Number of Ways to Multiply a Chain of M Matrices understanding the Problem: When multiplying a chain of matrices, the order in which we perform the multiplications can significantly impact the number of scalar multiplications required. This is because matrix multiplication is not commutative. The Catalan Number Connection: The number of ways to parenthesize a chain of M matrices is given by the Catalan number C(M-1) Catalan Numbers: Catalan numbers are a sequence of natural numbers that occur in many counting problems. The nth Catalan number, denoted as C_n, can be calculated using the following formula: C_n (1/(n+1)) (2n choose n) where (2n choose n) is a binomial coefficient. Why Catalan Numbers? Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial problem solved by Catalan numbers. Example: Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways: ((ABC)D (ABC))D ABC)D) A(BICD)) (ABCD) There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5. In Summary: The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number, C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal parenthesization to minimize the number of scalar multiplications.
Number of Ways to Multiply a Chain of M Matrices understanding the Problem: When multiplying a chain of matrices, the order in which we perform the multiplications can significantly impact the number of scalar multiplications required. This is because matrix multiplication is not commutative. The Catalan Number Connection: The number of ways to parenthesize a chain of M matrices is given by the Catalan number C(M-1) Catalan Numbers: Catalan numbers are a sequence of natural numbers that occur in many counting problems. The nth Catalan number, denoted as C_n, can be calculated using the following formula: C_n (1/(n+1)) (2n choose n) where (2n choose n) is a binomial coefficient. Why Catalan Numbers? Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial problem solved by Catalan numbers. Example: Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways: ((ABC)D (ABC))D ABC)D) A(BICD)) (ABCD) There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5. In Summary: The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number, C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal parenthesization to minimize the number of scalar multiplications.
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter2: Basic Linear Algebra
Section2.1: Matrices And Vectors
Problem 6P
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Transcribed Image Text:Number of Ways to Multiply a Chain of M Matrices
understanding the Problem:
When multiplying a chain of matrices, the order in which we perform the multiplications can
significantly impact the number of scalar multiplications required. This is because matrix
multiplication is not commutative.
The Catalan Number Connection:
The number of ways to parenthesize a chain of M matrices is given by the Catalan number
C(M-1)
Catalan Numbers:
Catalan numbers are a sequence of natural numbers that occur in many counting problems. The
nth Catalan number, denoted as C_n, can be calculated using the following formula:
C_n (1/(n+1)) (2n choose n)
where (2n choose n) is a binomial coefficient.
Why Catalan Numbers?
Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex
polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial
problem solved by Catalan numbers.
Example:
Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways:
((ABC)D
(ABC))D
ABC)D)
A(BICD))
(ABCD)
There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5.
In Summary:
The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number,
C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal
parenthesization to minimize the number of scalar multiplications.
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