lecture29

# lecture29 - IE170 Algorithms in Systems Engineering Lecture...

This preview shows pages 1–3. Sign up to view the full content.

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: IE170: Algorithms in Systems Engineering: Lecture 29 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University April 16, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 1 / 23 Taking Stock Last Time Matrix Review This Time Solving Triangular Systems Solving Symmetric Positive Definite Systems Least Squares Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 2 / 23 Systems of Equations: Ax = b From our previous discussion, we know that the system of equations Ax = b has a unique solution if and only if the matrix A is square and invertible This is true if the columns A j are linearly independent From now on, we will consider only invertible systems. In fact, today we will consider special versions of A The \$64 Question How do we solve a systems of equations? We factor the matrix A into a simpler form Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 3 / 23 Triangular Systems Let’s suppose that we are able to find two n × n matrices L , U such that A = LU where L is upper triangular. U is lower triangular with 1’s on the diagonal. How could use such a decomposition to solve the system Ax = b ? Jeff Linderoth (Lehigh University) IE170:Lecture 29 Lecture Notes 4 / 23 Using a Triangular Decomposition Once we have an triangular decomposition, we can use it to easily solve the system Ax = b . Note that the system Ax = b is equivalent to the original system, which is then equivalent to LUx = b . We can solve the system in two steps: First solve the system Ly = b (forward substitution)....
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

lecture29 - IE170 Algorithms in Systems Engineering Lecture...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online