By Joel S. Cohen
Mathematica, Maple, and related software program applications supply courses that perform subtle mathematical operations. using the information brought in laptop Algebra and Symbolic Computation: user-friendly Algorithms, this ebook explores the applying of algorithms to such equipment as computerized simplification, polynomial decomposition, and polynomial factorization. This publication contains complexity research of algorithms and different contemporary advancements. it's well-suited for self-study and will be used because the foundation for a graduate direction. conserving the fashion set by means of common Algorithms, the writer explains mathematical tools as wanted whereas introducing complex ways to deal with advanced operations.
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Additional info for Computer Algebra and Symbolic Computation: Mathematical Methods
This is the only place in the algorithm that involves a gcd calculation. 5 The factorial operator FactOp is not used in this section. It is used, however, in the simpliﬁer in Chapter 3. 7. The MPL Simplify RNE procedure. 8, the procedure Simplify RNE rec(u) determines the type of u (integer, FracOp, SumOp, DiﬀOp, ProdOp, or QuotOp) and then performs the appropriate operation. The unary operations are handled in lines 5–9, while the binary operations are handled in lines 10–23. 6 The actual calculations are obtained with the calls to procedures in lines 9, 16, 17, 18, 19, and 23.
A) Show that Ri = fn−(i+1) for i = −1, 0, . . , ρ − 1 and that the remainder sequence terminates with ρ = n − 1 and gcd(fn , fn−1 ) = 1. (b) For this sequence, what is the relation between mi and ni in the extended Euclidean algorithm and the Fibonacci numbers? 12. 8(3). 13. Suppose that a, b, u, and v are integers and a = u gcd(a, b), b = v gcd(a, b). Show that gcd(u, v) = 1. 14. Suppose that gcd(a, b) = 1 and t is a positive integer. Show that gcd(at , bt ) = 1. 15. Let a, b, and c be integers.
B) Denominator fun(u). (c) Evaluate sum(v, w). (d) Evaluate diﬀerence(v, w). (e) Evaluate product(v, w). The output of Numerator fun and Denominator fun is an integer, and the output for Evaluate sum, Evaluate diﬀerence, and Evaluate product is a fraction in function notation. 3 Fields A ﬁeld is a general mathematical system with axioms that describe the basic algebraic properties of number systems and other classes of expressions that arise in computer algebra. In this section we give a formal deﬁnition of a ﬁeld, give a number of examples, and show that many transformations routinely used in manipulations are logical consequences of the ﬁeld axioms.
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