Computational Physics - Department of Physics

164 6 Linear Algebra

while in Fortran we would have

DOj=1, n

DOi=1, n

a(i,j)=b(i,j)+c(i,j)

ENDDO

ENDDO

Fig. 6.3 shows how a 3 × 3 matrixAis stored in both row-major and column-major ways.

a 11 a 12 a 13

a 21 a 22 a 23

a 31 a 32 a 33

⇐= =⇒

a 11

a 12

a 13

a 21

a 22

a 23

a 31

a 32

a 33

a 11

a 21

a 31

a 12

a 22

a 32

a 13

a 23

a 33

Fig. 6.3Row-major storage of a matrix to the left (C++ way) and column-major to the right (Fortran way).

Interchanging the order ofiand jcan lead to a considerable enhancement in process
time. In Fortran we write the above statements in a much simpler waya=b+c. However, the
addition still involves∼n^2 operations. Matrix multiplication or taking the inverse requires
∼n^3 operations. The matrix multiplication of Eq. (6.4) of two matricesA=BCcould then take
the following form in C++

for(i=0 ; i < n ; i++){

for(j=0 ; j < n ; j++){