Electronic – What capacitor to get for relay coil

One approach would be to control N relays using N+1 half-H-bridges. One of them connects to one wire which is common to all the relays, and the others connect to one relay each. A variation on this would be to arrange the relays into a matrix, with a half-H driver for each row or column (the N+1-half-bridge solution is simply a 1xN matrix). One caveat with this approach is that other relays--especially those in the same row or column as the driven relays may receive a substantial fraction of the drive voltage. In an NxN matrix, the voltage fraction will be (N-1)/(2N-1), which would be 1/3 for a 2x2 matrix, 2/5 for a 3x3 matrix, and would approach 50% as a matrix got bigger. For a non-square matrix, things would be worse.

If you can be sure that relays will not fire unless they receive more than half the rated drive voltage, a square matrix approach might work. Even the relays which don't fire would gobble significant current (as the matrix gets bigger, the voltage across the relays in the same row and column as the driven row would approach half the drive voltage, so figure current similarly); if one used an 8x8 matrix, the total current gobbled by the inadvertently-driven relays would be about 7x the current going to the deliberately-driven one. Not great, but being able to drive 64 relays with 16 half-bridge drivers might be worth it.

The two capacitors used in the LM317 typical application are described on page five of this datasheet. An identical schematic to the one you provided is given:

Along with a helpful note about why the capacitors are needed:

Note: 3. \$C_i\$ is required when the regulator is located an appreciable distance from power supply filter. \$C_O\$ is not needed for stability; however, it does improve transient response. Since \$I_{ADJ}\$ is controlled to less than 100 μA, the error associated with this term is negligible in most applications.

The best practice, for either capacitor, is to always include them. Specifically for \$C_i\$, place it as close to the input pin on the LM317 as reasonable.

If you have determined that you know what you're doing, you can decide to omit or alter the suggested schematic. In determining that, for \$C_i\$, an appreciable distance in on the order of centimeters, but you should also factor in the level of noise you expect on the input supply and how much the load the LM317 will be changing. For \$C_O\$, you can likely omit it if you have decoupling capacitors at the inputs of any ICs and relatively high current draw devices, anywhere things are going to be changing how much current they're drawing in a short amount of time.

Do check out the related questions and answers regarding how to use decoupling capacitors and how to select the type of capacitors to use.