Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Numerical Methods

Lecture 2 – Taylor Series (Continuation
of Lecture 1)
Notes

Lecturer: Stephan Juricke
Date of lecture: February 10th, 2022
Author: Lirik Maxhuni

Compute 𝑙𝑛(2)
We can choose the Taylor series for 𝑙𝑛(𝑥 + 1) and evaluate at 𝑥 = 1
...
63452
Via calculator we get 𝑙𝑛(2) ≈ 0
...

In order to get a more accurate approximation, we can use another function’s
1+𝑥

Taylor series
...


From using the logarithm’s argument division rule, we have
ln (

1+𝑥
) = ln(1 + 𝑥) − ln(1 − 𝑥)
1−𝑥

1

If we choose 𝑥 = instead of 𝑥 = 1, we then get
3

1
1
1
ln(2) = 2 ( + 3
+ 5
+ ⋯)
3 3 ∙3 3 ∙5
This time, after only 4 terms we have a much better estimate of the result, at:
ln(2) ≈ 0
...
We change c to x and the old x becomes 𝑥 + ℎ,
where 𝑥, 𝑥 + ℎ ∈ [𝑎, 𝑏]
...

-Choose n such that the error bound is (not) met
...


Number representation
Error types:
A: Errors in data (partly because round-off
B: Round-off during calculations
1
...
5358 ≈ 1
...

C: Truncation error (Discretization)

We have many ways to calculate these errors
...
:
Let 𝕒 be an approximation of a
𝕒 − 𝑎: absolute error (|𝕒 − 𝑎|)
𝕒−𝑎
𝑎

: relative error

Error bound is the magnitude of admissible error
...
:
0
...
000004 = 0
...
So, there are 5 correct and 3 significant digits
2

(decimal) corresponding to round-off
...
001234 𝑤𝑖𝑡ℎ 𝑒𝑟𝑟𝑜𝑟 0
...
06 ∙ 10−4
1

1

2

2

Error is below ∙ 10−4 but larger then ∙ 10−5
...

i
...
, the error is between ±

1
2

∙ 10−𝑡 with corresponding t
...
4 → 0
0
...
004 → 0
...
005 → 0
...
05 ± 0
...
1
1
...
04 → 1
...


Theorem:
In addition and subtraction the bounds for the absolute errors are added up
...

Error of propagation: If 𝑦(𝑥) is a smooth (differentiable), |𝑦 ′(𝑥)| can be
interpreted as the sensitivity of 𝑦(𝑥) to errors in x
...


𝜕𝑦
| ∙ |∆𝑥𝑖 |
𝜕𝑥𝑖



---