Addition and Subtraction

In programming routines to add or subtract matrices it pays to remember how the matrix is stored in the computer. Unfortunately this varies from language to language: in C(++) and Pascal the first index varies slowest and the matrix is stored one complete row after another; whereas in FORTRAN the first index varies fastest and the matrix is stored one complete column after another. It is generally most efficient to access the matrix elements in the order in which they are stored. Hence the simple matrix algebra, $\bss{A} = \bss{B} + \bss{C}$, should be written in C^3.1 as

const int N =??;
int i, j;
double A[N][N], B[N][N], C[N][N];
for( i = 0; i < N; i++)
   for( j = 0; j < N; j++)
      A[i][j] = B[i][j] + C[i][j];

or its equivalent using pointers. In FORTRAN90 this should be written

      integer :: i,j
      integer, parameter :: N = ??
      real, double precision :: A(N,N), B(N,N), C(N,N)
      do i = 1,N
         do j=1,N
            A(j,i) = B(j,i) + C(j,i)
         end do
      end do

Note the different ordering of the loops in these 2 examples. This is intended to optimise the order in which the matrix elements are accessed. It is perhaps worth noting at this stage that in both C++ and FORTRAN90 it is possible to define matrix type variables (classes) so that the programs could be reduced to

        matrix A(N,N), B(N,N), C(N,N);
        A = B + C;

The time taken to add or subtract 2 $M\times N$ matrices is generally proportional to the total number of matrix elements, $MN$, although this may be modified by parallel architecture.