The suggestion on adding a low value (100-1000 pF) capacitor in parallel with the 20-1000 uF bypass capacitors if the best answer. It has to do with the frequency response of the capacitors and their ability to nibble away at the high frequency component.
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With linear regulators you need to have a few volts higher on the input when compared to the output so the regulator has something to regulate and drop voltage. Not too much because that voltage drop during regulation is converted to heat. For example; If you had a 18 volt input and a 12 volt regulated output at 2 amps that 6 volts of drop, multiplied by 2 amps would be 12 watts of (thermal) that would need a substantial heat sink to dissipate.
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The other way is bad too, a 12 volt regulator with a 12.5 volt supply. The 78xx series regulates best when the input voltage is > ~2.5V up to ~10V. Below that differential and the regulator cannot function correctly and you would be better off with just a diode string.
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Tisha Hayes
AA4HA
Tisha Hayes, AA4HA