# Numerical Analysis: 3. Approximations and Round-Off Errors

# Solution
1. analytical solution
2. numerical methods

# Error Types
1. Round-Off Error
 - computers can represent quantities with a finite number of digits
2. Truncation Error
 - approximate solution

# Significant Digits
- is the number of certain digits plus one estimated digit.
- Zeros are not always significant figures. They are just to locate a decimal point.
 - 0.00001845 -> four
- Four signicant figures may be acceptable.
- Omission of the remaining significant figures is called round-off error.

# Error
1. True error;
2. True percent relative error;
3. Approximation error
 - In real-world, we will obviously not know the true answer.
![image.png](https://cdn.hashnode.com/res/hashnode/image/upload/v1663116045967/VZ9c6DRYt.png align="left")
 - continue approximation until **e < prespecified acceptable level**.
4. (Scarborough, 1966) We can be assured that the result is correct to at least n significant figures if the approximation error falls below:
![image.png](https://cdn.hashnode.com/res/hashnode/image/upload/v1663116638482/3HSuvJYo4.png align="left")

# Floating-Point
## Representation
> - m b^e
 - m : matissa
 - b : base of the number system
 - e : exponent
  
- example: 7 bits
 - [0] : sign of number
 - [1] : sign of exponent
 - [2-3] : magnitude of exp
 - [4-6] : magnitude of mantissa
 - number range : 0011001 (1e-3) ~ 1111111 (7e+3)

## Approximation
- Rounding
 - nearby value
- Chopping
 - discard remains

## Error
- Quantizing errors
 - Overflow
 - Underflow

### Quantizing Errors
- machine epsilon : ε = b^(1-t)
 - b : number base
 - t : the number of mantissa bits

# Subtractive Cancellation
- round-off induced when subtracting two nearly equal floating-point numbers
- ![image.png](https://cdn.hashnode.com/res/hashnode/image/upload/v1663554811020/mqGibbqkF.png align="left")
 - subtractive cancellation occurs where b^2 >> 4ac