{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Slides_2010_03_03 - Applied linear algebra and numerical...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Applied linear algebra and numerical analysis Session 24 Prof. Ulrich Hetmaniuk Department of Applied Mathematics March 3, 2010
Image of page 1

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

View Full Document Right Arrow Icon
Normal Equations Consider a matrix A R m × n of full rank. Theorem A vector x minimizes the residual norm k b - Ax k 2 if and only if r R ( A ) , A T r = 0 or equivalently A T Ax = A T b The system has a unique solution if and only if A has full rank.
Image of page 2
Solving the Normal Equations Consider a matrix A R m × n of full rank. Algorithm Form the matrix A T A and the vector A T b . Compute the Cholesky factorization A T A = R T R . Solve the lower triangular system R T w = A T b . Solve the upper triangular system Rx = w . A T A is symmetric positive definite (invertible). Cost mn 2 + n 3 3 . If κ denotes the condition number of matrix A , then the matrix A T A has condition number κ 2 . x - x computed k x k = O ( κ 2 ε machine ) .
Image of page 3

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

View Full Document Right Arrow Icon
Solving the Normal Equations Consider a matrix A R m × n of full rank.
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern