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Solutions to CS 417/517 Computational Methods and Software HW 6, Assignments of Computer Science

Old Dominion UniversityComputer Science

The solutions to homework 6 for the cs 417/517 computational methods and software course offered in spring 2004. It includes the matrix of normal equations, the right hand vector of normal equations, the cholesky factor, and the least squares solution. The document also demonstrates the qr decomposition and the orthogonality of the matrix c.

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CS 417/517 Computational Methods and Software
Spring 2004
Solutions to HW 6
Assigned: Thurs Mar 25, 2004; Due: Thurs Apr 1, 2004
1. Matrix of normal equations = A0∗A
5.0000 0 2.5000
0 2.5000 0
2.5000 0 2.1250
.
Right hand vector of normal equations = A0∗b
4.0000
1.0000
3.2500
.
Cholesky factor is chol(A0∗A)
2.2361 0 1.1180
0 1.5811 0
0 0 0.9354
.
Least squares solution is
0.0857
0.4000
1.4286
.
2. (a) The linear system is
1 0
1 1
1 3
x=
1
2
3
.
(b) The System QRx ≈b[Q R] = qr(A)
Q =
−0.5774 0.6172 0.5345
−0.5774 0.1543 −0.8018
−0.5774 −0.7715 0.2673
.
pf2

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CS 417/517 Computational Methods and Software

Spring 2004 Solutions to HW 6 Assigned: Thurs Mar 25, 2004; Due: Thurs Apr 1, 2004

  1. Matrix of normal equations = A′^ ∗ A

 

 .

Right hand vector of normal equations = A′^ ∗ b  

 .

Cholesky factor is chol(A′^ ∗ A)  

 .

Least squares solution is (^) 



 .

  1. (a) The linear system is (^) 



  x =

 

 .

(b) The System QRx ≈ b [Q R] = qr(A) Q = (^) 

 

  .

and R = (^)  

 .

So (^)   

   x^ =^ Q′^ ∗^ b^ =

  

  .

(c) x = (^) (

  1. 1429
  2. 6429

) .

  1. c is the possible answer because it is orthongonal on every col of A.

(a) (Q 1 ∗ Q 2 ) ∗ (Q 1 ∗ Q 2 )T^ = (Q 1 ∗ Q 2 ) ∗ (QT 2 ∗ QT 1 ) = Q 1 ∗ (Q 2 ∗ QT 2 ) ∗ QT 1 = Q 1 ∗ I ∗ QT 1 = Q 1 ∗ QT 1 = I

(b) Since ‖Qv‖ 2 = ‖v‖ 2 then ‖Q‖ 2 = 1

Since ‖Q−^1 ‖ 2 = ‖Q‖ 2 = 1 then cond(Q) = 1