8

The need to get involved :



Numerical Methods cannot be read about, they must be used in order to be understood; Personal experience that the best test of whether one understands a method is not to carry out a hand calculation but to write a computer program; It is remarkable how hazy concepts can become clear under the resulting pressure to be completely precise and unambiguous.
2
f ( x ) ...
3!
O (h)
13
We shall employ the subscript notation:
f ( x h ) f i 1
f ( x) fi
Using this notation, then
f ( x ) f i 1 f i h O (h)
f ( x ) f ( a ) ( x a ) f ( a ) (x a) 3!
3
(x a) 2! (x a) n!
2
f ( a )
(n)
n
f ( a ) ...
f
( a ) ...
10
The Taylor Series

The Taylor Series is the foundation of Mathematical Problems: for x b
15

We now use the Taylor Series expression of about x to determine f ( x h )
f ( x h ) f ( x ) hf ( x ) h
2
f (x)
f ( x )
h
3
f ( x ) ...
4

There are many problems which are still impossible (in some cases we should say “impractical”) to solve using Numerical Methods :



for some of these problems no accurate and complete mathemetical model has yet been found; Other problems are simply so enourmous that their solution is beyond practical limits in terms of current computer technology; Of course, the entire question of practicality is strongly dependent upon how much one is willing to spend ...
9
The Taylor Series

The Taylor Series is the foundation of Mathematical Problems: If the value of a function f ( x ) can be expressed in a region of x closed to x a by the infinite power series
5

To study Numerical Methods :



No complex physical situation can be exactly simulated by a mathematical model; No numerical method is completely trouble-free in all situation; No numerical method is completely error-free; No numerical method is optimal for all situation.
We define the first forward difference of f at i ,
f i f i 1 f i
14
The expression for f ( x ) may now be written as
f ( x ) fi h O (h)
The term f i / h is called a first forward difference approximation of error order h to f ( x )
材料计算机数值模拟讲义
The Finite Difference Calculus
1
1、 Introduction to Numerical Methods 2 、the Taylor Series 3 、Difference Calculus
2
The Purpose and Power of Numerical Methods as well as their Limitations
7

The Verification Problem to Numerical Analysis:



One of the most vita and yet difficult tasks which must be carried out in obtaining a numerical solution to any problem is to verify that the computer program and the final solution are correct; The verification procedure can actually be more expensive and time consuming than obtaining the final desire answer; The process of verification for a general program or library subprogram, which would be employed by many users to solve a wide variety of problems, would be similar but necessarily even more extensive and painstaking…
We define the first backward difference of f at i ,
f i f i f i 1
f ( x h ) f ( x ) hf ( x ) f (x h) f (x) h
f ( x )
h2Βιβλιοθήκη f ( x ) h
3
f ( x ) ...
2! f ( x ) h 2!
f (x h) f (x) h
3! f ( x ) h
2! f ( x ) f (x) f (x h) h h 2! f ( x )
3! h
2
f ( x ) ...
3!
f ( x )
f (x) f (x h) h
O (h)
16
Using this notation, then
f ( x ) f i f i 1 h O (h )

Numerical Methods are a class of methods for Solving a wide variety of Mathematical Problems:



the Electronic Computers have been in widespread use since the middle 1950’s; Numerical Methods actually predate electronic computers by many years; Numerical Methods came of age with the introduction of the Electronic Computer.
n 1
d
f
m ax
(xa)
n 1
dx
n 1
( n 1) !
accurate to
O(x a)
n 1
f ( x ) f ( a ) ( x a ) f ( a )
(x a) 2!
2 3 f ( a ) O ( x a )
12
The Finite Difference Calculus
3

The Combination of Numerical Methods and digital computers has created a tool of immense power in Mathematical Analyses:



the Numerical Methods are capable of handling the nonlinearities, complex geometries, and large systems of coupled equations which are necessary for the accurate simulation of many real physical situations; Numerical Methods have displaced classical mathematical analysis in many industrial and research applications; Numerical Methods are so easy and iexpensive to employ and are often available as prepackaged Programs.