Algorithms using real numbers (continued) Noter ch.4 Algorithms - - PowerPoint PPT Presentation

Algorithms using “real” numbers (continued)

Noter ch.4

Algorithms using “real” numbers

• Representable numbers
• Rounding/truncation
• IEEE standard and Java
• Summation order
• Newton iteration
• More iteration
• Formula rewriting

Formula rewriting

• Example: computing by convergent series
• lim

→ , for being half the circumference of a

regular -gon inscribed in the unit circle. 1 .52 3 3.1

• Monotonicity: ⋯ ⋯
• Enough to compute side length , since

Formula rewriting

Computing from : From Pythagoras:

• 2
• 2
• 1 1

Leading to 1 2 4 and lim

→ ,

for

• Radius = 1

Pseudocode for computing approximation to

QUIZ

Why do we see this behaviour?

• 1. Stop criterium is bad, we should use while ( )
• 2. Stop criterium is bad, we should use while ( 10)
• 3. Stop criterium is ok, but

2 4 ; leads to catastrophic cancellation

• 4. Some other reason
• 5. I don’t know
• 4

2 4

• 3

1 3 6 1.732 .2680 .5177 3.106 12 1.932 .06800 .2607 3.128 24 1.983 .01700 .1304 3.129 48 1.995 .005000 .07071 3.394 96 1.998 .002000 .04473 4.296 192 1.999 .001000 .03162 6.070 384 1.999 .001000 .03162 12.14 ⋮ ⋮ ⋮ ⋮ ⋮ 6; 1; 3; do { 2 4 ; ; 2; } while ( ) Result from execution

Formula rewriting

• What goes wrong?

– It seems →∞ rather than → – 4 is so close to 2 that the subsequent subtraction 2 4 leads to catastrophic cancellation

• Rewrite formula

2 4 4 4 2 4

• 2

4

Catastrophic cancellation No Cancellation

Pseudocode for computing approximation to

QUIZ

What is the result of execution in F(10,4,-99,99)?

• 1. Infinite loop (execution never stops)
• 2. Stops with Infinity
• 3. Stops with incorrect approximation 3.128
• 4. Stops with correct approximation 3.141
• 5. Something else happens
• 6. I don’t know

Catastrophic cancellation 2

• 4

2 4

• 3

1 3 6 1.732 3.732 .5175 3.105 12 1.932 3.932 .2609 3.130 24 1.982 3.982 .1307 3.136 48 1.995 3.995 .06541 3.139 96 1.998 3.998 .03272 3.141 192 1.999 3.999 .01636 3.141 stop ⋮ ⋮ ⋮ ⋮ ⋮ 6; 1; 3; do { / 2 4 ; ; 2; } while ( ) Result from execution

?

• Try java double using

2 4

• The stopping criteria

while ( ) makes it stop, but only 8 out of 17 digits correct!

• More iterations does not

improve the approximation! (on the contrary)

• Try java double using

/ 2 4 ;

• The stopping criteria

while ( ) makes it stop, and the value is a good approximation of

Catastrophic cancellation

• Problem

– in some computation two terms of approximately identical size are subtracted: – where

• Warning:

– E will be computed with few or no significant digits – results are corrupted and loops may not terminate

• Advice:

– rewrite formula to avoid catastrophic cancellation – sometimes the following simple reformulation is useful:

QUIZ

We want to compute 2000 2000 30 in the number system F(10, 4,−99, 99) with truncation. The result is 1.000. We want to rewrite the underlying expression to minimize (catastrophic) cancellation. Which expression is expected to give a better result (and also mathematically equivalent)? 1.

• 2.
• 3.
• /
• 4. 1

1 /

• 5. None of these
• 6. I don’t know

Catastrophic cancellation 3 2000 2000 30 ?

Summary of advice

• Add terms in increasing order. If possible add terms of

equal size only.

• Don’t test whether two limited precision numbers are
• equal. Test whether numbers are sufficiently close. The

meaning of “sufficiently close” depends on the context.

• Mathematical expressions should be reformulated to

avoid catastrophic cancellation.

Computing

Some algorithms are faster than

• thers

Computing

• Buffon’s needle (Le Clerc, 1777)
• In/circumscribed polygon (Archimedes,

287-212, B.C.)

• Arithmetic-geometric mean (Salamin-Brent

1976)

http://crd.lbl.gov/~dhbailey/dhbpapers/pi-quest.pdf

Buffon’s needle

(Le Clerc, 1777)

Applet simulator at http://www.ventrella.com/Buffon/index.html

• Throw some needles of length
• n a

floor covered with planks of width 1

• Count those needles that land

across a crack between two planks

• Assuming needles are rotated

uniformly randomly, the probability

• f a needle landing across a gap is
• #needles

#crossings

• Experiment:
• 3.147663175
• Need 10 iterations to compute digits
• To get 1 additional digit throw 100 times

as many needles (no guarantees – when using randomness)

Inscribed/circumscribed polygon

(Archimedes, 287-212 B.C.)

• Let (and ) be length of

circumscribed (and inscribed) regular 6 ⋅ 2 -gons relative to a circle of diameter 1

• Mutually recursive formulae:

2 3 3 2

• Archimedes obtained
• using 4
• Zǔ Chōngzhī (429-500) obtained
• using 11
• Ludolph van Ceulen (1540-1610)

calculated 34 digits of

http://www.mathsisgoodforyou.com/applets/calculatingPi.htm

• Need iterations to compute digits
• To get 1 additional digit make 1 more

iterations

Inscribed/circumscribed polygon

2 3 3 2

• Need iterations to compute digits
• To get 1 additional digit make 1 more

iterations

Arithmetic-geometric mean

(Gauss-Legendre; Salamin-Brent, 1976)

1 1 2 1 2 2

• 2

2

• lim

→

• Need log iterations to compute digits
• To double the number of digits make 1

more iteration

http://www.apfloat.org/apfloat_java/applet/pi.html

• 1

3.187672642712109 2 3.141680293297657 3 3.14159265389546