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Use Strassen’s algorithm to compute the matrix product Show your work.
4.5-2 Professor Caesar wishes to develop a matrix-multiplication algorithm that is asymptotically faster than Strassen’s algorithm. His algorithm will use the divide- and-conquer method, dividing each matrix into pieces of size n/4 x n/4, and the divide and combine steps together will take O(n) time. He needs to determine how many subproblems his algorithm has to create in order to beat Strassen’s algo- rithm. If his algorithm creates a subproblems, then the recurrence for the running time T(n) becomes T(n)...
The problem: Compute AB, where A and B are both n×n matrices and n is a positive integer. The algorithms: standard matrix multiplication algorithm; a simple recursive algorithm; Strassen’s algorithm. Your task: Explain which of these three algorithms for this problem is fastest (asymptotically, in the worstcase). Explain how it achieves a performance increase over the other algorithms.
3. Use Euclid's algorithm to compute the following. Show all your steps 1. gcd(781, 994) 2. gcd(67457, 43521)
Is the following setup safe or unsafe according to the Banker's algorithm? Show your work and explain your result. R1 R2 R3 R1 R2 R3 P1 3 2 1 P1 2 1 0 P2 2 3 1 P2 0 1 1 P3 1 2 3 P3 0 0 1 Claim matrix Allocation matrix R1 R2 R3 R1 R2 R3 3 3 3 1 1 1 Resource vector Available vector
3. (a) Compute the following matrix-vector product. Express your answer in terms of a, b: a 2 090 1 (b) Use your answer from (a) to find the values of a, b, c such that: 620-(2) 1
Use dynamic programming algorithm to compute the binary coefficient C(7,5). Show all steps!
Show all work. Determine the matrix Need and the Available matrix for each step. Answer the following questions using the Banker’s algorithm: Allocation Max Available ABCD ABCD ABCD P0 0112 3412 2222 P1 1000 5755 P2 1354 7354 P3 3281 3682 P4 0222 1222 A. What is the content of the matrix Need? B. Is the system in a safe state?
Use the dynamic programming technique to find
an optimal parenthesization of a matrix-chain product whose
sequence of dimensions is <5, 8, 4, 10, 7, 50, 6>.
Matrix Dimension
A1 5*8
A2 8*4
A3 4*10
A4 10*7
A5 7*50
A6 50*6
You may do this either by implementing the MATRIX-CHAIN-ORDER
algorithm in the text or by simulating the algorithm by hand. In
either case, show the dynamic programming tables at the end of the
computation.
Using Floyd’s algorithm (See Dynamic Programming...
Let X be a kn × n matrix and Y by an n × kn matrix, for some integer k. (a) Describe an algorithm that computes the product XY using Strassen's algorithm as a subroutine, i.e., use it as a black-box without modi- fying it. Only describe your algorithm in words; pseudo-code is not required. Justify your answer, i.e., argue that your algorithrn does compute XY correctly. Establish its running time.
Work the following problem using the algorithm we covered in class. Show your work by listing all the steps, Use Example 13.8 on page 706. For the following reaction at 600 °C: 2SO2 (g) + O2 (g) ⇌ 2SO3(g) Kc = 4.32 What are the equilibrium concentrations of all species in a mixture that was prepared with [SO3] = 0.500 M, [SO2] = 0 M, and [O2] = 0.350 M?