Overview of Computer Algebra http://cocoa.dima.unige.it/ J. Abbott Universität Kassel J. Abbott Computer Algebra Basics Manchester, July 2018 1 / 12
Intro Characteristics of Computer Algebra or Symbolic Computation exact arithmetic (integers, rationals, algebraic numbers) polynomials (non-linear expressions with symbols), rational functions, matrices, ideals,... algebraic extensions (e.g. exact repr. of 2) inequalities, differential equations CoCoA cannot do this. Contrast with so-called Numerical Computing: faster but results are only approximate. good hardware support relatively easy to parallelize uniform data size J. Abbott Computer Algebra Basics Manchester, July 2018 2 / 12
Exact Arithmetic Pros and cons of exact arithmetic: (integer, rational, alg. extn) Pro: answer is correct Con: computation can be slow Example: let S = 1 + 1 2 + 1 3 + + 1 264. Compare S to 384211/62422 floating-point is insufficient. Example: let M be a 100 100 matrix with random integer entries (from 99 to +99). Let V be a vector with random integer entries. Solve M X = V. Takes about 2 sec. each entry in X has denominator with about 250 digits! J. Abbott Computer Algebra Basics Manchester, July 2018 3 / 12
Exact Arithmetic Linear Systems exact solution (also overdetermined or underdetermined) kernel basis, Hilbert basis (of non-neg kernel) det, rank, inverse, eigenvectors (and eigenfactors) No problems with ill-conditioning; not as fast as floating-point; answer may be cumbersome. J. Abbott Computer Algebra Basics Manchester, July 2018 4 / 12
Polynomials: univariate Univariate polynomials polynomials in 1 indeterminate Q[x] = the ring of polynomials in x with coefficients in Q. Examples: x, 3x 3 10x + 1 creation, basic arithmetic Q[x] GCD factorization square-free factors (cf. radical) irreducible factors coprime factor base (or GCD-free-basis) count real roots isolate and approximate real roots CoCoA cannot do: polynomial decomposition, isolate/approximate complex roots. J. Abbott Computer Algebra Basics Manchester, July 2018 5 / 12
Polynomials: multivariate Multivariate polynomials polynomials in many indeterminates Q[x, y, z] = the ring of polys in x, y, z with coeffs in Q. Examples: x, 3x 3 y 10xz 2 + y + 1 Q[x, y, z] Same operations (except for real roots ). Different ways to write one polynomial: x + y and y + x. Important extra ingredient, term-ordering: total ordering on power-products 1 = x 0 y 0 z 0 is smallest element compatible with multiplication: t 1 < t 2 = t t 1 < t t 2 Infinitely many term-orderings; CoCoA default is DegRevLex J. Abbott Computer Algebra Basics Manchester, July 2018 6 / 12
Polynomial systems Polynomial system: non-empty set of polynomials which must simultaneously vanish. Usually interested in the zero set: i.e. all x, y, z values such that f 1 (x, y, z) = 0 f 2 (x, y, z) = 0 f k (x, y, z) = 0 Poly system is zero-dim if the zero set is finitely many points. Example: many equivalent polynomial systems: {f 1, f 2, f 3,...} {f 1 3f 2 8f 3, f 2, f 3,...} {f 1 (f 2 + 1), f 2, f 3...} J. Abbott Computer Algebra Basics Manchester, July 2018 7 / 12
Polynomial systems Mathematical concept ideal = all equivalent polynomial systems. Definition of ideal generated by f 1, f 2,..., f k R: { } f 1, f 2,..., f k = gj f j : g j R An ideal is infinite, but noetherianity ensures there are finite bases. Example xy 1, x 2 + y 2 1 = x + y 3 y, y 4 y 2 + 1 Example: polynomial systems {x 2, y} and {x, y} are not equiv but have same zero set. Ideals x 2, y and x, y are different! Radical ideals correspond 1 1 to (complex) zero sets. J. Abbott Computer Algebra Basics Manchester, July 2018 8 / 12
Polynomial systems Gröbner basis: computationally useful ideal basis: depends on term-ordering explicit indication if system is unsolvable (over C) membership testing, solving, elimination, etc. radical membership (usu. faster than computing the radical) can be big and costly Zero-dimensional ideals: useful special case relatively fast (once G-basis has been computed) can compute exact/approximate solutions J. Abbott Computer Algebra Basics Manchester, July 2018 9 / 12
Algebraic Extensions An algebraic number is a root of a polynomial in Q[x]. Example: 3, 4 7, 2 and α such that α 3 3α + 4 = 0. Represent 2 as a new symbol satisfying ( 2) 2 2 = 0. Define algebraic extension Q( 2) a field! Typical element is a + b 2 with a, b Q. Example of division: 1 3 4 2 = 3 4 2 23 result is exact, and valid for both values of 2; also multiple algebraic extensions e.g. 2 and 3 3 together no need for approximation 2 1.414213562373 J. Abbott Computer Algebra Basics Manchester, July 2018 10 / 12
Heuristics CoCoA offers some features for heuristic programming : Time-out mechanism Time-out useful for Gröbner basis computation: computation might be quick, and if so, result is useful. useful for convergent algorithms (e.g. eval poly over interval) VerificationLevel some algorithms get probably right answer quickly, but checking is expensive. Examples: IsProbPrime, determinant, solving linear system. Twin-float arithmetic compromise between floating-point and rational arithmetic J. Abbott Computer Algebra Basics Manchester, July 2018 11 / 12
The End The End J. Abbott Computer Algebra Basics Manchester, July 2018 12 / 12