"Numerical Analysis"-0-Basic Knowledge

0. Background

The main purpose of numerical analysis is to illustrate and discuss methods for solving mathematical problems on a computer. The most basic arithmetic operations are addition and multiplication, and polynomials are the basis on which we construct computing techniques.

A major issue in numerical analysis is to make the user aware of the danger of calculations being unreliable due to small errors made by the computer, and to know how to avoid or minimize such danger.

 1. Polynomial evaluation

A general d-order polynomial can be solved by d times of multiplication and d times of addition, and the specific construction method is the nested multiplication/Horner method.

2. Decimal and binary

The easiest way to convert decimal to binary and binary to decimal is to treat the integer part of the number separately from the fractional part.

When converting binary to decimal, if the fractional part is not a finite expansion based on 2, it is necessary to use the 2-multiply translation feature conversion.

3. Truncation and rounding

Truncation: It is achieved by simply discarding the digits outside the end, but this method will move the result towards 0, which is a biased method.

Rounding: Determine whether to round up or round down by judging the first digit outside the given digit space. Randomly choose to round up or round up when there is a tie, avoiding unwanted slow shifts in long calculations due to biased rounding.

4. Missing significant figures

When calculating, sometimes it is unavoidable to cause the lack of significant figures, such as explicitly subtracting two approximately equal numbers, especially when solving quadratic equations. Faced with this situation, it can usually be avoided by rebuilding the equation. Multiplication by "conjugate equations" is a trick to refactoring computations. 

5. Taylor approximation

The Taylor approximation is fundamental to learning many computational techniques. The main idea is that if  the value of the function is known at the point, a lot of information about the function around the point can also be known (provided that the function is continuous, we can use all derivatives at the point to give all approximate estimates of the function around the point) . The difference from its approximation at the point is the Taylor remainder.