Here is a possible solution for the accelerometer problem in the iPhone 5S

  Last night I told you that Apple has chosen to replace STMicroelectronics with Bosch in terms of supplying the component that includes the accelerometer currently iPhone 5S, and this generates problems in displaying information in applications. Although a developer is working on software that will allow other developers to provide a simple method of recalibrating the accelerometer while using apps, those from macworld, probably with Apple's help, offers a somewhat simpler method that is already available in iOS 7.

  They talk about the application Compass which at the first opening already offers us a quick method to calibrate the microelectromechanical sensors of our terminals, the same recalibration can apparently also be used to solve the problem of an accelerometer that displays erroneous information. Basically, all you have to do is activate the GPS and close the application from the background Compass a iOS 7, and upon reopening it will ask you to recalibrate the sensors.

  To be sure that you have the most correct calibration, try to move away from other mobile terminals, magnets, etc., and do the calibration. Of course, this process does not guarantee that you will have 100% correct data and if the accelerometer problem is really of a hardware nature, you may not get a favorable result, but it is recommended to try.