# Marks starting from the initial iterate x 0 1 5 use

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2. [6 marks] Starting from the initial iterate x (0) = 1 . 5 , use Newton’s method to find the next two iterates x (1) and x (2) approximating a solution of the equation x 3 = x + 2 . 2.
MATH 2070U Midterm examination Page 6 of 7 3. Perform the following numeric conversions. Be sure to show your reasoning; no credit will be given for the answer alone. Write your answers to at least three digits using the round-to-nearest rule. Use normalised scientific notation with the leading nonzero digit to the right of the radix point, i.e., in the form (0 .d 1 d 2 . . . d t ) β × β e where e is a suitable exponent, d k ∈ { 0 , 1 , . . . , β - 1 } are digits base β and d 1 6 = 0 is the leading nonzero digit. (a) [2 marks] Convert (13 . 25) 10 to base 2. (a) (b) [2 marks] Convert (13 . 24) 8 to base 16. (b) (c) [2 marks] Convert ( E . 4 ) 16 to base 10. (c)
MATH 2070U Midterm examination Page 7 of 7 4. [5 marks] Find the LU factors of the matrix A = 4. 6 - 3 2 3 - 6 1 5 0 2 . Assume pivoting is not required.
MATH 2070U Midterm examination Page 3 of 7 1. [8 marks] Write in the space provided the output you would expect in an interactive M ATLAB session as a result of entering the given statements. For numerical values returned, assume format short , i.e., numerical values displayed rounded to 5 decimal digits. Where appropriate, be sure to distinguish row and column vectors. If the expression fails to evaluate, write ERROR with a short comment explaining the error. Assume the following statements have been executed, so the variables A , B , C , and D are available in the M ATLAB workspace. Exam solutions (Blue cover sheet) ] ] ] ] ] ] ] ] 4 ] ]
(b) The operator * is interpreted as matrix multiplication here. Since B is a 2x2 matrix and C is a 3x2 matrix, the matrix-matrix product is not defined. ??? Error using ==> mtimes Inner matrix dimensions must agree.
MATH 2070U Midterm examination Page 4 of 7 A = [1,-7; 4,0]; B = [-4,-1;7,-4]; C = [6,2;3,1;5,-8]; D = [8,-1,4;-1,7,4]; (c) A./B.ˆ2 (c) (d) A * D - 1 (d) The matrix-matrix product A*D is well-defined because A is a 2x2 matrix and D is a 2x3 matrix. Subtracting 1 is expanded out elementwise, i.e., one is subtracted from every element of the 2x3 matrix A*D. [ 14 -51 -25 ] A*D-1 = [ ] [ 31 -5 15 ] Here, we have to observe order of operations: ^ has higher precedence than / (division), so we interpret this as A./(B.^2) (rather than (A./B).^2 which has a different numerical value). [ 1/16 -7 ] A./B.^2 = [ ] [ 4/49 0 ] [ 1 -7 ] A = [ ] [ 4 0 ] [ -4 -1 ] B = [ ] [ 7 -4 ] [ 6 2 ] [ ] C = [ 3 1 ] [ ] [ 5 -8 ] [ 8 -1 4 ] D = [ ] [ -1 7 4 ]
MATH 2070U Midterm examination Page 5 of 7