Definition, understanding and calculation of significant figures in numerical analysis

One 、 Definition of significant numbers

To define a significant number , We need to give the following definitions first , In fact, they are all very simple and understandable things , But it's a bit of a detour to express it with a mathematical formula .

error ：

if $x^{\ast }$ Is the exact value $x$ An approximation of , said $e^{\ast }=x^{\ast }-x$ by $x^{\ast }$ The error of the .

This is the approximate value minus the actual value , And the error must be for the approximation , So it's called $e^{\ast }$ by $x^{\ast }$ The error of the

Error limits ：

if $\left|e^{\ast }\right|\le {\epsilon }^{\ast }$, said ${\epsilon }^{\ast }$ by $x^{\ast }$ The error limit of

The error and error limit mentioned above are absolute error limits , All have dimensions , for instance 100 Rice rope , The approximate length is 99 rice , The error is 1 rice . And such as 3 Rice rope , The approximate length is 2 rice , The error is also 1 rice . Same error , But for approximation , Obviously, the previous example is more approximate , Therefore, the concept of relative error is put forward , It's actually dividing the absolute error by the exact value . Generally speaking, errors are absolute errors , When you say relative error, you must use the word relative error , Otherwise it will be confused ....
Relative error ：

We take the error of the approximation $e^{\ast }$ And the exact value $x$ The ratio of the $$\frac{e^{\ast }}{x}=\frac{x^{\ast }-x}{x}$$ Called an approximation $x^{\ast }$ The relative error of , Write it down as $e_r^{\ast }$

But there's a problem , You don't know the exact value in the actual calculation ( Otherwise, there is no need for approximation ), Therefore, the approximate value is usually taken for calculation :
$$e_r^{\ast }=\frac{e^{\ast }}{x^{\ast }}=\frac{x^{\ast }-x}{x^{\ast }}$$
But on the condition that $e_r^{\ast }$ smaller .
Relative error limits ：

The relative error can be positive or negative , The upper bound of its absolute value is called the relative error limit , Write it down as : $${\epsilon }_r^{\ast }=\frac{{\epsilon }^{\ast }}{\left|x^{\ast }\right|}$$

That's where the error is defined , It's very simple , The absolute error is the approximate value minus the exact value , The error limit is the upper limit that the absolute value of the absolute error can reach , It's how much you can miss at most . The relative error is the absolute error divided by the exact value , But we usually don't know the exact value , So it's directly divided by the approximate value of the estimate . The relative error limit is the absolute error limit divided by the absolute value of the approximate value .

Let's talk about the definition of significant numbers . We learned significant numbers in primary school , It's time to learn numerical analysis , But you're dizzy with a definition , however , Mathematics is like this , In order to achieve some accuracy , Very complex formulas are often used to express a simple meaning . Let's take a look at the definition of significant numbers ：

Middle school definition ：

For an approximate number , The first one from the left is not 0 The number starts with , To the last digit , All numbers are called the significant numbers of this approximate number

Numerical analysis definition ：

If approximate $x^{\ast }$ The error limit of is half a unit on a bit , This bit ( Include this bit ) To $x^{\ast }$ The first non-zero significant digit of the share $n$ position , said $x^{\ast }$ have $n$ Significant digits

Two 、 Understanding of the definition of significant numbers

Let's think about what significant numbers are used for , In the process of people measuring , A tool for measuring 、 And the reading during measurement will produce errors , That is, the systematic error and accidental error learned in high school , For example, your ruler is broken , There must be an error between the measured value and the real value , This error is regular , For example, measure 2 The object of meters , Your ruler reads more than the real value 10cm, measuring 4 The object of meters , Your ruler reads more 20cm. There is also accidental error , You have a very accurate ruler , The scale fits the object very accurately , But different people measure , To measure , Everyone reads differently , Or the action when measuring is different , Resulting in errors , Is accidental error .
But when we measure the results , Always want to know to what extent the measurement results are accurate . But this does not mean that the significant number is from left to right. How many digits are the same as the real value , Now let's look at the definition of numerical analysis ：
If the error limit is half a unit on a bit , That means that the difference between your measured reading and the real value reading is no more than that of 1/2.
Example ：
A ruler with a millimeter scale measures a wooden stick , The measured reading is 876mm, The error limit is 0.5mm, The significant number is 3 Significant digits . A half unit of a millimeter unit , From the millimeter bit to the first non-zero significant digit , Yes 3 position .
Think about it , The error limit is 0.5mm, If the true value is 872.4mm, Then the reading may be 873mm Do you ？ impossible , Because the error is 0.6mm, The error limit is exceeded , The reading should be at 871.9 and 872.9 Between .

3、 ... and 、 How to calculate

To calculate significant numbers ：
Or you need to know the error limit ,
Or know the error （ Knowing the error, we can find the error limit ）.

Then count according to the estimated value .
Approximate value of pi 3.1415926 Valid number of
True value 3.14159265358979…
Error is 0.00000005358979…<0.5×10^(-6)
The significant figures are 7