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Basic Domains of Interest used in Computer Algebra Systems

Basic Domains of Interest used in Computer Algebra Systems. Lecture 2b. Some ideas just for representing integers. Integers are sequences of characters, 0..9. Integers are sequences of words modulo 10 9 which is the largest power of 10 less than 2 31 .

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Basic Domains of Interest used in Computer Algebra Systems

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  1. Basic Domains of Interest used in Computer Algebra Systems Lecture 2b Richard Fateman CS 282 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 Richard Fateman CS 282 Lecture 2

  3. 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. Richard Fateman CS 282 Lecture 2

  4. Yet more ideas Integers are stored in redundant form a+b+… 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, … Richard Fateman CS 282 Lecture 2

  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 = 1000+1+0.1 • 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. Richard Fateman CS 282 Lecture 2

  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. Richard Fateman CS 282 Lecture 2

  7. Modular arithmetic is really fast… all the arithmetic can be done without carry, in parallel. Not usually used because • You can’t 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. Richard Fateman CS 282 Lecture 2

  8. What’s a p-adic integer? For the moment assume p is a prime number. Consider representing an integer a = a0+a1.p+a2¢ p2 + … where ai are chosen from integers in the range 0 · ai < p For any finite positive integer a, either all the ak are zero in which case a = 0 and ||a||p is 0, or some initial set of the ai are zero. Let ar+1 be the first non-zero term. ||a||p = p-r. This replaces the absolute-value valuation |a| where we consider that x and y are close if x=y mod pk for many values of k=0,1,2,…. Richard Fateman CS 282 Lecture 2

  9. p-adic ordering is odd. 14 3-adically is 2*30+1*31+1*32= 2 + 3 + 9 = 14 Which is very close to 5, 3-adically because 5 is 2*30+1*31, and they are the same modulo 30 and 31. ||15-4||3 = 3-1 What about negative numbers? pk-1 is possible, but consider pk-1 = (p-1)¢(pk-1+pk-2…+p+1)= (p-1)*1+(p-1)*p+(p-1)*p2+… so -1 3-adically is 2*30+2*31+…. Infinite number of terms (2,2,2,2,2,….) Richard Fateman CS 282 Lecture 2

  10. How to multiply p-adically (a,b,c) (d,e,f)  Compute a*d , a*e, a*f; add columns b*d, b*e; b*f c*d, c*e, c*f Add modulo p, p2, p3 … with a carry. Richard Fateman CS 282 Lecture 2

  11. How to multiply p-adically approximately (a,b,c …) (d,e,f …)  Compute a*d , a*e, a*f; add columns b*d, b*e; b*f c*d, c*e, c*f 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 Richard Fateman CS 282 Lecture 2

  12. What other p-adic numbers are there?? -1/2 3-adically .. is 1*30+1*31+1*32= (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 ? (a0+3*a1+…)2-7=0 (mod 3i+1) Mod 30, a02-7=0 so a0 is 1 or –1. Let’s choose 1. Next solve (1+3*a1+…)2-7=0 (mod 31) = 1+2*3*a1 –7 =0 so a1=1 Eventually, p7 = (1,1,1,2,…) Richard Fateman CS 282 Lecture 2

  13. What does this buy us?? Not a great deal for integers, but… We’ll use p-adic representation where p is not a number, but an indeterminate (say x), or a polynomial (x^2-3). Or a polynomial in several variables (x+y+1). 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.) Richard Fateman CS 282 Lecture 2

  14. 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)=3x2+4x+5 where p(10) is 345. Except for the “carry”, the operation is the same. Richard Fateman CS 282 Lecture 2

  15. 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 Richard Fateman CS 282 Lecture 2

  16. GCD • Euclid’s 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 Richard Fateman CS 282 Lecture 2

  17. Reminder… A Ring R is Euclidean If there is a function y Richard Fateman CS 282 Lecture 2

  18. 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 Richard Fateman CS 282 Lecture 2

  19. 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. Richard Fateman CS 282 Lecture 2

  20. Interlude, regarding your homework (the last problem) • Usual representation in Lisp is trivial. • 3*x^2+4 is something like (+ (* 3 (^ x 2) 4) • In Lisp, * is a symbol. Times is a symbol.. Richard Fateman CS 282 Lecture 2

  21. 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(a:tree,b:tree):tree var temp:^tree; {hm, something like this..} begin temp:= new(tree); temp^.head:= asterisk; temp^.left:=a; temp^.right:=b, make_prod:=temp end {Pascal?? Similarly for C, Java, …} Richard Fateman CS 282 Lecture 2

  22. 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) Richard Fateman CS 282 Lecture 2

  23. 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 Richard Fateman CS 282 Lecture 2

  24. Preview: Extensions if x is an indeterminate, and if D is a domain, we can talk about D[x], polynomials in x with coefficients in D or D(x) ratios of polys in x. Or D[[x]] : (truncated) power series in x over D. Or Matrices over D. Richard Fateman CS 282 Lecture 2

  25. 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 p:D1D2or 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. Richard Fateman CS 282 Lecture 2

  26. Aside 2.1: canonical forms vs. mathematical equivalence Mathematically, we don’t distinguish between two equivalent elements in Z(x), say 1/(x+1) 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. Richard Fateman CS 282 Lecture 2

  27. 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 Richard Fateman CS 282 Lecture 2

  28. 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 ax2+bx+c?) Richard Fateman CS 282 Lecture 2

  29. 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: Q[r]. E.g. p(r)=r2-1=0 means r = p(-1) or i, and we have just constructed the complex rationals Q[r]. Z[i] is called “Gaussian integers" The set of elements a+bi, with a, b, integers. Q[i] 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?) Richard Fateman CS 282 Lecture 2

  30. 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 = a+b*p(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. We’ll motivate them when needed) Richard Fateman CS 282 Lecture 2

  31. 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: Richard Fateman CS 282 Lecture 2

  32. There’s 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. Richard Fateman CS 282 Lecture 2

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