~~~~~↵
Let's say we have 2 recurrence relations -↵
summation(a[i]*f[i] for i=1 to n1)+summation(b[i]*g[i] for i =1 to n2) = 0 --------> Equation 1↵
summation(c[i]*f[i] for i=1 to n3)+summation(d[i]*g[i] for i =1 to n4) = 0 --------> Equation 2↵
.↵
.↵
.↵
.↵
.↵
`-----------------------------------------------------------------------------------> Equation m↵
WHERE a[i], b[i], c[i] AND d[i] are constants.↵
↵
##################################################################################################↵
In what cases is this above system of recurrence relations separable?↵
I want to express: f[i] in terms of only f terms and g[i] in terms of only g terms.↵
Is there a method to do it? If not, what are some of the specific cases under which this is doable?↵
↵
I know of 1 such method when:↵
f[i]=a*f[i-1]+b*g[i-2] --------------------------------------------------Equation A↵
g[i]=c*f[i-5]+d*g[i-4] --------------------------------------------------Equation B↵
-> g[i-2]=(f[i]-a*f[i-1])/b -------------------------------------------- by rearranging A↵
-> f[i-5]=(g[i]-d*g[i-4])/c -------------------------------------------- by rearranging B↵
But now once you have say g[i-2]=(f[i]-a*f[i-1])/b, then by replacing i with i+2, we get,↵
-> g[i+2-2]=(f[i+2]-a*f[i+2-1])/b↵
-> g[i]=(f[i+2]-a*f[i+1])/b↵
Similarly g[i+5]=(f[i+7]-a*f[i+6])/b↵
now substitute these in Equation B.↵
Solve similarly for f.↵
Thus, separated.↵
##################################################################################################↵
Any general method?↵
Thanks in advance.↵
~~~~~↵
↵
Let's say we have 2 recurrence relations -↵
summation(a[i]*f[i] for i=1 to n1)+summation(b[i]*g[i] for i =1 to n2) = 0 --------> Equation 1↵
summation(c[i]*f[i] for i=1 to n3)+summation(d[i]*g[i] for i =1 to n4) = 0 --------> Equation 2↵
.↵
.↵
.↵
.↵
.↵
`-----------------------------------------------------------------------------------> Equation m↵
WHERE a[i], b[i], c[i] AND d[i] are constants.↵
↵
##################################################################################################↵
In what cases is this above system of recurrence relations separable?↵
I want to express: f[i] in terms of only f terms and g[i] in terms of only g terms.↵
Is there a method to do it? If not, what are some of the specific cases under which this is doable?↵
↵
I know of 1 such method when:↵
f[i]=a*f[i-1]+b*g[i-2] --------------------------------------------------Equation A↵
g[i]=c*f[i-5]+d*g[i-4] --------------------------------------------------Equation B↵
-> g[i-2]=(f[i]-a*f[i-1])/b -------------------------------------------- by rearranging A↵
-> f[i-5]=(g[i]-d*g[i-4])/c -------------------------------------------- by rearranging B↵
But now once you have say g[i-2]=(f[i]-a*f[i-1])/b, then by replacing i with i+2, we get,↵
-> g[i+2-2]=(f[i+2]-a*f[i+2-1])/b↵
-> g[i]=(f[i+2]-a*f[i+1])/b↵
Similarly g[i+5]=(f[i+7]-a*f[i+6])/b↵
now substitute these in Equation B.↵
Solve similarly for f.↵
Thus, separated.↵
##################################################################################################↵
Any general method?↵
Thanks in advance.↵
~~~~~↵
↵
