Step 1: At first I stop the MySQL service from Linux root user.
Step 2: Then I start again MySQL service with --skip-grant-tables, so that at the time of login into the MySQL service system should not ask for authentication.
mysqld_safe --skip-grant-tables &
Step 3: In this step I first login into MySQL root user without password.
mysql -u root
Then I issue the following MySQL command to change the root password and quit to Linux shell..
update user set password=password('password') where user='root';
Step 4: The last stage is to restart MySQL server using default option.
The password for root user is changed now. I have done all these procedures on the OS Debuan 5.