Basic Domains of Interest used in Computer Algebra Systems

1
Basic Domains of Interest used in Computer
Algebra Systems

• Lecture 2b

2
Some ideas just for representing integers

• Integers are sequences of characters, 0..9.
• Integers are sequences of words modulo 109 which
  is the largest power of 10 less than 231.
• Integers are sequences of hexadecimal digits.
• Integers are sequences of 32-bit words.
• Integers are sequences of 64-bit double-floats
  (with 8 bits wasted).
• Sequences are linked lists
• Sequences are vectors
• Sequences are stored in sequential disk locations

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Aside using half the bits in a word is common
strategy

• Integers are sequences of 32-bit words, but only
  the bottom 16 are used There is some lack of
  uniformity in architectural support for unsigned
  32X32 bit multiply. But 16X16 ? 32 bit is
  supported, so programs can be portable.
• Downside if you multiply two n-bigit numbers in
  time n2 then with half-length bigits you need
  twice as many of them and so you take time (2n)2,
  or 4 times slower and 2X the space.

4
Yet more ideas

• Integers are stored in redundant form ab
• Integer are stored modulo set of different primes
• Integers are stored in p-adic form as a sequence
  of x mod p, x mod p2,

5
Redundant (big precision) floats

• Sequences (xn .x2 x1)
• Non-overlapping means each the lsb of xk is more
  significant than the msb of xk-1. May be big
  gaps.
• Not unique.
• (binary values..) 1100 -10.1 1001 0.1
  100010.1
• Easy to tell the sign. Look at the leading term.
• Adding must restore non-overlapping property.
• Important use by Jonathan Shewchuk (UCB) in
  geometric predicate calculations.

6
Modular (mod a set of primes q1, q2, qn)

• images unique only within multiple of product of
  the primes Q q1 q2 qn.
• CRA (Chinese Remainder Algorithm) provides a way
  of going from modular to conventional positional
  notation, but takes O(n2) in practice.
• This and its generalizations heavily used in
  computer algebra systems and this course.

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Modular arithmetic is really fast

• all the arithmetic can be done without carry, in
  parallel.
• Not usually used because
• You cant tell for sure if a number is , -, 0 or
  Q or 3 Q.
• Parallelism is almost always irrelevant ?
• If you must see the answer converted to decimal,
  the conversion is O(n2)
• Conversion to decimal may be very common if your
  application is a bignum calculator.

8
Whats a p-adic integer?

• For the moment assume p is a prime number.
  Consider representing an integer a a0a1.pa2
  p2 where ai are chosen from integers in the
  range 0 ai lt p
• For any finite positive integer a, either all
  the ak are zero in which case a 0 and ap
  is 0, or some initial set of the ai are zero.
  Let ar1 be the first non-zero term. ap
  p-r. This replaces the absolute-value valuation
  a where we consider that x and y are close if
  xy mod pk for many values of k0,1,2,.

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p-adic ordering is odd.

• 14 3-adically is 230131132 2 3 9 14
• Which is very close to 5, 3-adically because
• 5 is 230131, and they are the same modulo 30
  and 31. 15-43 3-1
• What about negative numbers? pk-1 is possible,
  but consider
• pk-1 (p-1)(pk-1pk-2p1)
• (p-1)1(p-1)p(p-1)p2 so
• -1 3-adically is 230231. Infinite number of
  terms (2,2,2,2,2,.)

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How to multiply p-adically

• (a,b,c)
• (d,e,f) ?
• Compute ad , ae, af add
  columns
• bd, be bf
• cd, ce, cf
• Add modulo p, p2, p3 with a carry.

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How to multiply p-adically approximately

• (a,b,c )
• (d,e,f ) ?
• Compute ad , ae, af add
  columns
• bd, be bf
• cd, ce, cf
• Add modulo p, p2, p3 with a carry. Dropping
  off extra terms is like 3.14159 vs 3.14, but
  with respect to p-adic distances.

Ignore these
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What other p-adic numbers are there??

• -1/2 3-adically .. is 130131132
  (1,1,1,1,1). Proof
• Multiply by 2, which is (2,0,0,0,.) to get
  (2,2,2,2) which is -1
• What about p 7 ?
• (a03a1)2-70 (mod 3i1)
• Mod 30, a02-70 so a0 is 1 or 1. Lets choose 1.
• Next solve
• (13a1)2-70 (mod 31)
• 123a1 7 0 so a11
• Eventually, p7 (1,1,1,2,)

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What does this buy us??

• Not a great deal for integers, but
• Well use p-adic representation where p is not a
  number, but an indeterminate (say x), or a
  polynomial (x2-3). Or a polynomial in several
  variables (xy1). If we compute a gcd of 2
  polynomials p-adically approximately to high
  enough degree, we will know the exact GCD. If we
  can do this computation faster than other means,
  we have a winner. (This is the case.)

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Multiplication, usual representation

• Extremely well studied.
• The usual method takes O(n2),
• Karatsuba style O(n1.585)
• or FFT style O(n log n).
• These will be studied in the context of
  multiplying polynomials.
• Note that 345 can be mapped to p(x)3x24x5
  where p(10) is 345.
• Except for the carry, the operation is the
  same.

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Integer Division

• This is too tedious to present in a lecture.
• Techniques for guessing the next big digit
  (bigit) of a quotient within 1 are available
• For exact division, consider Newton iteration is
  an alternative
• FFT / fast multiplication helps

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GCD

• Euclids algorithm is O(n2 log n) but is hard to
  beat in practice, though see analysis of HGCD
  (Yap) for an O(n log2 n) algorithm..
• HGCD is portrayed as a winner for polynomials,
  but only by complexity analysts who (especially
  in this case) assume that
• certain costs are constant when in fact they
  grow exponentially.
• Multiplication cost n log n when, for relevant
  cases n2 algorithms are much faster than the n
  log n ones

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Reminder A Ring R is Euclidean
If there is a function y
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Shows the tendency to obfuscate
What Rings do we use, and what is y? For
integers, absolute value y For p-adic
numbers, p norm For polynomials in x, degree
in x y
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Where next?

• We could spend a semester on integer arithmetic,
  but this does not accomplish any higher goals of
  CAS
• We can proceed to build models of real numbers by
  approximation (e.g. as limit of rational
  intervals). We may return to this..
• We proceed to polynomials, typically with integer
  coefficients or finite field coeffs.

20
Interlude, regarding your homework (the last
problem)

• Usual representation in Lisp is trivial.
• 3x24 is something like ( ( 3 ( x 2) 4)
• In Lisp, is a symbol. Times is a symbol..

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Trivial parts..

• Given a, b, a program to produce a product is
• (defun prod(a b)(list a b)) common lisp
  syntax
• (define (prod a b)(list a b)) scheme
  syntax
• function make_prod(atree,btree)tree
• var temptree hm, something like
  this..
• begin
• temp new(tree) temp.head asterisk
  temp.lefta temp.rightb, make_prodtemp
• end Pascal?? Similarly for C, Java,

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Harder parts..

• If you need to know that ( x 0), ( x ( -1 x))
  and 0 are the same, how much work must you do?
• (Schwartz-Zippel polynomial identity testing)

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Reminder The Usual Coefficients

• Z natural numbers (ring, integers, - not
  closed under division!)
• Zp integers modulo p, p a prime usually.
• Q rational numbers (a field). What is the
  difference between 3/4 and 6/8?
• R real includes irrationals p 2
  ,transcendentals (e, p) mention continued fracs,
  intervals
• C complex
• approximation
• special case of algebraic extension
• Analysis functions of a complex variable

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Preview Extensions

• if x is an indeterminate, and
• if D is a domain, we can talk about Dx,
  polynomials in x with coefficients in D
• or D(x) ratios of polys in x.
• Or Dx (truncated) power series in x over D.
• Or Matrices over D.

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Beware of notation

• A particular polynomial expression might be
  referred to as p(x), which notation is also used
  to denote a function or mapping pD1?D2 or a
  function application if x is not an indeterminate
  but a element in a field as p(3). Confused? The
  notation is unfortunate. Not confused? You are
  probably used to it.

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Aside 2.1 canonical forms vs. mathematical
equivalence

• Mathematically, we dont distinguish between
  two equivalent elements in Z(x), say 1/(x1) and
  (x-1)/(x2-1).
• Computationally these can be distinguished, and
  generally they must be distinguished. Often we
  must compute a canonical form for an expression
  by finding a particular simplest form in an
  equivalence class.

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Aside 2.2 Simplification is almost everything in
this business..

• Trivial reduction. All computational problems in
  computer algebra can be reduced to
  simplification
• simplify (ProblemStatement) to CanonicalSolution

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Aside 2.3 Computer representation and canonical
forms

• A computer might distinguish between two strings
  abc and abc if they are stored in different
  locations in memory. Or might not.
• Usually it is advantageous to store an object
  only once in memory, but not always. (Should we
  store 43 just once? How about 3.141592654 ? How
  about ax2bxc?)

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Back to extensions

• if r is a root of an irreducible polynomial p,
  that is, p(r)0, we will also talk about a ring
  or field extended by r Qr. E.g. p(r)r2-10
  means r p(-1) or i, and we have just
  constructed the complex rationals Qr.
• Zi is called Gaussian integers" The set of
  elements abi, with a, b, integers.
• Qi would allow rational a, b. (remember
  rationalizing denominators?)
• Given such a field, you can extend it again.
• IF you want to represent Q extended by sqrt(2),
• and then THAT extended by sqrt(3), you can do so.
  Don't extend it again by sqrt(6). (why?)

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More on extensions

• In nice cases (primitive element), algebraic
• arithmetic can be done by "reducing" modulo r.
  This is accomplished by dividing by p(r) and
  discarding the remainder if E abp(r) then
  E a
• You may need reminders of shortcuts. e.g.
  remainder of p(x) / (x-a) is the same as
  substituting a for x in p.
• (other terms we will use on occasion Euclidean
  domains, unique factorization domains, ideals,
  differential fields, algebraic curves. Well
  motivate them when needed)

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Other extensions

• Differential fields have what amounts to log()
  and exp() extensions.
• And an operation of differentiation such that
• D(exp(x)) exp(x), D(log(x)) 1/x.
• Exp and log can be nested, and you can make trig
  functions

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Theres more Other kinds of symbolic computation

• Whole careers have been made out of other kinds
  of symbolic computation
• theorem proving, string manipulation, group
  representations, geometric computation, type
  theory/programming language representations,
  etc.
• (J. Symbolic Computation publishes broadly)
• We will not probably not get to any of these
  areas in this course, although I could be swayed
  by student interest also projects involving
  these topics are generally appropriate.