Question

P1. (This is similar to Exercise 3.3.7 from the textbook) Work out the following problem by hand. Consider the overdetermined system Calculate a full QR decomposition of the coefficient matrix, where Q is 2 x 2 and R is 2 x 1. Use this QR decomposition to calculate the least squares solution (the minimizer). Use the formulas given in lecture or textbook to calculate the norm of the residual with the help of Q (the minimum).

Answer 1

solution The Given, over determined system is [:] (13 = [:)

Now. We have findout QR de composition Assoming, and R Now, . QQT = 12 a Iq (2²+62 ac+bd .. 12 2/ ac +bd

let us consider, a2+6²=1, c²+d² = 1 → ac+bd = 0 ac =-bd a= dk , b=ack (de)?+ Eckej? 21 d²2² + c26²=1 k² ( c²4d²)=1 since, c+d= :: K = I1

Then, au le ) - (1) am = 1 cm=1 m (a-c) = 0 mo a = C . a d and b= - > b= -c b=-6=- 7.bec, b=0=03- NOW Cue COD 3 uile

R- ::: ORX = 6) – RX = Q (9 RX - O Alla wis > Al -52 X x= -14 x /

x= 124 125 a=7 And, 452 = 252 Hence, 622) = residual of the least square 252 . which equident form Equation (A).

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