Summon

Summons are among the most well known and powerful abilities in the Golden Sun series. They consist of the user calling upon a powerful spirit to assist the user's party. The use of any summon requires that the party have the necessary number of Djinn of the proper element on standby at the time the character attempts the summon. These Djinn are subsequently placed into recovery. In general, the power of the summon increases with the number of Djinn required to use it. A few enemies have attacks that are treated as summons.

Unlike Psynergy, summons always hit all enemies (with the exception of Coatlicue, which heals the party rather than attacking the enemy). They also temporarily enhance the summoner's elemental power in the elements corresponding to the Djinn necessary for the summon. Summons deal more damage against enemies with more HP, making them very effective against bosses.

Damage
The damage from a summon is calculated as follows:

$$\mathrm{damage = \left \lfloor \left( \frac{\left \lfloor 256 \cdot dimRatio \right \rfloor}{256}\right) \left(base + \frac{\left \lfloor 256 \cdot multiplier \cdot hp \right \rfloor}{256}\right)\left(1 + \frac{\left \lfloor 256 \cdot \left( \frac{power - resistance}{200} \right)\right \rfloor}{256}\right) \right \rfloor + randDamage}$$


 * damage: final damage.
 * $$\lfloor x \rfloor$$ is the greatest integer less than or equal to $$x$$.
 * dimRatio: This is always 1, .7, .4, .3, .2, or .1 for summons. Its value depends on the location of the enemy of interest relative to the main target. For example, the main target will take 100% of the damage, the enemies next to him will take 70%, and so on.
 * base: base damage of the attack. The value depends on the summon; see the list below.
 * hp: HP of the target.
 * multiplier: the HP multiplier of the summon. This factor is what makes summons so effective against bosses; the more HP the target has, the more damage a summon will deal. For the party's summons, the HP multiplier is usually equal to $$(3 \cdot \text{number of Djinn used})/100$$. The only exceptions are the multipliers for Iris and Daedalus.
 * power: the user's elemental power in the relevant element.
 * resistance: the target's resistance to the relevant element.
 * randDamage: a randomly generated integer between 0 and 3, inclusive.

In Golden Sun
There are a total of sixteen summons in the first Golden Sun. All of them use one to four Djinn of a single element and are available as soon as the party has enough Djinn to use them. These summons have no secondary effects.

In Golden Sun: The Lost Age
Golden Sun: The Lost Age has all the summons from Golden Sun plus thirteen new ones. The new summons all use Djinn of two different elements and are acquired by picking up special stone tablets. Each of these summons has secondary effects.

In Golden Sun: Dark Dawn
After viewing trailers for Golden Sun: Dark Dawn many summons from the first two games appear to be returning for the new installation in the Golden Sun series. However many seemingly similar summon sequences have had their animations changed significantly enough for viewers to question the identity of the summon. One summon that was instantly recognizable was the Neptune summon. Currently few facts are definitively confirmed except for summons and djinni returning as a battle strategy.

From the 2010 E3 Playable demo, it does appear that Ramses, Kirin and Atlanta will be returning, but it is not yet known if the player is able to summon individual djinni as they could in previous games.1