Evaluating an expression with integers, +, and *, as well as -, / - revision 3
From your second revision:


  I don't consider 2 * -3 a valid equation. It should look like this:
  
  
    -2*3
  


What if you need to multiply two negative integers? :p Hold that thought for awhile...

Validation logic

Even if we suppose you have made a case for your existing validation logic, it should belong to computeWithAllOperations(), possibly throwing an IllegalArgumentException exception containing a message to let the caller know what went wrong. You may want to go one step further to give a brief documentation about this on the method's Javadoc.

Handling -

Your code transforms the - sign based on a preceding digit, which is a slightly odd twist in logic. Instead, I think you should be checking whether a - sign is preceded by any other mathematical operators ignoring whitespaces, which will yield the following change:

final String parts[] = input.replaceAll("\\s+", "")
                            .replaceAll("--", "+")
                            .replaceAll("((?&lt;![/*+])-)", "+$1");


Just like your original code, we strip out whitespaces, then transform a 'minus negative' operation into an addition, and finally replace only the - signs that do not follow other mathematical operators: /, * and +. These are the only cases when we need to treat it as a subtraction rather than to signify a negative value. More can be added here to support % and/or ^.

However, this change does introduce a slight quirk, which changes expressions starting with a negative integer to "+-1*5...". To mitigate that, we can do a simple short-circuiting in computeMultAndDivision():

private static final Pattern REGEX = Pattern.compile("(?=[/*%^])|(?&lt;=[/*%^])");

public static double computeMultAndDivision(final String part) {
    final String[] parts = REGEX.split(part);
    if (parts[0].isEmpty()) {
        return 0;
    }
    ...
}


Unit testing

Last but not least, I'll suggest doing proper unit testing for your utility method to assert its correctness. :)