Computer components

• CPU - Central Processor Unit (e.g., G3, Pentium III, RISC)

• RAM - Random Access Memory (generally lost when power cycled)

• VRAM - Video RAM (amount sets screen size and color depth)

• ROM -Read Only Memory (used for boot)

• IO Input/Output: IO through interfaces such as SCSI (Small Computer System Interface), USB (Universal Serial Bus), Firewire(video standard), Ethernet

• HD - Hard Disk Drive (permanent storage); CDROM - Compact Disk Read-only-memory; CD-RW; DVD Read/Write

Computer Architecture

• Computer Organization

– The von Neumann architecture– Same storage device for both instructions and data

Computer Architecture

• Device Controllers– Memory mapped I/O– Direct Memory Access (DMA)

• Instruction Set– Data transfer operations– Arithmetic / logic operations– Control flow instructions

The architecture of a hypothetical machine

Computer basics• Smallest thing a computer knows is 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 50-100, machine language)

Basics• The 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 more than one instruction per cycle

• All programs end up as 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, or flops

Bits/Bytes and words

• Bits are grouped into larger units– 8 bits = 1 byte (still common)– 2/4/8 bytes = word (varies between CPU)– Most desktop machines are 32-bit words but

64 bits machines are becoming more common (set by data bus)

– Why important? Sets minimum size unit you can access in program, and often precision

Grouping 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”

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=1101102

• Prefixes: kilo = 1024; mega=1048576; giga=10737741824 (approximately 103,106,109)

Program Execution“The machine cycle”

The composition of an instructionfor a machine

Stored Program

Performing the fetch step of themachine cycle I

Performing the fetch step of themachine cycle II

Decoding the instruction 35A7

Mnemonics

• It is hard to remember commands as numbers• Use words associated with the numbers

Some Assembly language

Components: OS

• The Operating System (OS)– Controls everything in the way the computer works.– Not Specific to a CPU type but often some OSs 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

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

Storage in memory• Memory may be treated as a linear array of

bytes from 1 to <size of memory>• The OS keeps track of which parts of memory

are being used and which parts are still free for use by programs and data

• Some computers do “byte-swapping”, i.e., the bytes are not counted linearly but rather are switched. The 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)

Hard disks• Hard disks contain the

computers “file system” (allows access through file names)

• Directory structure points to where files are located (reason for having less space available than the size of disk + some calibration tracks)

• Actual content of HD and directories depend on OS used (different standards exist, e.g., FAT16, FAT32, NTFS for Windows, EXT2 for Linux

• In general, OS can only use their own file-system

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) 3200

DUAL DDR PC3200 6400

DUAL DDR2-400 8600

DUAL DDR2-533 10600

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”

Integers

• Integer numbers can be represented exactly (up to the range allowed by the number of bytes)

• A 2-byte integer, unsigned 0-65535, signed ±32767 (sometimes called short)

• A 4-byte integer, unsigned 0-4294967295, signed ±2147483827 – (With a 32-bit address bus, can have 4Gbytes of

memory—reason max memory is limited in computers)

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)

The IEEE standard for floating point arithmetic

• Single precision (32 bits=4 bytes)S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFF0 1 8 9 31

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

Single precision floating point• In particular

0 00000000 00000000000000000000000 = 01 00000000 00000000000000000000000 = -0

0 11111111 00000000000000000000000 = Infinity1 11111111 00000000000000000000000 = -Infinity

0 11111111 00000100000000000000000 = NaN1 11111111 00100010001001010101010 = NaN

0 10000000 00000000000000000000000 = +1 * 2(128-127) * 1.0 = 20 10000001 10100000000000000000000 = +1 * 2(129-127) * 1.101 = 22*(23+22+1)/23 = 6.51 10000001 10100000000000000000000 = -1 * 2(129-127) * 1.101 = -6.5

0 00000001 00000000000000000000000 = +1 * 2(1-127) * 1.0 = 2(-126)

0 00000000 10000000000000000000000 = +1 * 2(-126) * 0.1 = 2(-127)

0 00000000 00000000000000000000001 = +1 * 2(-126) * 0.00000000000000000000001 = 2(-149) (Smallest positive value)

Double precision floating pointS EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 1 11 12 63

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

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

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛19971999

9989999991000

yx

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛01.199799.1998

ˆˆ

9989999991000

yx

ExampleNote

What went wrong?

Recall an n-by-n matrix A is symmetric iff A=AT.Fact (“spectral decomposition”):

If A is symmetric, it may be written asA=UDUT,

where D is the diagonal matrix, U is unitary (i.e., UUT=I, the identity matrix).

99.1897.20

ˆˆ

but 11

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛yx

yx

ExampleFact: 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,x is the corresponding right eigenvector iffAx=tx.

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 asχ(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.

χ(A) shows how far off the solution to Ax=b may be from the solution to Ax=b+Δb (a measure of “relative singularity”).

Assignment 2MATLAB and C code are posted on the web. The MATLAB script is slightly modified in an

attempt to produce a more accurate graph: the C call is referred to several times using reps counter.

1. 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.

2. Condition number I: for the class example, answer the following:a) find the matrix spectral decomposition and compute the matrix condition number,b) find perturbations Δb of the right-hand side of the equation with (the Euclidean norm) ||Δb||=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,

c) given the equation for the curve ||Δb||=1, find its image under A-1 and plot it,d) find the explicit expression for the condition number χ(A-γ I) for γ > 0.

3. Condition number II: given a 2x2 matrix with double-precision entries, what is the worst condition number this matrix might have? Explain your answer.

4. 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.

5. Bonus question: find all the distinct spectral decompositions of the identity matrix.