{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lectures.notes6-5

# lectures.notes6-5 - A x = b EXAMPLE Find a least-squares...

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

SECTION 6.5 LEAST-SQUARES PROBLEMS, OR BEST APPROXIMATIONS THE BASIC PROBLEM. We want to solve A x = b but the system is inconsistent. So we want to ﬁnd x so that A x is as close as possible to b . Where is A x ? No matter what x is, it is in Col A . So, we already know what A x must be – it must be the projection ˆ b of b onto Col A . Suppose ˆ x satisﬁes A ˆ x = ˆ b . Then b - A ˆ x is orthogonal to each column of A . Express this by making the columns of A into rows. We get A T A ˆ x = A T b (called . the normal equations ), so this is the system we need to solve to produce a least-squares solution of

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: A x = b . EXAMPLE. Find a least-squares solution of A x = b when A = 1-2-1 2 3 2 5 and b = 3 1-4 2 . Now ﬁnd the least-squares error associated with this least-squares solution. This is the distance from b to A x . FACT. If A has linearly independent columns and A = QR is a QR factorization of A , then A x = b has a unique least-squares solution given by ˆ x = R-1 Q T b . HOMEWORK: SECTION 6.5...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

lectures.notes6-5 - A x = b EXAMPLE Find a least-squares...

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

View Full Document
Ask a homework question - tutors are online