Dual boot with Windows 10 broken after update
Solution 1:
I see this question was asked a couple of years ago. Since it comes up fairly high in the search results, I'll put in a late answer anyway. I have a dual-boot Windows/Ubuntu machine that I've been using since Win10 came out. Every major Windows update kills the dual-boot.
When this happens, open a Windows "command prompt" as Administrator. Then enter the following command:
bcdedit /set "{bootmgr}" path \EFI\ubuntu\shimx64.efi
For me, this has reliably fixed the problem every time. Obviously, for other distros, the path may be different. This article provides some good background information.
Solution 2:
My solution has been different. And, please, do not reinstall Windows or do some other drastic stuff of the same kind! If you have a dual boot, you are likely to have at least four partitions: NTFS System Reserved, Main Windows NTFS, Linux Swap, Linux Ext4 (I am using SysLinux, but if you prefer Grub and alike you may also have a Linux Boot Partition). Normally the Linux Ext4 (or Linux Boot) partition would have a boot flag set. All you need to do is set boot flag to the Windows System Reserved partition, install updates and then change the boot flag back (after windows has restarted and installed its updates fully). I always have a live CD/USB handy (Knoppix or antiX), so I can boot from it, open gparted and change partitions boot flag around at will...