Converting AC to DC at mains frequency, and regulating with a linear regulator, at 20A output current, are both very inefficient and require some very large and heavy duty components.
Your bridge rectifier needs to be rated for an average current of at least 25A. I would use a 35A one such as
http://www.digikey.com/product-detail/en/GBJ3504-BP/GBJ3504-BPMS-ND/3191596 which is cheap but will need a heatsink.
Your mains transformer will have a peak output voltage of around 70V. Subtract 4V to allow a 2V drop across each diode in the bridge and your peak voltage into the smoothing capacitors would be about 66V. Allow 4V dropout for the regulator, from a 52V output voltage, and the trough voltage at the smoothing capacitors must be at least 56V. That leaves (66 - 56) = 10V maximum ripple depth.
At 50 Hz mains frequency the ripple drop period will be about 9 ms. So the smoothing capacitance has to drop 10V maximum in a 9 ms period at 20A load current. You can calculate the smoothing capacitance:
dV / dT = I / C
C = I dT / dV
= 20 × 0.009 / 10
= 18,000 µF
So your smoothing capacitor would need to be at least 18,000 µF (I would use 27,000 µF or 33,000 µF) and rated for 100V. Something like
http://www.digikey.com/product-detail/en/LNR2A333MSE/493-7699-ND/3930079 which will set you back USD 57. But that capacitor is rated for an RMS ripple current of 10A and the actual ripple current will be about 53A!
Now the mean voltage across a series regulator will be about (61 - 50) = 11V. At 20A that's a power dissipation of 220W. It would be best to split this across several transistors or MOSFETs; I would use a bank of about four 2N3055s since they're cheap. But with 220W dissipation you'll need something pretty big, and/or forced air cooled (i.e. with a fan). Something at least as big as
http://uk.mouser.com/ProductDetail/Wakefield/395-2AB/?qs=sGAEpiMZZMttgyDkZ5WiulP/BNP%2bTZRYdZPVZTJEIlM= which would set you back about USD 43 plus the cost of the fan(s).
I hope you see that this is a lot more impractical than using switching power supplies!