1.In the following, we are to construct a LS problem by choosing A R104 and b R101.

a). Choose A and b with which the dierence between the solution vectors xhouse and xne is large. Print A, b, and d1 = kxhousexnek2, rhouse and rne. The larger d1 is, the better it is. Explain how you chose the input A and b and what properties of A or b, and the methods, make two solutions dierent.

b). Choose A and b with which the dierence between the solution vectors xhouse and xmgs is large. Print A, b, and d2 = kxhouse xmgsk2, rhouse and rmgs. The larger d2 is, the better it is. Explain how you chose the input A and b and what properties of A or b, and the methods make two solutions dierent

c). Choose A and b so that the dierence in the orthonormality in Q and Q1 by houseql and mgsql is large. Print A, b, eh = kQTQ I10kF/100 where Q is from houseql, and em = kQT 1 Q1 I4kF/16 where Q1 is from mgsql. Also print d3 = |eh em|. The larger d3 is, the better it is. Explain how you chose the input A and b and what properties of A or b, and the methods make the orthonormality dierent.

Note xhouse is x found by householders method

xne is x found by cholesky in normal equations

xmgs is x found by modified gram schmidt

houseq is QL algorithm using householders method