1. BASICS

2. Hardware components

3. Computer Architecture
Computer Organization
The von Neumann architecture
Same storage device for both instructions and data
Processor components
Arithmetic Logic Unit
Control Unit
Registers

4. Computer Architecture
Device Controllers
Memory mapped I/O
Direct Memory Access (DMA)
Instruction Set
Data transfer operations
Arithmetic / logic operations
Control flow instructions

5. Computer components Central Processor Unit Random Access Memory
e.g., G3, Pentium III, RISC
Random Access Memory
generally lost when power cycled
Video RAM
amount sets screen size, color depth
Read Only Memory
used for boot
Input/Output, through interfaces such as
Small Computer System Interface
Universal Serial Bus
Firewire (video standard)
Ethernet
Hard Disk Drive (permanent storage), Compact Disk Read-Only-Memory, CD-Read/Write, DVD R/W, etc.

6. A typical architecture (fragment)

7. CPU basics Smallest thing a computer knows
a bit 0 or 1 (false/true)
CPU knows how to perform and, or, xor (exclusive or) operations
And returns true if both same
Or returns true if either true
Xor returns true if different
CPU is a massive collection of and and or gates
A specific CPU has a set of instructions it can execute (usually , machine language)

8. CPU basics
Number of instructions per seconds is set by the “clock speed”
e.g., 500 MHz Pentium III
One clock tick is called a cycle
modern CPUs can often execute >1 instruction per cycle
Programs are set of instructions to be executed by the CPU
compilers/linkers or interpreters do this for you
Floating point speed is measured in floating point operations per seconds (flops)

9. Data

10. Bits/Bytes and Words Bits are grouped into larger units
8 bits = 1 byte (still common)
2/4/8 bytes = word (varies between CPU’s)
Most desktop machines are 32-bit words, bit machines are becoming more common
set by data bus
Why important?
Sets minimum size unit you can access in program, and often the precision for computations

11. Words and Bytes
Number of unique values that can be represented depends on number of bits
With n bits one can have 2n unique values
For n=8 have (Byte) = 256
grouped into larger units to represent different data
ASCII – American Standard Code for Information Interchange
Basic version is 7 bit (127 characters)
A-Z, a-z, 0-9 and special characters
Values <32 are “control characters”

12. Numbers and bases Numbers represented in different base systems
Binary base 2 (0-1)
Octal base 8 (0-7)
Hexadecimal base 16 (0-15, with A-F representing 10-15)
E.g, 5410=3616=668=
Prefixes: kilo = 1024; mega= ; giga= (approximately 103,106,109)

13. Instructions

14. Program execution “The machine cycle”

15. Instruction composition

16. Stored program

17. Fetch step of the machine cycle I

18. Fetch step of the machine cycle II

19. Decoding the instruction

20. Mnemonics It is hard to remember commands as numbers
Use words associated with the numbers

21. Some Assembly language

22. Operating System

23. OS The Operating System (OS)
Controls everything in the way the computer works.
Not Specific to a CPU type but often some OS’s are associated with specific CPUs
G3/4/5 68x series MacOS
Pentium, x86 DOS (Windows)
SPARC Solaris (Unix)
OS controls IO and memory management
Program implementations are dependent on OS

24. Programming interface to OS
Depending on language used, OS interface may or may not be important
For Fortran, C, C++ when program is linked OS routines are needed
How to read from keyboard or file?
How to write to screen or disk?
In your program you do not need to go into the low-level (OS) details

25. Storage in memory
Memory treated as a linear array of bytes, from 1 to <size of memory>
OS keeps track of used and free memory, for use by programs and data
Some computers do “byte-swapping”
the bytes are not counted linearly but rather are switched
main (but not only) styles are Big Endian (HP, Sun, Macs) and Little Endian (PC)
affects ability to transfer binary data; TCP knows this and will accommodate this up to a certain degree

26. Basics revisited

27. Hard disks Contain the computer “file system”
allows access through file names
Directory structure points to file location
reason for having less space available than the size of disk + some calibration tracks
Actual content of HD and directories depend on OS
e.g., FAT16, FAT32, NTFS for Windows, EXT2 for Linux
In general, OS can only use their own file-system

28. Accessing RAM vs. HDD
The highest possible bandwidth  (peak bandwidth) for the various types of RAM
However, RAM also has to match the motherboard, chipset and the CPU system bus
HDD ~ only 80MB/s
In MATLAB: try save, load, pack, clear
Module type
Max. Transfer, MB/s
SD RAM, PC100
800
SD RAM,  PC133
1064
Rambus, PC800
1600
Rambus, Dual PC800
3200
DDR 266 (PC2100)
2128
DDR 333 (PC2700)
2664
DDR 400  (PC3200)
DUAL DDR PC3200
6400
DUAL DDR2-400
8600
DUAL DDR2-533
10600

29. RAM and “fast” RAM/cache
A CPU cache is a cache used by the central processing unit of a computer to reduce the average time to access memory
Access time: roughly speaking “CPU speed against the bus speed”

30. Integers
Integer numbers can be represented exactly (up to the range allowed by the number of bytes)
A 2-byte integer, unsigned , signed ±32767 (sometimes called short)
A 4-byte integer, unsigned , signed ±
(With a 32-bit address bus, can have 4Gbytes of memory—reason max memory is limited in computers)

31. Floating point
Representations vary between machines (often reason binary files can not be shared)
Precise layout of bits depends on machine and format; all formats are (mantissa)*2(exponent)

32. The IEEE standard for floating point arithmetic
Single precision (32 bits=4 bytes)
S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFF
The value V represented by the word may be determined as follows:
If E=255 and F is nonzero, then V=NaN ("Not a number")
If E=255 and F is zero and S is 1, then V=-Infinity
If E=255 and F is zero and S is 0, then V=Infinity
If 0<E<255 then V=(-1)S * 2(E-127) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point
If E=0 and F is nonzero, then V=(-1)S * 2 (-126) * (0.F). These are "unnormalized" values.
If E=0 and F is zero and S is 1, then V=-0
If E=0 and F is zero and S is 0, then V=0

33. Single precision floating point
In particular
= 0
= -0
= Infinity
= -Infinity
= NaN
= NaN
= +1 * 2( ) * 1.0 = 2
= +1 * 2( ) * = 22*( )/23 = 6.5
= -1 * 2( ) * = -6.5
= +1 * 2(1-127) * 1.0 = 2(-126)
= +1 * 2(-126) * 0.1 = 2(-127)
= +1 * 2(-126) * =
2(-149) (Smallest positive value)

34. Double precision floating point
S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
The value V represented by the word may be determined as follows:
If E=2047 and F is nonzero, then V=NaN ("Not a number")
If E=2047 and F is zero and S is 1, then V=-Infinity
If E=2047 and F is zero and S is 0, then V=Infinity
If 0<E<2047 then V=(-1)S * 2(E-1023) * (1.F) where "1.F" is intended to represent the binary number created by prefixing F with an implicit leading 1 and a binary point.
If E=0 and F is nonzero, then V=(-1)S * 2(-1022) * (0.F) These are "unnormalized" values.
If E=0 and F is zero and S is 1, then V=-0
If E=0 and F is zero and S is 0, then V=0

35. Is the finite precision an issue?
An extended example: condition number of a symmetric matrix.
Consider a system of linear equations
and the perturbed system

36. Example Note but ? What went wrong?
Recall an n-by-n matrix A is symmetric if A=AT.
Fact (“spectral decomposition”):
If A is symmetric, it may be written as
A=UDUT,
where D is the diagonal matrix,
U is unitary (i.e., UUT=I, the identity matrix).

37. Example Fact: U is unitary is “almost the same” as
U is a rotation matrix
“almost the same” because U might include reflections
Fact: For A=UDUT as before, the diagonal elements of D are the eigenvalues of A and columns of U are the right eigenvectors of A.
Recall t is an eigenvalue of A iff det(A-tI)=0,
u is the corresponding right eigenvector iff
Au=tu.

38. (A)=|t|max(A) / |t|min(A)
Example
How does A act on x, step-by-step:
Ax=UDUTx=UD(UTx)=U(D(UTx)),
that is, “rotate, scale, rotate back”.
Define the condition number of A as
(A)=|t|max(A) / |t|min(A)
where |t|max and |t|min are the largest and the smallest in absolute value eigenvalues of A.
(A) shows how far off the solution to Ax=b may be from the solution to Ax=b+E (a measure of “relative singularity”).

39. Assignment 2 MATLAB and C code are posted on the web.
Using the MATLAB script from the class, try to identify the cache size on your machine. Note that usually a number in MATLAB occupies 8 bytes (double precision floating point), and that the function requires roughly 2-times the memory needed to store the x vector (x in the input, x_new in the output). Explain your results. As a sanity check, you might want to use a CPU info tool.
Condition number I: for the class example, answer the following:
find the matrix spectral decomposition and compute the matrix condition number,
find perturbations E of the right-hand side of the equation with (the Euclidean norm) ||E||=1 that give the largest and the smallest errors in the solution vector; plot each of the two perturbations, together with their image under UT, under D-1UT and, finally, under A-1,
given the equation for the curve ||E||=1, find its image under A-1 and plot it,
find the explicit expression for the condition number (A-g I) for g > 0.
Condition number II: given a 2x2 matrix with double-precision entries, what is the worst condition number this matrix might have (as a real number)? Explain.
Condition number III: for the question 2 of the first assignment, is there any ill-conditioning? (Ill-conditioning refers to a matrix having large condition number, hence often resulting in numerical instability; if you happen to encounter complex eigenvalues, you are on the wrong track; for the definition in class, matrix must be symmetric.) Explain your answer.
Bonus question: find all the distinct spectral decompositions of the identity matrix.