Final Fantasy XIII/Components

Components are used to upgrade your weapons and accessories by adding experience to them. You can get to the upgrade menu from save stations when you obtain the Omni-kit Key in ../Chapter 4/. When you've selected the weapon or accessory to upgrade, you'll go to the component list where you can choose which component to use and in what quantity, if you have more than one. When you select a component, you'll see how much experience it provides. Each component has a set amount of experience and a hidden multiplier that affects how big your experience gain will be in conjunction with the hidden rank of the item you're upgrading.

In general, organic components give greater experience multipliers and less experience, while the opposite is true of mechanical components. If you just want to upgrade by feel and not worry about min/maxing for efficiency, try to put as many organic components as you can on the item until you reach an experience multiplier of 3, then as many of your highest experience-paying mechanical component as you need, in one go. The reason you want to upgrade with a single mechanical component, is they actually reduce your experience multiplier, making each individual upgrade less and less effective. This means you will get more experience for upgrading with five of a single component at once than upgrading five times with one of the component.

The other function of components is to make you money. Plenty of components will be worthless as far as upgrading goes, but some of them can be sold for a lot of Gil. Most everything you need can be found on enemies or in item spheres, but occasionally you'll want to make a purchase, and farming the correct component will save you a lot of time.

Upgrading
Each weapon or accessory has a certain amount of levels until it maxes out. Once it has reached its max level, denoted by a ★ replacing its level number, you can upgrade it to its next incarnation using a catalyst. At this point the item gets a new name and is generally more powerful and has more levels so you can continue upgrading it. Weapons can be upgraded twice and accessories can be upgraded once or numerous times depending on the item. All level two and Omega level weapons are only available by upgrading level one weapons, and the same goes for the highest levels of accessories.

When you start the game, you'll likely not come across the weapon you want for your long-term play right away, so don't bother upgrading anything until the later chapters where you'll have access to most weapons and have the freedom to get to good farming locations. It's more worth your while in the beginning to upgrade a few accessories instead, as they upgrade quicker and you can dismantle them when you've found something better. Don't dismantle or sell any of your weapons though, as you won't really know which one you want to use until very far into the game.

Advanced upgrading
Once you understand the science behind the component upgrade system, you'll want to find components based on availability and cost. The factors involved in upgrading are the item's rank and level, the component's rank and experience multiplier and if transforming the item, its catalyst. The rank of the item you're upgrading will determine how much experience you get from the component, with lower ranked items getting more experience, and higher ranked items getting less. This means that better items get progressively harder to upgrade.

The first thing you want to do is max out the experience multiplier, so that when you use a high experience component you get three times the experience. Building the experience multiplier is done in the background by keeping a running total of what you've used to upgrade with. Components with positive and negative multiplier values are added together until certain thresholds are met, at which point your multiplier will go up or down. To calculate how many components you'll need, simply divide 500 by the multiplier value of the component. Obviously, the best components to use to get your multiplier up will be ones with a high multiplier values that are easy and cheap to come by.
 * Boosting the experience multiplier

Once you have your item at ×3 experience, you'll want to get as many of a high-experience component as you can and upgrade with all of them, all at once. When you select a component, you'll be able to set how many to use if you have more than one. Because the inventory cap is at 99, you'll want to use a component with high enough experience to go straight to the max level in one shot. Keep in mind that if you don't have enough high-experience components, you'll want to boost your multiplier again before continuing to upgrade, as your multiplier will become tanked with mechanical components' negative multiplier values.
 * Maxing the experience

The experience values in the tables below are based on an item with an average rank of 6. You'll find that a component may award more or less experience if the item has a lower or higher rank. To find out exactly how much experience it will take, you can upgrade the item once and see how much the required experience grows by, which stays constant between levels. For instance, if it requires 1,000 experience to upgrade the item to level 2, and then it requires 1,500 the next time, you know the requirement will increase by 500 experience points for every level. You can then use the following formula to determine the exact amount of experience needed:
 * $$Exp=\frac{n}{2}[2a + (n-1)b] \mbox{ where }

\begin{cases} n=\mbox{upgrades needed} \\ a=\mbox{exp. for first upgrade} \\ b=\mbox{increment} \end{cases}$$

Taking the example above, if the item's max level is 30, then the upgrades needed would be 28 (starting by upgrading to level 3, since you upgraded to level 2 to find the increment). The experience needed for the first upgrade (to level 3) is 1,500 and the increment of each level is 500. That gives:
 * $$Exp=\frac{28}{2}[2(1,500) + (27)500]=14(3,000 + 13,500)=231,000$$

So you need 231,000 experience to go from level 2 to level 30. Once you have the experience you need, divide it by three (for the multiplier you'll be enjoying) and then divide again by the experience granted by the component. This will tell you how many of the component you need to use.

Alternatively, you can just save first, then guesstimate and reload if you're off. For this method, use bracketing to close in on the actual amount. Bracketing is simply choosing a number and if you're high or low, try another number that is likely to be on the other side of the number so you can keep breaking the bracket down in half. For instance, in the above example, you might start with the amount of components that give you 200,000 experience, and find yourself short. Reload and try 250,000, showing that the number you need is between it. The next time you reload, use 225,000 and continue going halfway up or down depending on where you are until you get the appropriate amount.