        exit(1);
    }
    return add(mat1, multiplyScalar(mat2, -1.0));
}

Finally we can define matrix multiplication:

Mat* multiply(Mat* mat1, Mat* mat2){
    if(mat1->cols != mat2->rows){
        printf("A matrix of size (%d, %d) cannot be multiplied with a matrix with size (%d, %d)", mat1->rows, mat1->cols, mat2->rows, mat2->cols);
    }
    Mat * res = initializeMatrix(mat1->rows, mat2->cols);
    for(int i = 0; i < res->rows; ++i){
        for(int j = 0; j < res->cols; ++j){
            for(int k = 0; k < mat1->cols; ++k){
                res->data[i][j] += mat1->data[i][k] * mat2->data[k][j];
            }
        }
    }
    return res;
}