Hi guys,
This is a voltage tracking regulator. The output tracks the input (5v) signal. The output however is twice that of the input, so with 5v input you get 10v output. Of course you need a decent input voltage supply to the regulator too, of at least 12.5v to get 10v out.
Compensation is not too hard to achieve. Change the NPN transistor to a PNP type with emitter to the ADJ pin collector to ground, reverse the connections to the input of the op amp.
Alternately, remove the transistor entirely, and replace with maybe a 10 ohm resistor from ADJ to the output of the op amp, again reversing the input connections to the op amp.
The whole idea with a tracking regulator is to use an op amp as the error amplifier so that it can detect very small deviations of the output and compensate by adjusting the signal going to the ADJ pin of the LM317. The LM317 has some minor problems which the op amp stage helps to correct. The op amp also allows for a variable control voltage to output voltage ratio by simply changing one resistor (one of the 10k units, for example to get a 1 to 4 ratio change the top 10k to 30k). Of course it also can 'track' another signal, and also helps with the temperature characteristic.
Whatever you do you should also do a quick simulation to make sure it works right