First: The supply voltage is 9 V (+/- 10%). This is all I have, so this part cannot be changed. I can tweak it down to 8.1V, so that is my starting point.9v +/- 10% means your input could already be 8.1-9.9v. If you tweak it down to 8.1v, you're probably still going to have some variation, depending on power supply temperature, current load etc.
Are you sure it's sane to use 9v with 20A of current? For 20A, you need good quality power cable between the power supply and the input of the regulators, and proper connectors ... if you plan on powering your product through a regular round DC jack as seen on laptops and other small devices, 20A may be a problem.
Regarding power cable, with 20A there's going to be some voltage loss in your cables.. for example, with two AWG18 wires (common on ATX power supplies, to make it easier for you to compare sizes), with 1 meter of cable and 9v @20 A at the end of the cable you'll have 8.16v (there's about 0.8v drop in the cable alone), see
here.
That's why a 6pin pci-express cable meant for 75w (12v x 6.25A) is designed with 3 positive and 3 negative cables, for redundancy and minimal voltage drop.
It would make more sense to reduce the input current by using higher voltage, for example by using a 16-18v laptop adapter, then using your own dc-dc converter to get 5v at 20A. Or if you really want smooth dc voltage, to use a dc-dc converter to get about 6.5v @ 20-25A and then use linear regulators to cut the switching noise.
Diodes in the TO- packages look tempting, but all have voltage drop that varies with temperature and current. If you're going to put them on the same heatsink with the linear regulators, they're going to get hot and (most of the time) their forward voltage drop will lower a bit, which would only make the linear regulator dissipate more power as heat, so you're not gaining anything. Pretty much the same amount of heat is dissipated.
By using two linear regulators, you're dissipating the same amount of heat as a diode+regulator combo, but the regulators themselves have circuitry inside which insures the amount of voltage drop on each regulator remains the same regardless of temperature and load... solving exactly the problem you want to solve.
One more note... pay attention to datasheets of linear regulators, in particular to power dissipation. Some linear regulators can only dissipate about 15w of power, some can go up to 25-30w ... these matter when choosing the heatsink and when calculating the maximum input voltage acceptable.
If your 1084 in only guaranteed at up to 15w that means that at 5A current output, your input voltage - output voltage can be maximum 3V.