Lecture 23
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Lecture 23
Numeric and Symbolic
Computing
Numeric & Symbolic Computing
Computer Networks
(S&G, §§11.3
–11.4, 12.1–
§§11.3–
12.1–12.2)
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Numeric Computation
• Performance:
–
–
–
–
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Symbolic Computing
( x −1)
– automate processes that are mechanical,
tedious, and error-prone
€
• Examples: Macsyma, Mathematica, Maple,
MatLab
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Example: Simplification
• Manipulate mathematical formulas,
equations, etc. much the way a
mathematician would
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better algorithms
accessing of data in memory hierarchies
parallel computation
data communication in networks
• Mathematical software libraries
• Accuracy and stability of numerical
approximations
 a teraflop machine performs at least 1012 (a trillion)
floating-point operations per second
 36 Tflops already achieved (Japan’s Earth Simulator,
which cost $350–500M)
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Computer Science Issues
• Applications that make heavy use of real
arithmetic
• Especially used in science, engineering,
economics, statistics, animation
• The motivation for the first computers
• Still drives the development of supercomputers
and parallel computers
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2
+ ( x + 2) + (2x − 3) + x
• Simplify[(x-1)^2 + (x+2) +
(2x-3)^2 + x]
• 12 - 12x + 5x2
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Example: Solving Equations
Example: Expansion
(1+ x + 3y )
4
2x + y = 11
6x − 2y = 8
• Expand[(1 + x + 3y)^4]
• Solve[ {2x + y == 11,
6x - 2y == 8},
{x,
€ y}]
• 1 + 4x + 6x2 + 4x3 + x4 + 12y
€
+ 36xy + 36x2y + 12x3y + 54y2
+ 108xy2 + 54x2y2 + 108y3
+108xy3 + 81y4
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• {{x -> 3, y -> 5}}
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Typical Expansion Rules
Digression
Expand[ X × (Y + Z )] ⇒ X × Y + X × Z
Expand[( X + Y ) × Z ] ⇒ X × Z + Y × Z
• Recall our discussion of formalized
mathematics, and the idea of reducing
mathematics to the mechanical application
of formal rules
• Formal rules: depend on the form of
expressions, not their meaning
• Symbolic computation is an application of
the idea of a calculus
Expand[ X 2 ] ⇒ X × X
Hence,
Expand[(n + 1)2]
€ ⇒Expand[(n + 1)(n + 1)]
⇒Expand[(n + 1)n + (n + 1)1]
⇒Expand[(n + 1)n + (n + 1)1]
⇒Expand[n×n + 1×n + n×1 + 1×1]
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Computer Science Issues
• Symbolic computation systems are:
– very high-level languages
– problem-specific
– nonprocedural
Computer Networks
(S&G ch. 12)
• Depend on many algorithms, e.g.:
– pattern matching
– efficient management of complex data structures
representing formulas
• Results should be presented in a form familiar and
useful to the mathematically literate
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Kinds of Networks
Communication Links
• Local Area Network (LAN)
• Switched dial-up phone lines
– limited range (building, campus)
– typically owned by owners of premises
– example: Ethernet
– require modem (modulator-demodulator) to
interface between digital interface and analog
phone line
• Wide Area Network (WAN)
• Dedicated links
– worldwide (and beyond!)
– typically owned by telecommunications
services
– example: Internet
– wired (wire proper & fiber optic cable)
– wireless
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Single-Cable Ethernet
Ethernet
intended transmission
• Developed at Xerox PARC in mid-70s
• Communication protocols govern use of the
network
• Ethernet uses contention-based distributed
control
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Single-Cable Ethernet
(1) Listen
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Single-Cable Ethernet
(2) Transmit
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Single-Cable Ethernet
(3) Broadcast
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Single-Cable Ethernet
(4) Receive
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Single-Cable Ethernet
(5) Inactive
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Single-Cable Ethernet
(1) Listen
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Single-Cable Ethernet
(2) Transmit
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Single-Cable Ethernet
(3) Collision
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Single-Cable Ethernet
(4) Wait Randomly
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Single-Cable Ethernet
(5) Listen
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Single-Cable Ethernet
(6) Rebroadcast
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Single-Cable Ethernet
(7) Inactive
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Single-Cable Ethernet
(8) Listen
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Single-Cable Ethernet
(9) Rebroadcast
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Collision Recovery
Wide Area Networks (WANs)
• LANs typically broadcast messages
• WANs typically use point-to-point
communication
• Typically use store and forward packetswitching
A:
wait
B:
wait
C:
wait
– packet = fixed-size quantity of information
– large messages divided into packets
time
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Store-and-Forward Packet-Switching
A
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What We Need
C
• A way of finding a computer (“server”) on
the Internet
• A way of locating a Web page on that
computer
• A way of displaying that Web page on my
computer (“client”)
ACK
D: packet
D: packet
B
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Point-to-point Communication:
Node Failure
Point-to-point Communication
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(slide < S. Levy)
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Locating Computers
on the Internet
IP Format
• TCP/IP (Transmission Control Protocol/Internet
Protocol) – a set of protocols (“rules of the
game”) for making transactions
• IP address – standard for identifying a computer
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• 4 groups of 3 digits
• Range from 0 to 255 (8 bits)
• Total of 2564 or over 4 billion computers
• TCP – describes how packets are sent and
retrieved
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• Example: 204.202.129.230
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(slide < S. Levy)
The Internet and the Web
The Internet and the Web
• The Web sits on top of the Internet and uses
its protocols (HTTP, FTP, MAILTO, etc.)
• URL (Uniform Resource Locator) – allows
different computers to use the same
protocols for Web-based transactions
• HTTP (HyperText Transfer Protocol)
format: http://where/what
e.g.,
http://www.cs.utk.edu/~cs100
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Internet
URL
HTTP, FTP,
MAILTO, ...
TCP
IP
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• Standard protocol (URL)
• Provide a name and get the computer
• Standard language for describing Web
(Domain Name Server) to translate it to the IP
pages (HTML – HyperText Markup
address
Language)
• HTML: A set of commands for formatting
Web pages
http://www.cs.utk.edu/~cs100
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Two Standards for Web Pages
• It’s hard to remember 12 digits
• e.g, http://160.36.56.64/~cs100 =
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Web
(slide < S. Levy)
Named Address (Domain Name)
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How HTML Works: Client/Server
How HTML Works: Client-Side
• The spirit of HTML:
Server’s disk
1.
– Content (text, images, videos) + “suggestions”
on how to format it
– Hyper-links to other web pages
Network
connection
Client’s browser
The client sends a request for a page to the server.
2.
HTML code is sent as text to the client.
3.
4.
HTML code is stored in the client’s cache (temporary memory).
The client’s browser interprets the HTML code and formats
the page.
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• Actual formatting (fonts, positions, colors)
should be up to client’s browser (e.g., lynx
entirely text-based)
• In practice, web designers want more
control (frames, tables, blank spaces)
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(slide < S. Levy)
Typical Network Services
• Resource sharing
– print servers, file , compute servers
Read S&G, ch. 13
(Artificial Intelligence)
for next class
• Information sharing
– data interchange, information utilities
• Communication
– e-mail, bulletin boards, news groups, chat
rooms
• Electronic commerce
• Security issues
– authentication, secrecy, fault tolerance
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