Confusion is arising here. The impedances known as "iterative impedances" have been known as such for about 100 years. You are describing the result of your square root equation as an "iterative impedance value" and it is not the same as the classical "iterative impedance". When I am discussing these things, I will use the word "iterative" only in connection with the classical concepts. I will use the phrase "repeated impedance value" to denote the number(s) you have calculated, and which are the same as the classical "image impedances". You can see the possibilities for confusion here, when the impedances that you are calculating, the "repeated impedances", are the same as the classical "image impedances", but not the same as the classical "iterative impedances" (explained further below).
The image impedances (Zi1 and Zi2) are these: when Zi2 is connected as a load to a 2-port, the impedance looking into the input of the 2-port will be Zi1; when Zi1 is connected to the input of a 2-port, the impedance looking into the output of the 2-port will be Zi2. The image impedances are the impedances which simultaneously match the input and output of the 2-port.
This page shows it well:
https://www.transtutors.com/homework-help/Networks-Systems/two-port-network/image-impedance/
The classical "iterative impedance" (not the same as the "image impedance") is the result of cascading a lot of identical 2-ports and calculating the input and output impedances of the cascade; this is why they called it "iterative impedance". Imagine the last 2-port in the cascade; its input impedance becomes the load for the next to last 2-port. The input impedance of the next to last 2-port is affected by the fact that its load is the input impedance of the last 2-port. As you work your way back the cascade, eventually the input impedances of each 2-port approaches a particular value. That value is the impedance (it is the output iterative impedance), which, if it is the load of the 2-port, will cause the input impedance of the 2-port to be the same as that load. For the transistor circuit under discussion, its value is 10316.2 ohms.
And, of course, it works the same in the other direction. The input iterative impedance is that impedance, which, when connected as the driving impedance for the 2-port, will cause the output impedance to have the same value as that driving impedance (the input iterative impedance).
From this thread:
https://www.electro-tech-online.com/threads/audio-transformers-and-two-ports.124261/
"Given a two port, measure the input impedance with the output port open-circuited; call that impedance Zoc. Then measure the input impedance with the output port short-circuited; call that impedance Zsc. Then Zi1 = SQRT(Zsc*Zoc); this value is the geometric mean of Zsc and Zoc. The same procedure can be followed to determine the output image impedance, Zi2.
This same procedure is well known as a method to determine the characteristic impedance of a transmission line. A length of transmission line is essentially a two-port; it has an input and output port.
"
I would use the word "repeated" where the red "iteratively" is above. It is being used to mean something other than what the classical "iterative impedance" means.
I used the word "iterative" in "iterative impedance"; I didn't use "iterative
ly".