DO ¯I = A, B ... J = J+H ... 
DO ¯I = A, B ... J = J+K polynomial ... K = K+H linear ... J = J+1 2^{nd} update ... P = 2*P geometric ... 

DO ¯I = 0, N ... J = J+K ... K = K+H ... J = J+1 ... P = 2*P ... 

¯ J = 0 K = 1 DO ¯I = 0, N S_{1}: J = J+K S_{2}: A[J] = B[K] S_{3}: K = K+2 S_{4}: B[2*I] = ... ENDDO 
¯ J = 0 K = 1 DO ¯I = 0, N S_{1}: J = J+K S_{2}: A[I*(I+2)+1] = B[2*I+1] S_{3}: K = K+2 S_{4}: B[2*I] = ... ENDDO 
J = 0 K = 1 DO ¯I = 0, N J = J+K A[J] = B[K] K = K+2 B[2*I] = ... ENDDO 
DO ¯I = 0, N A[I*(I+2)+1] = B[2*I+1] B[2*I] = ... ENDDO 
DO ¯I = 0, N T = a a = b b = c+2*bT+2 c = c+d d = d+I ENDDO 
A = 1 B = 1 DO ¯I = 0, N T=A A=B B=T ... ENDDO 
Note: loop unrolling might help!


Loop variable I=A..B is replaced by {A,+,1}_{I}
DO ¯I = A, B L = 2*I+H P = I*II Q = I*(I1) R = 2**I 

DO ¯I = A, B J = J+K K = K+H J = J+1 ... 
DO ¯I = A, B J = J+K+1 K = K+H ... 

Replace all uses of V with the CR of V
DO ¯I = A, B K = {K_{0},+,H}_{I} J = J+{K_{0},+,H}_{I}+1 ... 
DO ¯I = A, B K = {K_{0},+,H}_{I} J = {J_{0}+1,+,K_{0},+,H}_{I} ... 
DO ¯I = A, B K = {K_{0},+,H}_{I} J = {J_{0}+1,+,K_{0},+,H}_{I} ... 
J_{0} = J K_{0} = K DO ¯I = 0, BA K = K_{0}+H*I J = J_{0}+(H*I*(I1))/2+I*K_{0}+1 ... 

J = M DO ¯I = M, N A[J] = B[K] J = I+1 K = J ENDDO 
J_{0} = M DO ¯I = 0, NM A[¯{J_{0}M,*,0}_{I}+{M, +, 1}_{I}] = B[{KM,*,0}_{I}+{M,+,1}_{I}] ENDDO 
DO ¯I = 0, NM A[I+M]=B[(0**I)*(KM)+I+M] ENDDO 
DO ¯I = 1, N X[J] = ... J = J+K K = K*P P = P*4 ENDDO 
A[0] = 0 DO ¯I = 1, N1 A[I]=A[I1]+(P**I)*(2**(I*(I1))) ENDDO DO ¯I = 0, N1 X[J+K*A[I]] = ... ENDDO 
CR of a closed form c(I) (0 £ I £ N) is

cr_{0} = f_{0} cr_{1} = f_{1} : cr_{k} = f_{k} DO ¯I = 0, N c(I) = cr_{0} cr_{0} = cr_{0}Ä_{1} cr_{1} cr_{1} = cr_{1}Ä_{2} cr_{2} : cr_{k1} = cr_{k1}Ä_{k} cr_{k} ENDDO 