一个数值计算中通常很有用的数值的定义和计算

matlab中直接用eps,默认是针对double双精度类型的.

在微软的msdn中是这么提的:

numeric_limits::epsilon

The function returns the difference between 1 and the smallest value greater than 1 that is representable for the data type.

The difference between 1 and the smallest value greater than 1 that is representable for the data type.

代码例子:

// numeric_limits_epsilon.cpp
// compile with: /EHsc
#include <iostream>
#include <limits>

using namespace std;

int main( )
{
   cout << "The difference between 1 and the smallest "
        << "value greater than 1\n for float objects is: " 
        << numeric_limits<float>::epsilon( ) 
        << endl;
   cout << "The difference between 1 and the smallest "
        << "value greater than 1\n for double objects is: " 
        << numeric_limits<double>::epsilon( ) 
        << endl;
   cout << "The difference between 1 and the smallest "
        << "value greater than 1\n for long double objects is: " 
        << numeric_limits<long double>::epsilon( ) 
        << endl;
}

输出结果是:

The difference between 1 and the smallest value greater than 1

for float objects is: 1.19209e-007

The difference between 1 and the smallest value greater than 1

for double objects is: 2.22045e-016

The difference between 1 and the smallest value greater than 1

for long double objects is: 2.22045e-016

在维基中有这样一个表格:

Values for standard hardware floating point arithmetics

http://eigen.tuxfamily.org/index.php?title=Main_Page

The following values of machine epsilon apply to standard floating point formats:

IEEE 754 - 2008	Common name	C++ data type	Base	Precision	Machine epsilon[a]	Machine epsilon
binary16	half precision	short	2	11 (one bit is implicit)	2−11 = 4.88e-04	2−10 = 9.77e-04
binary32	single precision	float	2	24 (one bit is implicit)	2−24 = 5.96e-08	2−23 = 1.19e-07
binary64	double precision	double	2	53 (one bit is implicit)	2−53 = 1.11e-16	2−52 = 2.22e-16
binary80	extended precision	_float80[1]	2	64	2−64 = 5.42e-20	2−63 = 1.08e-19
binary128	quad(ruple) precision	_float128[1]	2	113 (one bit is implicit)	2−113 = 9.63e-35	2−112 = 1.93e-34
decimal32	single precision decimal	_Decimal32[2]	10	7	5 × 10−7	10−6
decimal64	double precision decimal	_Decimal64[2]	10	16	5 × 10−16	10−15
decimal128	quad(ruple) precision decimal	_Decimal128[2]	10	34	5 × 10−34	10−33

[b]a according to Prof. Demmel,LAPACK,Scilabb
according to Prof. Higham; ISO C standard;C,

C++ and
Python language constants;
Mathematica,
MATLAB and
Octave; various textbooks - see below for the latter definition

如果使用了GMP/MPIR/MPFR之类的扩展软件工具来提高实际参与计算的精度, 用Pavel的C++ wrapper mpfrC++(很奇怪他的个人主页只涉及技术相关信息也被屏蔽了), 结合Eigen
C++ template library是我的最爱.
https://code.google.com/p/gmpy/downloads/detail?name=full-src-mpir-mpfr-mpc-gmpy2-2.0.2.zip&can=2&q=
在Linux, MacOS上这类库的源代码直接编译通常没有问题, 在windows下稍微有些麻烦. 这里有一个修改后的,可以在visual studio下面直接编译成动态和静态链接库的项目文件的包,mpir,mpfr,mpc,gmpy都包括了. 我把下载所需资源分数设置为10,实际上,正常评论之后根据规则这个资源分都会自动返还而且会额外加分的.
所以下载的时候别有心理压力. 实在接受不了,可以到code.google.com的上面的链接中去下载. 没有任何问题.