ECE 204 Numerical Methods for Computer Engineers MIDTERM EXAMINATION /4:30-6:00

Transcription

1 ECE 4 Numerical Methods for Computer Engineers ECE 4 Numerical Methods for Computer Engineers MIDTERM EXAMINATION --7/4:-6: The eamination is out of marks. Instructions: No aides. Write your name and student ID number on each booklet. Turn off all electronic media and store them under your desk. Write all your answers in your booklets. Answer the questions order in which they appear on the eamination. If you need to write your answer to a question elsewhere as a result of space considerations, please indicate this. You may ask only two questions during the eamination:. May I go to the washroom?. May I have another booklet? At the end of the eam, place the eam paper and any additional booklets into the first booklet. Do not leave during the first minutes of the eamination. Do not leave during the last minutes of the eamination. Do not stand up until all eams have been picked up. Attention: The questions are in the order of the course material, not in order of difficulty. I have read and understood all of these instructions: Name: Signature: Page of 4

2 ECE 4 Numerical Methods for Computer Engineers Error Analysis and Numeric Representation. [] Multiply the two binary numbers. and.. Round your answer to bits.. [] Using decimal digits of precision for each arithmetic operations, evaluate the two equivalent epressions ( ) and for =.9. Which is more accurate?. [] What number does the headecimal double-precision floating-point number c 98 represent? 4. [] Using four decimal digits of precision, given a =., b =.4, and c =.4, show that (a + b) + c? a + (b + c). Linear Algebra. [6] Find the three matrices P T, L, and U of the LUP-decomposition of the matri [4] Find the Cholesky decomposition of the matri [] Show that if A is invertible, that is Av = if and only if v =, then A T A is positive definite. You may need one or more of: (A T B T ) = (BA) T T A > Av = A v v = v T v Page of 4

3 ECE 4 Numerical Methods for Computer Engineers 8. [] Suppose that you wanted to calculate the maimum eigenvalue of M T M for a given matri M. Using asymptotic analysis (big-o), which of the following two functions would you use, and why? function lambda = eigen( M ) v = rand( length( M ), ); for i=: v = (M *M)*v/norm(v); end; lambda = norm( v ); end function lambda = eigen( M ) v = rand( length( M ), ); for i=: v = M *(M*v)/norm(v); end; lambda = norm( v ); end 9. [] Perform three iterations of the algorithm for finding the maimum eigenvalue of a 4 matri M = starting with the vector v =. Will this iteration process 4 converge?. [] Given the vector v = (,,, ) T, which has v =, use the appropriate matri norm to find the maimum possible value of Mv when 4 M = 7 4 Given that Mv = 8, find the relative error of your approimation.. [] Perform one iteration of the Jacobi method and one iteration of the Gauss-Seidel method to solve the system of linear equations given by = starting with the vector = (,, ) T. Interpolation. [] Write down the Vandermonde matri which must be solved to find the quadratic polynomial interpolating the points (-, ), (, 4), (, 7). Page of 4

4 ECE 4 Numerical Methods for Computer Engineers. [] Write down the Lagrange polynomial which interpolates the three points (-, ), (, ), (4, ). 4. [] Use the table of divided differences 4 6 to help you find the Newton polynomial which interpolates the four points (, ), (4, ), (, 6), and (6, ).. [] Use Horner s rule to evaluate at = 4 the Newton polynomial defined by the coefficient vector (-, -, ) corresponding to the values (,, ). Least Squares Regression 6. [] Write down the generalized Vandermonde matri used to find the best-fitting straight line which passes through the five points (-, ), (, 4), (, ), (7, ), (9, ). 7. [] Given n points (, y ),..., ( n, y n ), assume that the column vector of -values is assigned to the variable and the column vector of y-values is assigned to the variable y. Write down the Matlab code necessary to find a solution vector c which represents the best-fitting quadratic polynomial and evaluate that solution vector at the point =.. 8. [] Find the best-fitting line passing through the origin of the five points (, 6.), (, 8), (, 7), (, 8.), (, ). Matlab 9. [] Recalling that the Matlab function ma optionally returns two values, e.g., >> [m, posn] = ma( [ - -] ) m = posn = 6 Given an n n matri M, assign to the variable s that entry in the matri which is in the column which has the ma imum column sum and which is in the row which has the minimum row sum.. [] What is the result of the Matlab command >> v = :. :.7 Page 4 of 4

5 ECE 4 Numerical Methods for Computer Engineers Page of and -.8. The first is more accurate and ,...., T A T A = (A) T (A) = A >. 8. Answer: the second, as M *(M*v) is two O(n ) operation while (M *M)*v is an O(n ) and an O(n ) operation. 9. (4, ) T, (, ) T, (4, ) T. The sequence does not converge.. = 6, relative error is... and.... V = 4. 4) ( 4) )( ( ) ( 4) ( ( ) + ( )( 4) + ( )( 4)( ). s = s = s(4 ) = -

6 ECE 4 Numerical Methods for Computer Engineers s = s(4 ) = - 6. V = V = vander( ) or V = [.^ ones( n, )] V \ y or (V * V) \ (V * y) 8. y =. 9. [m, cl] = ma( sum( M ) ); [m, rw] = min( sum( M ) ); s = M(rw, cl). (,.,,.) Page 6 of 4