while updating my apple mac it shut down and will not finish installing new update.

Boot the computer. Immediately after the chime press and hold down the COMMAND and R keys. When the Utility Menu appears select Disk Utility, and press the Continue button. Select your Macintosh HD volume on the left, then click on the First Aid tab in the tool bar. If all is well quit Disk Utility. Select the option to Install Mac OS X, then press the Continue button.

You are installing OS X on your Macintosh HD volume. You will be performing a reinstallation over an existing one. Your files are not involved in the process.

Reinstall Sierra or El Capitan Without Erasing Drive

Please be sure you back up.

1. Restart the computer. Immediately after the chime hold down the Command and R keys until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the side list.
4. Click on the First Aid icon in the toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Reinstall OS X and click on the Continue button.

Also, see this User Tip: Basic Steps for Upgrading OS X.

During the Startup process, your Mac loads a large "Blob" of software off the boot drive from a pre-defined place on the drive. When initialized, this software draws the solid Apple in the center of the screen. If you get the solid Apple, the drive Hardware and the process of getting stuff into memory is working.

The next part of the boot-up process requires many Files. So the File System is initialized and the boot software attempts to Mount the drive to read those Files (not just stuff from a known location). The drive Directory is key to this process, since it contains the names and locations of all those files.

If the Directory is damaged, you Mac stops normal loading to run the equivalent of Disk Utility "Repair" or "First Aid". This usually takes extra time -- sometimes as much as five minutes more time.

--If it succeeds in fixing the Directory in one pass, and the drive can be Mounted, booting continues.

--If it fails to completely repair the Directory, and the drive can NOT be mounted, Boot-up is stuck, and can do nothing more. Your Mac will power off at this point.

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To make progress using this drive as the Boot drive, the drive Directory MUST be fixed. You can neither continue boot-up nor Install a new MacOS unless and until the Directory is good.

You can try multiple additional passes of Repair (run from Recovery), in hopes that this will fix the Directory.

If the drive will not come clean, you may have to use a different drive. An External drive can be used to Install an alternate copy of MacOS, which can then be used in an attempt to salvage YOUR files off the damaged drive. Over 350,000 files on the damaged drive are MacOS, and you can re-create them exactly as they were with a re-Install on an external drive.

If you have no other alternatives, you can say goodby to ALL the files on the drive and erase it and re-Install MacOS. This WILL Delete all your files.