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under the Creative Commons Attribution License (BY)

Due to rounding errors, most floating-point numbers end up being slightly imprecise. As long as this imprecision stays small, it can usually be ignored. However, it also means that numbers expected to be equal (e.g. when calculating the same result through different correct methods) often differ slightly, and a simple equality test fails. For example:

` float a = 0.15 + 0.15 float b = 0.1 + 0.2 if(a == b) // can be false! if(a >= b) // can also be false! `

The solution is to check not whether the numbers are exactly the same, but whether their difference is very small. The error margin that the difference is compared to is often called *epsilon*. The most simple form:

` if( Math.abs(a-b) < 0.00001) // wrong - don't do this `

This is a bad way to do it because a fixed epsilon chosen because it “looks small” could actually be way too large when the numbers being compared are very small as well. The comparison would return “true” for numbers that are quite different. And when the numbers are very large, the epsilon could end up being smaller than the smallest rounding error, so that the comparison always returns “false”. Therefore, it is necessary to see whether the *relative error* is smaller than epsilon:

` if( Math.abs((a-b)/b) < 0.00001 ) // still not right! `

There are some important special cases where this will fail:

- When both
`a`

and`b`

are zero.`0.0/0.0`

is “not a number”, which causes an exception on some platforms, or returns false for all comparisons. - When only
`b`

is zero, the division yields “infinity”, which may also cause an exception, or is greater than epsilon even when`a`

is smaller. - It returns
`false`

when both`a`

and`b`

are very small but on opposite sides of zero, even when they’re the smallest possible non-zero numbers.

Also, the result is not commutative (`nearlyEquals(a,b)`

is not always the same as `nearlyEquals(b,a)`

). To fix these problems, the code has to get a lot more complex, so we really need to put it into a function of its own:

` public static boolean nearlyEqual(float a, float b, float epsilon) { final float absA = Math.abs(a); final float absB = Math.abs(b); final float diff = Math.abs(a - b); if (a == b) { // shortcut, handles infinities return true; } else if (a == 0 || b == 0 || diff < Float.MIN_NORMAL) { // a or b is zero or both are extremely close to it // relative error is less meaningful here return diff < (epsilon * Float.MIN_NORMAL); } else { // use relative error return diff / (absA + absB) < epsilon; } } `

This method passes tests for many important special cases, but as you can see, it uses some quite non-obvious logic. In particular, it has to use a completely different definition of error margin when `a`

or `b`

is zero, because the classical definition of relative error becomes meaningless in those cases.

There are some cases where the method above still produces unexpected results (in particular, it’s much stricter when one value is nearly zero than when it is exactly zero), and some of the tests it was developed to pass probably specify behaviour that is not appropriate for some applications. Before using it, make sure it’s appropriate for your application!

There is an alternative to heaping conceptual complexity onto such an apparently simple task: instead of comparing `a`

and `b`

as real numbers, we can think about them as discrete steps and define the error margin as the maximum number of possible floating-point values between the two values.

This is conceptually very clear and easy and has the advantage of implicitly scaling the relative error margin with the magnitude of the values. Technically, it’s a bit more complex, but not as much as you might think, because IEEE 754 floats are designed to maintain their order when their bit patterns are interpreted as integers.

However, this method does require the programming language to support conversion between floating-point values and integer bit patterns. Read the Comparing floating-point numbers paper for more details.

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$sudo sh -c 'echo "greeter-show-manual-login=true" >> /etc/lightdm/lightdm.conf'

restart computer and choose to login as the desired user.

reference link: Liberian Geek Login as Root in Ubuntu 12.04 (Precise Pangolin)

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enable the root account by running the commands below:

$ sudo passwd -u root

When prompted for password, enter your personal password to continue.

2) Then

reset/add the root password by running the commands below:

$sudo passwd root

When you run the commands above, you'll get prompted to enter a new password.

reference link: Liberian Geek [Question] What is the Root Default Password in Ubuntu 12.04?

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Precedence | Operator | Description | Associativity |
---|---|---|---|

1
highest |
`::` |
Scope resolution (C++ only) | None |

2 | `++` |
Suffix increment | Left-to-right |

`--` |
Suffix decrement | ||

`()` |
Function call | ||

`[]` |
Array subscripting | ||

`.` |
Element selection by reference | ||

`->` |
Element selection through pointer | ||

`typeid()` |
Run-time type information (C++ only) (see typeid) | ||

`const_cast` |
Type cast (C++ only) (see const_cast) | ||

`dynamic_cast` |
Type cast (C++ only) (see dynamic_cast) | ||

`reinterpret_cast` |
Type cast (C++ only) (see reinterpret_cast) | ||

`static_cast` |
Type cast (C++ only) (see static_cast) | ||

3 | `++` |
Prefix increment | Right-to-left |

`--` |
Prefix decrement | ||

`+` |
Unary plus | ||

`-` |
Unary minus | ||

`!` |
Logical NOT | ||

`~` |
Bitwise NOT (One's Complement) | ||

`(` |
Type cast | ||

`*` |
Indirection (dereference) | ||

`&` |
Address-of | ||

`sizeof` |
Size-of | ||

`new` , `new[]` |
Dynamic memory allocation (C++ only) | ||

`delete` , `delete[]` |
Dynamic memory deallocation (C++ only) | ||

4 | `.*` |
Pointer to member (C++ only) | Left-to-right |

`->*` |
Pointer to member (C++ only) | ||

5 | `*` |
Multiplication | Left-to-right |

`/` |
Division | ||

`%` |
Modulo (remainder) | ||

6 | `+` |
Addition | Left-to-right |

`-` |
Subtraction | ||

7 | `<<` |
Bitwise left shift | Left-to-right |

`>>` |
Bitwise right shift | ||

8 | `<` |
Less than | Left-to-right |

`<=` |
Less than or equal to | ||

`>` |
Greater than | ||

`>=` |
Greater than or equal to | ||

9 | `==` |
Equal to | Left-to-right |

`!=` |
Not equal to | ||

10 | `&` |
Bitwise AND | Left-to-right |

11 | `^` |
Bitwise XOR (exclusive or) | Left-to-right |

12 | `|` |
Bitwise OR (inclusive or) | Left-to-right |

13 | `&&` |
Logical AND | Left-to-right |

14 | `||` |
Logical OR | Left-to-right |

15 | `?:` |
Ternary conditional (see ?:) | Right-to-left |

16 | `=` |
Direct assignment | Right-to-left |

`+=` |
Assignment by sum | ||

`-=` |
Assignment by difference | ||

`*=` |
Assignment by product | ||

`/=` |
Assignment by quotient | ||

`%=` |
Assignment by remainder | ||

`<<=` |
Assignment by bitwise left shift | ||

`>>=` |
Assignment by bitwise right shift | ||

`&=` |
Assignment by bitwise AND | ||

`^=` |
Assignment by bitwise XOR | ||

`|=` |
Assignment by bitwise OR | ||

17 | `throw` |
Throw operator (exceptions throwing, C++ only) | Right-to-left |

18
lowest |
`,` |
Comma | Left-to-right |

reference url: http://en.wikipedia.org/wiki/Operators_in_C_and_C%2B%2B

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ASCII 码表，及扩展表内容如下：

Extended ASCII table

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