I am looking for an Android screencast video recorder that doesn't need root (e.g. unlike the Screencast Video Recorder app). If possible, free and HD. I use a Samsung Galaxy S3 if the application is device-dependent.
Generally speaking, what you want is not possible for security reasons -- that's why everything needs root.
On an Android 4.4+ device, you can use
adb. This is time-limited to a maximum of three minutes and will not record audio.
For devices with HDMI capability, you can use try hardware HDMI recorders.