# Modern Computer Algebra

# Modern Computer Algebra

## Joachim von zur Gathen, Jürgen Gerhard

Language: English

Pages: 811

ISBN: B00CMNTKZ4

Format: PDF / Kindle (mobi) / ePub

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

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possible. The early Roman calendar was of a mixed lunisolar type. It consisted of originally 10 and later 12 months, and occasionally one extra month was added in order to keep in step with the seasons. The Julian calendar, named after Julius Caesar (and which had been invented by the Egyptian Sosigenes), started on 1 January 45 BC. Since the Romans before Caesar had badly neglected the management of the calendar, the year 46 BC, the annus confusionis, had 445 days! Caesar used 84 4.

• •• •••••••••••••••••••••• ✲ • 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 n • Error in the nth approximation small, where log is the binary logarithm. This is a Diophantine approximation problem and best solved by the continued fraction expansion of log ri . For log r5 = log(6/5) = 0.2630344058 . . ., we find from the continued fraction expansion of Table 4.1 the approximations 1 3 1 4 5 19 111 422 = 0.3333333333 . . . , = 0.2500000000, = 0.2631578947 . . . , = 0.2630331753 . . . . The

terminology, the theoretical basis for many of our algorithms. 5.4. The Chinese Remainder Algorithm 105 T HEOREM 5.2. χ is surjective with kernel m . P ROOF. Let f ∈ R. Then f ∈ ker χ ⇐⇒ χ( f ) = ( f mod m0 , . . . , f mod mr−1 ) = (0, . . . , 0) ⇐⇒ mi | f for 0 ≤ i < r ⇐⇒ lcm(m0 , . . . , mr−1 ) | f ⇐⇒ m | f , and ker χ = m . For the surjectivity, it is sufficient to show that a “Lagrange interpolant” li ∈ R with χ(li ) = ei exists for 0 ≤ i < r, where ei = (0, . . . , 0, 1, 0, . . . , 0) ∈

terminology, the theoretical basis for many of our algorithms. 5.4. The Chinese Remainder Algorithm 105 T HEOREM 5.2. χ is surjective with kernel m . P ROOF. Let f ∈ R. Then f ∈ ker χ ⇐⇒ χ( f ) = ( f mod m0 , . . . , f mod mr−1 ) = (0, . . . , 0) ⇐⇒ mi | f for 0 ≤ i < r ⇐⇒ lcm(m0 , . . . , mr−1 ) | f ⇐⇒ m | f , and ker χ = m . For the surjectivity, it is sufficient to show that a “Lagrange interpolant” li ∈ R with χ(li ) = ei exists for 0 ≤ i < r, where ei = (0, . . . , 0, 1, 0, . . . , 0) ∈

that the Bézier spline is continuously differentiable with respect to the parameter u if and only if Bi − Ai = Ai −Ci−1 for 1 ≤ i < n. Show that in this case, Bi − Ai is the tangent vector to the curve at the point Ai for 0 ≤ i < n. What is the tangent vector at the point An ? (iii) Consider the following control points for n = 4: i Ai Bi Ci 0 (−1.8, 0) (−1.8, 2) (−3.8, 5.83) 1 (−3.8, 8.5) (−3.8, 11.17) (−2, 12.8) 2 (0, 12.8) (2, 12.8) (3.8, 11.17) 3 (3.8, 8.5) (3.8, 5.83) (1.8, 2) 4 (1.8,