## 8.8. Matrices

A matrix is a 2D array. A matrix in the C language can be statically allocated as a 2-dimensional array (`M[n][m]`), dynamically allocated with a single call to `malloc()`, or dynamically allocated as an array of arrays. Let’s consider the array of arrays implementation. The first array contains `n` elements (`M[n]`), and each element `M[i]` in our matrix contains an array of `m` elements. The following code snippets each declare matrices of size 4 x 3:

``````//statically-allocated matrix (allocated on stack)
int M1;

//dynamically-allocated matrix (programmer friendly, allocated on heap)
int **M2, i;
M2 = malloc(4 * sizeof(int*));
for (i = 0; i < 4; i++) {
M2[i] = malloc(3 * sizeof(int));
}``````

In the case of the dynamically allocated matrix, the main array contains a contiguous array of `int` pointers. Each integer pointer points to a different array in memory. Figure 1 illustrates how we would normally visualize each of these matrices: Figure 1. Illustration of a statically-allocated (M1) and dynamically-allocated (M2) 3x4 matrix.

For both of the above matrix declarations, element (i,j) can be accessed using the double indexing syntax `M[i][j]`, where `M` is either `M1` or `M2`. However, the organization of these matrices are different in memory. While both store the elements in their primary array contiguously in memory, our statically allocated matrix also stores all the rows contiguously in memory: Figure 2. Matrix M1’s memory layout in row-major order.

This contiguous ordering is not guaranteed for M2. Recall that to contiguously allocate an n x m matrix on the heap, we should use a single call to `malloc()` that allocates n x m elements:

``````//dynamic matrix (allocated on heap, memory efficient way)
#define ROWS 4
#define COLS 3
int *M3;
M3  = malloc(ROWS*COLS*sizeof(int));``````

Recall that with the declaration of `M3`, element (i,j) cannot be accessed using the `M[i][j]` notation. Instead, we must index the element using the format: `M3[i*cols + j]`.

### 8.8.1. Contiguous Two Dimensional Arrays

Consider a function `sumMat()` that takes a pointer to a contiguously allocated (either statically allocated or a memory efficient dynamically allocated) matrix as its first parameter, along with a number of rows and columns and returns the sum of all the elements inside the matrix.

We use scaled indexing in the code snippet below, since it applies to both statically and dynamically allocated contiguous matrices. Recall that the syntax `m[i][j]` does not work with the memory efficient contiguous dynamic allocation previously discussed.

``````int sumMat(int *m, int rows, int cols) {
int i, j, total = 0;
for (i = 0; i < rows; i++){
for (j = 0; j < cols; j++){
total += m[i*cols + j];
}
}
}``````

Here is the corresponding assembly. Each line is annotated with its English translation:

```<sumMat>:
0x08048507 <+0>:   push  %ebp                   #save ebp
0x08048508 <+1>:   mov   %esp,%ebp              #update ebp (new stack frame)
0x0804850a <+3>:   sub   \$0x10,%esp             #add 4 more spaces to stack frame
0x0804850d <+6>:   movl  \$0x0,-0xc(%ebp)        #copy 0 to ebp-12 (total)
0x08048514 <+13>:  movl  \$0x0,-0x4(%ebp)        #copy 0 to ebp-4 (i)
0x0804851b <+20>:  jmp   0x8048555 <sumMat+78>  #goto <sumMat+78>
0x0804851d <+22>:  movl  \$0x0,-0x8(%ebp)        #copy 0 to ebp-8 (j)
0x08048524 <+29>:  jmp   0x8048549 <sumMat+66>  #goto <sumMat+66>
0x08048526 <+31>:  mov   -0x4(%ebp),%eax        #copy i to eax
0x08048529 <+34>:  imul  0x10(%ebp),%eax        #multiply i with cols, place in eax
0x0804852d <+38>:  mov   %eax,%edx              #copy i*cols to edx
0x0804852f <+40>:  mov   -0x8(%ebp),%eax        #copy j to %eax
0x08048534 <+45>:  lea   0x0(,%eax,4),%edx      #multiply (i*cols+j) by 4,put in edx
0x0804853b <+52>:  mov   0x8(%ebp),%eax         #copy m pointer to eax
0x08048540 <+57>:  mov   (%eax),%eax            #copy m[i*cols+j] to eax
0x08048545 <+62>:  addl  \$0x1,-0x8(%ebp)        #increment j by 1 (j+=1)
0x08048549 <+66>:  mov   -0x8(%ebp),%eax        #copy j to eax
0x0804854c <+69>:  cmp   0x10(%ebp),%eax        #compare j with cols
0x0804854f <+72>:  jl    0x8048526 <sumMat+31>  #if (j < cols) goto <sumMat+31>
0x08048555 <+78>:  mov   -0x4(%ebp),%eax        #copy i to eax
0x08048558 <+81>:  cmp   0xc(%ebp),%eax         #compare i with rows
0x0804855b <+84>:  jl    0x804851d <sumMat+22>  #if (i < rows) goto sumMat+22
0x0804855d <+86>:  mov   -0xc(%ebp),%eax        #copy total to eax
0x08048560 <+89>:  leave                        #prepare to leave the function

Local variables `i`, `j`, and `total` are loaded at addresses `%ebp-4`, `%ebp-8` and `%ebp-12` on the stack, respectively. Input parameters `m`, `row`, and `cols` are located at locations `%ebp+8`, `%ebp+12`, and `%ebp+16` respectively. Using this knowledge, let’s zoom in on the component that just deals with the access of element (i,j) in our matrix:

```0x08048526 <+31>:  mov   -0x4(%ebp),%eax        #copy i to eax
0x08048529 <+34>:  imul  0x10(%ebp),%eax        #multiply i with cols, place in eax
0x0804852d <+38>:  mov   %eax,%edx              #copy i*cols to edx```

The first set of instructions computes `i * cols` and places the result in register `%edx`. Recall that for a matrix `matrix`, the pointer notation `matrix + (i * cols)` is equivalent to the access `&matrix[i]`.

```0x0804852f <+40>:  mov   -0x8(%ebp),%eax       #copy j to eax
0x08048534 <+45>:  lea   0x0(,%eax,4),%edx     #multiply (i*cols+j) by 4, put in edx```

The next set of instructions computes `(i * cols + j) * 4`. The compiler multiplies the index `(i * cols) + j` by 4 since each element in the matrix is a 4 byte integer and this multiplication enables the compiler to calculate the correct offset.

The last set of instructions adds the calculated offset to the matrix pointer and dereferences it to yield the value of element (i,j):

```0x0804853b <+52>:  mov   0x8(%ebp),%eax         #copy m pointer to eax
0x08048540 <+57>:  mov   (%eax),%eax            #copy m[i*cols+j] to eax

The first instruction loads the address of matrix `m` into register `%eax`. The `add` instruction adds the offset `(i*cols + j)*4` to the address of `m` to correctly calculate the address of element (i,j), and places this address in register `%eax`. The third instruction dereferences `%eax` and places the resulting value in register `%eax`. The last instruction adds the value in `%eax` to the accumulator `total`, which is located at stack address `%ebp-0xc`.

Let’s consider how element (1,2) is accessed in Figure 2 (reproduced below for convenience): Figure 3. Matrix M1’s memory layout in row-major order.

Element (1,2) is located at address `M1 + (1 * cols) + 2`. Since `cols` = `3`, element (1,2) corresponds to `M1+5`. To access the element at this location, the compiler must multiply `5` by the size of the `int` data type (`4` bytes), yielding the offset `M1 + 20`, which corresponds to byte x20 in the figure. Dereferencing this location yields element 5, which is indeed element (1,2) in the matrix.

### 8.8.2. Non-contiguous Matrix

The non-contiguous matrix implementation is a bit more complicated. Figure 4 visualizes how `M2` may be laid out in memory: Figure 4. Matrix M2’s non-contiguous layout in memory.

Notice that the array of pointers is contiguous, and that each array pointed to by an element of `M2` (i.e. `M2[i]`) is contiguous. However, the individual arrays are not contiguous with each other.

The `sumMatrix()` function below takes an array of integer pointers (called `matrix`) as its first parameter, and a number of rows and columns as its second and third parameters:

``````int sumMatrix(int **matrix, int rows, int cols) {
int i, j, total=0;
for (i = 0; i < rows; i++) {
for (j = 0; j < cols; j++) {
total += matrix[i][j];
}
}
}``````

While this function looks nearly identical to the `sumMat()` function shown earlier, the matrix accepted by this function consists of a contiguous array of pointers. Each pointer contains the address of a separate contiguous array, which corresponds to a separate row in the matrix.

The corresponding assembly for `sumMatrix()` follows. Each line is annotated with its English translation.

```0x080484ad <+0>:   push  %ebp                      # save ebp
0x080484ae <+1>:   mov   %esp,%ebp                 # update ebp (new stack frame)
0x080484b0 <+3>:   sub   \$0x10,%esp                # add 4 more spaces to stack frame
0x080484b3 <+6>:   movl  \$0x0,-0xc(%ebp)           # copy 0 to %ebp-12 (total)
0x080484ba <+13>:  movl  \$0x0,-0x4(%ebp)           # copy 0 to %ebp-4 (i)
0x080484c1 <+20>:  jmp   0x80484fa <sumMatrix+77>  # goto <sumMatrix+77>
0x080484c3 <+22>:  movl  \$0x0,-0x8(%ebp)           # copy 0 to %ebp-8 (j)
0x080484ca <+29>:  jmp   0x80484ee <sumMatrix+65>  # goto <sumMatrix+65>
0x080484cc <+31>:  mov   -0x4(%ebp),%eax           # copy i to %eax
0x080484cf <+34>:  lea   0x0(,%eax,4),%edx         # multiply i by 4, place in %edx
0x080484d6 <+41>:  mov   0x8(%ebp),%eax            # copy matrix to %eax
0x080484d9 <+44>:  add   %edx,%eax                 # add i*4 to matrix, place in %eax
0x080484db <+46>:  mov   (%eax),%eax               # copy matrix[i] to %eax
0x080484dd <+48>:  mov   -0x8(%ebp),%edx           # copy j to %edx
0x080484e0 <+51>:  shl   \$0x2,%edx                 # multiply j by 4, place in %edx
0x080484e5 <+56>:  mov   (%eax),%eax               # copy matrix[i][j] to %eax
0x080484ee <+65>:  mov   -0x8(%ebp),%eax           # copy j to %eax
0x080484f1 <+68>:  cmp   0x10(%ebp),%eax           # compare j with cols
0x080484f4 <+71>:  jl    0x80484cc <sumMatrix+31>  # if (j < cols) goto <sumMatrix+31>
0x080484fa <+77>:  mov   -0x4(%ebp),%eax           # copy i to %eax
0x080484fd <+80>:  cmp   0xc(%ebp),%eax            # compare i with rows
0x08048500 <+83>:  jl    0x80484c3 <sumMatrix+22>  # if (i < rows) goto <sumMatrix+22>
0x08048502 <+85>:  mov   -0xc(%ebp),%eax           # copy total to %eax
0x08048505 <+88>:  leave                           # prepare to leave the function

Once again, variables `i`, `j`, and `total` are at stack addresses `%ebp-4`, `%ebp-8` and and `%ebp-12` respectively. Input parameters `m`, `row`, and `cols` are located at stack addresses `%ebp+8`, `%ebp+12`, and `%ebp+16` respectively.

Let’s zoom in on the section that deals specifically with an access to element (i,j), or `matrix[i][j]`:

```0x080484cc <+31>:  mov   -0x4(%ebp),%eax      # copy i to %eax
0x080484cf <+34>:  lea   0x0(,%eax,4),%edx    # multiply i by 4, place in %edx
0x080484d6 <+41>:  mov   0x8(%ebp),%eax       # copy matrix to %eax
0x080484d9 <+44>:  add   %edx,%eax            # add i*4 to matrix, place in %eax
0x080484db <+46>:  mov   (%eax),%eax          # copy matrix[i] to %eax```

The five instructions between `<sumMatrix+31>` and `<sumMatrix+46>` compute `matrix[i]`, or `*(matrix+i)`. Note that the compiler needs to multiply `i` by 4 prior to adding it to `matrix` to calculate the correct offset (recall that pointers are 4 bytes in size). The instruction at `<sumMatrix+46>` then dereferences the calculated address to get the element `matrix[i]`.

Since `matrix` is an array of `int` pointers, the element located at `matrix[i]` is itself an `int` pointer. The jth element in `matrix[i]` is located at offset `j*4` in the `matrix[i]` array.

The next set of instructions extract the jth element in array `matrix[i]`:

```0x080484dd <+48>:  mov   -0x8(%ebp),%edx      # copy j to %edx
0x080484e0 <+51>:  shl   \$0x2,%edx            # multiply j by 4, place in %edx
0x080484e3 <+54>:  add   %edx,%eax            # add j*4 to matrix[i], place in %eax
0x080484e5 <+56>:  mov   (%eax),%eax          # copy matrix[i][j] to %eax
The first instruction in this snippet loads variable `j` into register `%edx`. The compiler uses the left shift (`shl`) instruction to multiply `j` by 4 and stores the result in register `%edx`. The compiler then adds the resulting value to the address located in `matrix[i]` to get the address of `matrix[i][j]`.
Let’s revisit Figure 4 to consider an example access of `M2` (reproduced below for convenience). Note that `M2` starts at memory location x0. The compiler first computes the address of `M2` by multiplying 1 by 4 (`sizeof(int *)`) and adding it to the address of `M2` (x0), yielding the new address x4. A dereference on this address yields the address associated with `M2`, or x36. The compiler then multiplies index 2 by 4 (`sizeof(int)`), and adds the result (`8`) to x36, yielding a final address of x44. The address x44 is dereferenced, yielding the value `5`. Sure enough, the element in [DynamicMatrix6] that corresponds to `M2` has the value `5`.