You have to make a spreadsheet for this or you'll drive yourself crazy going through the iterations. Are you just trying to get the moment capacity of a doubly reinforced beam? I will typically use PCACol (but you can use any concrete column software that gives you an interaction diagram) and just pick off the pure moment value.
I'll do you one better. Here is a spreadsheet that will find Icr and Ieff directly. I developed this for input into RAM Advanse for the Ie/Ig factor.
I developed it for T-beams, but if you have a rectangular beam, just use b=0. The top part of the spreadsheet is for a quick strength check. The bottom is what you're interested in. Only input the boxes shaded yellow.
Iterations can be very simple when using goal seek to solve for T=C. I would ignore the extra capacity that you get for beams reinforced near the compression face. Basically all that steel is going to do is raise your neutral axis but I would not include it in the design.
It has to be a fairly good distance above the NA to do anything for the moment capacity, not just above it. More importantly, however, is it's role in deflections, specifically long-term deflections. I've rarely had a situation where I needed compression steel for strength, but often need it for deflections.
It does two things. First it provides a higher Ieff. This is benficial for immediate deflections as well as the long term, since the long term is a multiple of the immediate deflections. The real benefit, however, is that the presence of compression steel directly impacts the magnitude of what that long-term multiplier is. If there is no compression steel, then the multiplier is 3 (for long term loads). If you have 1% compression steel, the multiplier is 2.33 (for long term loads) - this essentially knocks down the long term deflections by 22%. That's pretty significant in my book.
Additionally, I just want to point out that there is a difference (even if it's only a small one) between the NA at ultimate strength and the NA at service loads.
SEIT and apsix have given you code based answers. Compression reinforcement in practice makes a concrete beam act more like a steel beam. So it removes some of the creep deflection. Steel beams don't creep.
"Additionally, I just want to point out that there is a difference (even if it's only a small one) between the NA at ultimate strength and the NA at service loads"
I believe in ultimate strength it will shift more up?
You are correct. So bars that may not be compression bars for strength considerations could conceivably be compression bars for serviceability considerations. I'm not sure I would want to cut the gnat's behind that close on a regular basis, but I thought I would mention it because I was faced with this about 1.5 years ago.
