1. Chapter 2 A Survey of Simple Methods and Tools

2. 2.1 Horner’s Rule and Nested Multiplication
For example

3. Horner’s rule for polynomial evaluation
多項式最高次項的係數
多項式的係數

4. Horner’s rule for polynomial derivative evaluation
Polynomial first derivative:
For example:

5. Horner’s rule for polynomial derivative evaluation

6. A more efficient implementation of Horner’s rule
If the intermediate values in the computation of p (x) are saved, then the subsequent computation of the derivative can be done more cheaply.
Define
So that
Then
and, in particular,
Define
Therefore
Since
注意bk亦為x的函數

8. 2.2 Difference Approximations to the Derivative—one-side difference
The definition of the derivative:
Taylor’s Theorem:
So that we have
Thus the error is roughly proportional to h.
Can we do better?

9. 2.2 Difference Approximations to the Derivative—centered difference
Consider the two Taylor expansions:

10. Example 2.1

11. Further illustrate these differences in accuracy
Let’s continue computing with the same example, but take more and smaller values of h.
Let
with the corresponding errors

13. Nearly 4
Error increase. why?

14. Rounding Error
Let denote the function computation as actually done on the computer.
Define as the error between the function as computed in infinite precision and as actually computed on the machine.
The approximate derivative that we compute is constructed with , not f.
Define

15. Rounding Error
We have
which we write as

16. Rounding Error

17. Nearly 4

19. 2.3 Application: Euler’s Method for Initial Value Problems
General form:
One-side difference (Eq. 2.1)
Euler’s method

21. Example 2.2

26. 2.4 Linear Interpolation
Given a set of nodes xk, if for all k, then we say the function p interpolates the function f at these nodes.
Linear interpolation: using a straight line to approximate a given function
For example: the equation of a straight line that passes through the two points

28. The accuracy of linear interpolation

29. Example 2.3
f (0.2)
f (0.1)

31. Piecewise linear interpolation
Example 4.2

34. 2.5 Application: the trapezoid rule
Define the integration of interest as I(f ):

36. Error analysis
Apply the Integral Mean Value Theorem
thus

37. The n-subinterval trapezoid rule

39. This theorem tells us:
The numerical approximation will converge to the exact value
How fast this convergence occurs h 2

40. Example 2.5

42. Example 2.6

43. The stability of the trapezoid rule
We conclude that the trapezoid rule is a stable numerical method.
In fact, almost all methods for numerically approximating integrals are stable.

44. 2.6 Solution of tri-diagonal linear systems

45. If A is tri-diagonal, then
For example:

46. Make a notational simplification:
where
Then the augmented matrix corresponding to the system is

47. Gaussian elimination The elimination step where
The backward solution step

48. Example 2.7

49. After a single pass through the first loop:
We cannot continue the process, for we would have to divide by zero in the next step.
However, the solution of the system indeed exist:

50. Diagonal dominance for tri-diagonal matrices
For example

51. 2.7 Application: Simple Two-point Boundary Value Problems
Two-point boundary value problem (BVP)

52. Use Taylor expansions similar to (2. 2) and (2
Use Taylor expansions similar to (2.2) and (2.3) (just take more terms) to derive an approximation to the second derivative, by adding them. Then we get
This is a tri-diagonal system of linear equations.

53. In matrix-vector form
It is diagonally dominant, so we can apply the algorithm developed in the precious section to produce solutions.

54. Example 2.8