Ask Question, Ask an Expert

+1-415-315-9853

info@mywordsolution.com

Ask Math Expert


Home >> Math

Q1) This problem is about determining when Strassen's Algorithm is faster than ordinary matrix multiplication, assuming that scalar addition and scalar multiplication take the same amount of time. Show all of your workings. Don't use the Master Method in this problem.

(i) describe why ordinary matrix multiplication of two n by n matrices involves 2n3-n2 calculations. (Multiplication+Addition).

(ii) In Strassen's Algorithm, T(n), the total number of calculations (Multiplication+Addition) involved in multiplying two n by n matrices satisfies T(n) = 7T(n/2 ) + 9/2n2 given T(1) = 1. Suppose n = 2m. By considering the transformation G(m) = T(2m) obtain a recursion equation for G(m) along with a condition on G(0).

(iii) Show that the solution to the recursion equation is G(m) = 7m+1- 6 *4m.

(iv) Hence show that T(n) = 7nlog27 -6n2.

(v) Determine the smallest value of m for which Strassen's Algorithm involves fewer calculations than ordinary matrix multiplication.

Q2) In this problem you will need to determine which of the 5 algorithms V to Z is quickest at solving a problem of input size n. The running time for each algorithm satisfies its own recursion equation given by:
Algorithm V: V (n) = 3V (n-1) + 4n with V (1) = 1
Algorithm W: W(n) = 3W(n-1) + 10n2 + 20n + 40 with W(1) = 1
Algorithm X: X(n) = 3X(n -1) + 3n with X(1) = 1
Algorithm Y: Y (n) = 5Y (n - 1) - 4Y (n- 2) + 6n - 12 with Y (1) = 1; Y (2) = 2
Algorithm Z: Z(n) = 8Z(n - 1) - 15Z(n - 2) + 2n with Z(1) = 1; Z(2) = 2

(i) Obtain a closed form solution for the running time for each algorithm.
(These should be derived by hand without using Maple).

(ii) Use Maple to plot a suitable function of the running time against n for each algorithm on the same graph to illustrate how the choice of the most efficient algorithm changes with n. (Plotting multiple graphs may help to illustrate your argument).

(iii) By considering the running time for algorithm Z, simplify this expression 5n - 5(3n) + 2n+3 mod 6 for n   1
(Just state the answer and say why - no need to prove anything)

Q3) The running times of algorithms J, K, L and M are given by

Algorithm J: J(n) = 3J(n/4 ) + n with J(1) = 1

Algorithm K: K(n) = 5K(n/2 ) + n2 with K(1) = 1

Algorithm L: L(n) = 6L(n/6 ) + n ln n with L(3) = 1

Algorithm M: M(n) = 2M(n/5 ) +M(n/3 ) + √n with M(1) = 1

(i) describe how algorithms K & M divide up a problem of size n.
(ii) Sketch a recursion tree for algorithm J.
(iii) Using the Master Method, determine the asymptotic running times for each of the algorithms J, K & L.
(iv) Prove the 3 results that you obtained in (iii) are upper bounds on the running times of algorithms J, K & L, using proof by induction.
(v) Consider algorithm M and suppose p > 0.5 such that M(n) = O(np).
(a) By seeking the tightest possible upper bound on M determine p.
(b) Using proof by induction show that M(n) <=cnp + d √n where c and d are chosen to make this upper bound as tight as possible.
(vi) Which of the algorithms J, K, L and M is asymptotically fastest?

Q4) This problem is about solving nonlinear equations. You should ignore repeated roots. You cannot use any of Maple's built in `solve' type commands.

(i) prepare a Maple procedure to numerically solve the coupled system of equations f(x; y) = 0 and g(x; y) = 0 with initial guesses x=x0 and y=y0 by extending the Newton-Raphson method to a system of 2 nonlinear equations.
(Use  nite di erences to evaluate the partial derivatives).

(ii) Using an initial guess of x0 = 1 and y0 = -1 test your procedure by obtaining one
solution, accurate to 5 decimal places, to this nonlinear system:
sin(5x) + 9- 3y2 = 0
e7y-4x + 6x- 8 = 0

Q5) In this problem you are going to prepare a Maple program to determine the centre and radius of the smallest circle that encloses a set of points.
(i) Create a set A of 20 points located in a two dimensional space. The x and y coordinates of each point should be randomly assigned such that |x| <=100 and |y| <=100.

(ii) Determine the maximum and minimum values of x and y over the points in A. Define x0 = 1/2(max(x) - min(x)) and y0 = 1/2(max(y)- min(y)).
(iii) find out the radius, r0, of a circle centred at the point (x0; y0) that encloses the set of points in A. Define dx = 1/10r0.

(iv) Redefine dx as dx=1:1. By considering the resulting radius for each of the points (x0- dx; y0), (x0, y0), (x0 + dx, y0), (x0, y0- dx) and (x0, y0 + dx) choose a new value of x0 and y0 to reduce the radius.
(If none of the four new points reduces the radius, then leave x0 and y0 unchanged.)

(v) Iterate step (iv) 30 times.

(vi) Plot the points in set A along with a circle centred at the point (x0, y0) of the appropriate radius on the same graph.

(vii) Discuss any problems associated with successively reducing the value of dx.

Math, Academics

  • Category:- Math
  • Reference No.:- M9897

Have any Question? 


Related Questions in Math

How much money should be deposited today in an account that

How much money should be deposited today in an account that earns 8% compounded quarterly so that it will accumulate to $20,000 in six years?

As you prepare to write this weeks discussion post first

As you prepare to write this week's discussion post, first review the article: How to Neutralise Aggressive Negotiators Tactics.  Then, fill out the Negotiation Gambits Chart. Include the following information: Identify ...

Sund intensity known as decibels d is calculated usingd

Sound intensity, known as decibels, d, is calculated using:d = 9.8log IaI is the intensity per unit of area. a is faintest audible sound of the average human ear. Find the decibels of a motorcycle that has a sound intens ...

In a set of 105act scores where the mean is 22and the

In a set of 105ACT scores, where the mean is 22and the standard deviation is 5.39,(a) how many scores are expected to be lower than 16.61(one standard deviation below the mean)? (b) How many of the 105scores are expected ...

1nbspyou deposit 6000 in an account earning 6 interest

1. You deposit $6000 in an account earning 6% interest compounded monthly. How much will you have in the account in 10 years? 2. You deposit $3000 in an account earning 8% interest compounded  continuously . How much wil ...

The average number of prisoners in a county jail is modeled

The average number of prisoners in a county jail is modeled by y = 0.15x 2  + 10.34x + 122.54, where x is the number of years after 2000. a. Graph this function from 0 to 30. b. Approximate, to the nearest whole number, ...

Use successive differences to classify the function

Use successive differences to classify the function represented in the table as linear, quadratic, or other. Show all your work. x -3 -2 -1 0 1 2 3 g(x) -29 -10 -3 -2 -1 6 25

Think of a recent event in the news where an ethical issue

Think of a recent event in the news where an ethical issue or social responsibility was highlighted. Describe the event, and share your thoughts on how it was handled by the individual or business.

Mpl project- pharmaceutical clinical trial distribution a

MPL Project- Pharmaceutical Clinical Trial Distribution A pharmaceutical company, named "LongerLife", plans to deliver clinical trials to three different clinics (C1, C2, and C3). The trials are used for the emergency tr ...

Analyzenbspit collaboration technology and ecommerce

Analyze  IT collaboration technology and eCommerce techniques that can be applied to your project based on this week's readings ( Essentials of MIS , Chs. 9 and 10, and Week 3 Electronic Reserve Readings).  Create  a Mic ...

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro

Describe what you learned about the impact of economic

Describe what you learned about the impact of economic, social, and demographic trends affecting the US labor environmen