By Joel S. Cohen
Mathematica, Maple, and related software program applications supply courses that perform refined mathematical operations. utilising the guidelines brought in laptop Algebra and Symbolic Computation: user-friendly Algorithms, this e-book explores the appliance of algorithms to such equipment as automated simplification, polynomial decomposition, and polynomial factorization. This publication comprises complexity research of algorithms and different contemporary advancements. it really is well-suited for self-study and will be used because the foundation for a graduate path. retaining the fashion set through trouble-free Algorithms, the writer explains mathematical tools as wanted whereas introducing complex easy methods to deal with complicated operations.
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Additional info for Computer Algebra and Symbolic Computation: Mathematical Methods
By mathematical convention, the degree of the 0 monomial is −∞. If u is a GPE in xi that is a sum of monomials, then deg(u, xi ) is the maximum of the degrees of the monomials. If the generalized variable xi is understood from context, we use the simpler notation deg(u). The MPL operator Degree gpe(u, x) returns deg(u, x). For example, Degree gpe(sin2 (x)+b sin(x)+c, sin(x)) → 2. • Coeﬃcient gpe(u, x, j). Let u be a GPE in x, and let j be a nonnegative integer. The MPL operator Coeﬃcient gpe(u, x, j) returns the sum of the coeﬃcient parts of all monomials of u with variable part xj .
2. Let b, c > 0, and m = 0 be integers. (a) Show that irem(c b, c m) = c · irem(b, m). (b) Show that the relationship in Part (a) may not hold if c < 0. 3. Let m be an integer. Show that iquot(m, 2) + iquot(m − 1, 2) + 1 = m. 4. 4. 4. An MPL procedure that obtains the solution to the system of remainder equations described in the Chinese remainder theorem. ) 5. Let u be a rational number. The ﬂoor of u (notation u ) is the largest integer ≤ u. The ceiling of u (notation u ) is the smallest integer ≥ u.
The proof is obtained with mathematical induction on the number of equations r. For the base case r = 1, integer division shows that x = x1 is a solution to the ﬁrst remainder equation. For the induction step, let’s assume there is an integer s that satisﬁes the remainder equations irem(s, mi ) = xi , i = 1, . . 44). 45) implies i = 1, . . 46) s = qi mi + xi , where qi = iquot(s, mi ). 29) implies gcd(n, mr ) = 1, and using the extended Euclidean algorithm, we obtain integers c and d such that c n + d mr = 1.