Here is a short description of issues that can arrive from a weak battery.
I found this in a service manual:
Vehicles equipped with engine or transmission/transaxle computers may require a computer relearn procedure after the vehicle battery is disconnected. Vehicle computers memorize and store vehicle operation patterns for optimum driveability and performance. When the vehicle battery is disconnected, this memory is lost, resulting in a driveability problem. Depending on the vehicle and how it is equipped, the following driveability problems may exist:
- Rough or unstable idle.
- Hesitation or stumble.
- Rich or lean running.
- Poor fuel mileage.
- Harsh or poor transmission/transaxle shift quality.
Default data is used until NEW data from each key start is stored. As the computer restores its memory from each new key start, driveability is restored.
Driveability problems may occur during the computer relearn stage. To accelerate computer relearn process after battery removal and installation, specified computer relearn procedures should be performed.