1 / 9

Linear Least Squares

Linear Least Squares. Paul Heckbert Computer Science Department Carnegie Mellon University. Orthogonal and Hermitian Matrix. A square matrix Q is orthogonal iff Q -1 = Q T Q T Q = QQ T = I its rows are orthonormal  its columns are orthonormal

kaylee
Download Presentation

Linear Least Squares

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Linear Least Squares Paul Heckbert Computer Science Department Carnegie Mellon University 15-859B - Introduction to Scientific Computing

  2. Orthogonal and Hermitian Matrix A square matrix Q is orthogonal iff Q-1=QT • QTQ = QQT = I • its rows are orthonormal  its columns are orthonormal note: orthogonal matrices are often named Q generalization: a matrix is Hermitian iff Q-1=QH where superscript H denotes complex conjugate transpose 15-859B - Introduction to Scientific Computing

  3. Householder Transformations The Householder transformation determined by vector v is: To apply it to a vector x, compute: outer product, nn matrix inner product, scalar scalar 15-859B - Introduction to Scientific Computing

  4. Householder Geometry • Hx is x reflected through the hyperplane perpendicular to v (p : pTv=0) 15-859B - Introduction to Scientific Computing

  5. Householder Properties • H is symmetric, since • H is orthogonal, since 15-859B - Introduction to Scientific Computing

  6. Householder to Zero Matrix Elements We’ll use Householder transformations to zero subdiagonal elements of a matrix. Given any vector a, find the v that determines an H such that, Now solve for v: 15-859B - Introduction to Scientific Computing

  7. Choosing the Vector v 15-859B - Introduction to Scientific Computing

  8. Applying Householder Transforms • Don’t compute Hx explicitly, that costs 3n2 flops. • Instead use the formula given previously, which costs 4n flops (if you pre-compute vTv or pre-normalize vTv=2). • Typically, when using Householder transformations, you never compute the matrix H; it’s only used in derivation and analysis. 15-859B - Introduction to Scientific Computing

  9. QR Decomposition • Householder transformationsare a good way to zero out subdiagonal elements of a matrix. • A is decomposed: • where QT=Hn…H2H1 is the orthogonal (prove!) product of Householders and R is upper triangular. • Overdetermined system Ax=b is transformed into the easy-to-solve 15-859B - Introduction to Scientific Computing

More Related