quiz2 and ans

# quiz2 and ans - invariant under multiplication by Q so k y...

This preview shows pages 1–2. Sign up to view the full content.

AMSC/CMSC 660 Quiz 2 , Fall 2006 1. (10) Suppose we have factored the m × n matrix A = QR ( m n ), and let ˆ x be the solution to the least squares problem min x k Ax - b k . Show that k A ˆ x - b k 2 = k c 2 k 2 , where c 2 is the vector consisting of the last m - n components of Q * b . Answer: (Note that we must assume that A is full rank.) De±ne c = Q * b = c 1 c 2 , R = R 1 0 where c 1 is n × 1, c 2 is ( m - n ) × 1, R 1 is n × n , and 0 is ( m - n ) × n . Then k Ax - b k 2 = k Q * ( Ax - b ) k 2 = k Rx - c k 2 = k R 1 x - c 1 k 2 + k 0x - c 2 k 2 = k R 1 x - c 1 k 2 + k c 2 k 2 . To minimize this quantity, we make the ±rst term zero by taking x to be the solution to the n × n linear system R 1 x = c 1 , so we see that the minimum value of k Ax - b k is k c 2 k . Note: The derivation above is based on three fundamental facts: Minimizing the norm of Ax - b gives the same solution as minimizing the square of the norm. For any vector y and any unitary matrix Q the norm of the vector is

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: invariant under multiplication by Q , so k y k = k Q * y k . • Suppose we partition the vector y into two pieces: y = • y 1 y 2 ‚ . Then k y k 2 = k y 1 k 2 + k y 2 k 2 . 1 2. (10) Write a column-oriented algorithm to solve Ux = b where U is an n × n nonsingular upper triangular matrix. (If you can’t do this, you can get 5 points for any correct algorithm to solve this problem, but you may not use the backslash operator or an inverse matrix.) Answer: (Note that this is like the 4th exercise.) x = b; for j=n:-1:1, x(j) = x(j) / U(j,j); x(1:j-1) = x(1:j-1) - U(1:j-1,j)*x(j); end 2...
View Full Document

## This homework help was uploaded on 02/05/2008 for the course CMSC 660 taught by Professor Oleary during the Fall '06 term at Maryland.

### Page1 / 2

quiz2 and ans - invariant under multiplication by Q so k y...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online