Introduction to Computer Algebra Carlos D'Andrea Oslo, December 1st 2016 Carlos D'Andrea Introduction to Computer Algebra What is Computer Algebra? Carlos D'Andrea Introduction to Computer Algebra What is Computer Algebra? Carlos D'Andrea Introduction to Computer Algebra Computer Algebra Carlos D'Andrea Introduction to Computer Algebra Computer Algebra Carlos D'Andrea Introduction to Computer Algebra Computer Algebra Carlos D'Andrea Introduction to Computer Algebra Computer Algebra Carlos D'Andrea Introduction to Computer Algebra Computer Algebra Carlos D'Andrea Introduction to Computer Algebra \...is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects..." From Wikipedia (Symbolic Computation) Carlos D'Andrea Introduction to Computer Algebra algorithms and software for manipulating mathematical expressions and other mathematical objects..." From Wikipedia (Symbolic Computation) \...is a scientific area that refers to the study and development of Carlos D'Andrea Introduction to Computer Algebra for manipulating mathematical expressions and other mathematical objects..." From Wikipedia (Symbolic Computation) \...is a scientific area that refers to the study and development of algorithms and software Carlos D'Andrea Introduction to Computer Algebra From Wikipedia (Symbolic Computation) \...is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects..." Carlos D'Andrea Introduction to Computer Algebra make it effective and to discover efficient algorithms to implement this effectiveness” From Wikipedia (Symbolic Computation) \A large part of the work in the field consists in revisiting classical algebra in order to Carlos D'Andrea Introduction to Computer Algebra From Wikipedia (Symbolic Computation) \A large part of the work in the field consists in revisiting classical algebra in order to make it effective and to discover efficient algorithms to implement this effectiveness” Carlos D'Andrea Introduction to Computer Algebra Boolean decisions:= ; =6 ; (>; <) Arithmetic Operations over \computable" rings: Z; Q; Fq; Q[x1;:::; xn];::: Finite-Dimensional Linear Algebra “Effective” Operations Carlos D'Andrea Introduction to Computer Algebra Arithmetic Operations over \computable" rings: Z; Q; Fq; Q[x1;:::; xn];::: Finite-Dimensional Linear Algebra “Effective” Operations Boolean decisions:= ; =6 ; (>; <) Carlos D'Andrea Introduction to Computer Algebra Finite-Dimensional Linear Algebra “Effective” Operations Boolean decisions:= ; =6 ; (>; <) Arithmetic Operations over \computable" rings: Z; Q; Fq; Q[x1;:::; xn];::: Carlos D'Andrea Introduction to Computer Algebra “Effective” Operations Boolean decisions:= ; =6 ; (>; <) Arithmetic Operations over \computable" rings: Z; Q; Fq; Q[x1;:::; xn];::: Finite-Dimensional Linear Algebra Carlos D'Andrea Introduction to Computer Algebra Given A; B 2 Z;compute the product A · B 2 Z What is \an algorithm" for CA? Carlos D'Andrea Introduction to Computer Algebra compute the product A · B 2 Z What is \an algorithm" for CA? Given A; B 2 Z; Carlos D'Andrea Introduction to Computer Algebra What is \an algorithm" for CA? Given A; B 2 Z;compute the product A · B 2 Z Carlos D'Andrea Introduction to Computer Algebra What is \an algorithm" for CA? Given A; B 2 Z;compute the product A · B 2 Z Carlos D'Andrea Introduction to Computer Algebra Output: A · B 2 Z Procedure: Example: The Multiplication Algorithm Input: A; B 2 Z Carlos D'Andrea Introduction to Computer Algebra Procedure: Example: The Multiplication Algorithm Input: A; B 2 Z Output: A · B 2 Z Carlos D'Andrea Introduction to Computer Algebra Example: The Multiplication Algorithm Input: A; B 2 Z Output: A · B 2 Z Procedure: Carlos D'Andrea Introduction to Computer Algebra Effective: the procedure must finish after a finite number of operations, and give the right answer Efficient: the procedure must be as short as possible and use as less space as possible Effective and Efficient Carlos D'Andrea Introduction to Computer Algebra the procedure must finish after a finite number of operations, and give the right answer Efficient: the procedure must be as short as possible and use as less space as possible Effective and Efficient Effective: Carlos D'Andrea Introduction to Computer Algebra and give the right answer Efficient: the procedure must be as short as possible and use as less space as possible Effective and Efficient Effective: the procedure must finish after a finite number of operations, Carlos D'Andrea Introduction to Computer Algebra Efficient: the procedure must be as short as possible and use as less space as possible Effective and Efficient Effective: the procedure must finish after a finite number of operations, and give the right answer Carlos D'Andrea Introduction to Computer Algebra the procedure must be as short as possible and use as less space as possible Effective and Efficient Effective: the procedure must finish after a finite number of operations, and give the right answer Efficient: Carlos D'Andrea Introduction to Computer Algebra and use as less space as possible Effective and Efficient Effective: the procedure must finish after a finite number of operations, and give the right answer Efficient: the procedure must be as short as possible Carlos D'Andrea Introduction to Computer Algebra Effective and Efficient Effective: the procedure must finish after a finite number of operations, and give the right answer Efficient: the procedure must be as short as possible and use as less space as possible Carlos D'Andrea Introduction to Computer Algebra of N digits Output: A · B 2 Z Output's size is around2 N digits High school algorithm takes around O(N2) operations It is effective, but is it efficient? 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Short and less Space Input: A; B 2 Z of N digits Output: A · B 2 Z Output's size is around2 N digits High school algorithm takes around O(N2) operations Carlos D'Andrea Introduction to Computer Algebra , but is it efficient? Short and less Space Input: A; B 2 Z of N digits Output: A · B 2 Z Output's size is around2 N digits High school algorithm takes around O(N2) operations It is effective Carlos D'Andrea Introduction to Computer Algebra Short and less Space Input: A; B 2 Z of N digits Output: A · B 2 Z Output's size is around2 N digits High school algorithm takes around O(N2) operations It is effective, but is it efficient? Carlos D'Andrea Introduction to Computer Algebra Efficient Multiplication Karatsuba's algorithm (1960): O(N1:585) Carlos D'Andrea Introduction to Computer Algebra Sch¨onhage-Strassen'salgorithm (1971): O(N log N log log N) Efficient Multiplication Toom-Cook's algorithm (1966): O(Nlog(2k−1)= log k); k ≥ 3 Carlos D'Andrea Introduction to Computer Algebra Efficient Multiplication Toom-Cook's algorithm (1966): O(Nlog(2k−1)= log k); k ≥ 3 Sch¨onhage-Strassen'salgorithm (1971): O(N log N log log N) Carlos D'Andrea Introduction to Computer Algebra Is it the most efficient?? Efficient Multiplication F¨urer'salgorithm (2007): O(N log N 2O(log∗ N))) Carlos D'Andrea Introduction to Computer Algebra Efficient Multiplication F¨urer'salgorithm (2007): O(N log N 2O(log∗ N))) Is it the most efficient?? 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