In the algorithm described above we encountered expressions of the form A₁⊗A₂⊗⋯⊗A_n which we denote by To calculate the product (⊗_iA_i)x it is computationally advantageous to factor ⊗_iA_i into terms of the form I⊗A_i⊗I1. Then each term represents a set of copies of A_i. First, recall the following property of Kronecker products

This property can be used to factor ⊗_iA_i in the following way. Let the number of rows and columns of A_i be denoted by r_i and c_i respectively. Then

(5.2)

But we can also write

(5.3)

Note that in factorization Equation 5.2, copies of A₂ are applied to the data vector x first, followed by copies of A₁. On the other hand, in factorization Equation 5.3, copies of A₁ are applied to the data vector x first, followed by copies of A₂. These two factorizations can be distinguished by the sequence in which A₁ and A₂ are ordered.

Lets compare the computational complexity of factorizations Equation 5.2 and Equation 5.3. Notice that Equation 5.2 consists of r₂ copies of A₁ and c₁ copies of A₂, therefore Equation 5.2 has a computational cost of r₂Q₁+c₁Q₂ where Q_i is the computational cost of A_i. On the other hand, the computational cost of Equation 5.3 is c₂Q₁+r₁Q₂. That is, the factorizations Equation 5.2 and Equation 5.3 have in general different computational costs when A_i are not square. Further, observe that Equation 5.2 is the more efficient factorization exactly when

or equivalently, when

(5.5)

Consequently, in the more efficient factorization, the operation A_i applied to the data vector x first is the one for which the ratio is the more negative. If r₁>c₁ and r₂<c₂ then Equation 5.4 is always true (Q_i is always positive). Therefore, in the most computationally efficient factorization of A₁⊗A₂, matrices with fewer rows than columns are always applied to the data vector x before matrices with more rows than columns. If both matrices are square, then their ordering does not affect the computational efficiency, because in that case each ordering has the same computation cost.

We now consider the Kronecker product of more than two matrices. For the Kronecker product ⊗ⁿ_i=1A_i there are n! possible different ways in which to order the operations A_i. For example

(5.6)

Each factorization of ⊗_iA_i can be described by a permutation g(·) of {1,...,n} which gives the order in which A_i is applied to the data vector x. A_g(1) is the first operation applied to the data vector x, A_g(2) is the second, and so on. For example, the factorization Equation 5.6 is described by the permutation g(1)=3, g(2)=1, g(3)=2. For n=3, the computational cost of each factorization can be written as

In general

(5.8)

Therefore, the most efficient factorization of ⊗_iA_i is described by the permutation g(·) that minimizes C.

It turns out that for the Kronecker product of more than two matrices, the ordering of operations that describes the most efficient factorization of ⊗_iA_i also depends only on the ratios . To show that this is the case, suppose u(·) is the permutation that minimizes C, then u(·) has the property that

(5.9)

for k=1,⋯,n–1. To support this, note that since u(·) is the permutation that minimizes C, we have in particular

where v(·) is the permutation defined by the following:

(5.11)

Because only two terms in Equation 5.8 are different, we have from Equation 5.10

(5.12)

which, after canceling common terms from each side, gives

Since v(k)=u(k+1) and v(k+1)=u(k) this becomes

which is equivalent to Equation 5.9. Therefore, to find the best factorization of ⊗_iA_i it is necessary only to compute the ratios and to order them in an non-decreasing order. The operation A_i whose index appears first in this list is applied to the data vector x first, and so on

As above, if r_u(k+1)>c_u(k+1) and r_u(k)<c_u(k) then Equation 5.14 is always true. Therefore, in the most computationally efficient factorization of ⊗_iA_i, all matrices with fewer rows than columns are always applied to the data vector x before any matrices with more rows than columns. If some matrices are square, then their ordering does not affect the computational efficiency as long as they are applied after all matrices with fewer rows than columns and before all matrices with more rows than columns.

Once the permutation g(·) that minimizes C is determined by ordering the ratios , ⊗_iA_i can be written as

(5.15)

where

(5.16)

(5.17)

and where γ(·) is defined by

(5.18)

function [g,C] = cgc(Q,r,c,n)
% [g,C] = cgc(Q,r,c,n);
% Compute g and C
% g : permutation that minimizes C
% C : computational cost of Kronecker product of A(1),...,A(n)
% Q : computation cost of A(i)
% r : rows of A(i)
% c : columns of A(i)
% n : number of terms
f = find(Q==0);
Q(f) = eps * ones(size(Q(f)));
Q = Q(:);
r = r(:);
c = c(:);
[s,g] = sort((r-c)./Q);
C = 0;
for i = 1:n
   C = C + prod(r(g(1:i-1)))*Q(g(i))*prod(c(g(i+1:n)));
end
C = round(C);

function y = kpi(d,g,r,c,n,x)
% y = kpi(d,g,r,c,n,x);
% Kronecker Product : A(d(1)) kron ... kron A(d(n))
% g : permutation of 1,...,n 
% r : [r(1),...,r(n)]
% c : [c(1),..,c(n)]
% r(i) : rows of A(d(i))
% c(i) : columns of A(d(i))
% n : number of terms
for i = 1:n
   a = 1;
   for k = 1:(g(i)-1)
      if i > find(g==k) 
         a = a * r(k);
      else
         a = a * c(k);
      end
   end
   b = 1;
   for k = (g(i)+1):n
      if i > find(g==k)
         b = b * r(k);
      else
         b = b * c(k);
      end
   end
   % y = (I(a) kron A(d(g(i))) kron I(b)) * x;
   y = IAI(d(g(i)),a,b,x);
end

function y = IAI(A,r,c,m,n,x)
% y = (I(m) kron A kron I(n))x
% r : number of rows of A
% c : number of columns of A
v = 0:n:n*(r-1);
u = 0:n:n*(c-1);
for i = 0:m-1
   for j = 0:n-1
      y(v+i*r*n+j+1) = A * x(u+i*c*n+j+1);
   end
end