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Dean Eckles
Dean Eckles

Reputation: 173

Finding full QR decomposition from reduced QR

What's the best way to find additional orthonormal columns of Q? I have computed the reduced QR decomposition already, but need the full QR decomposition.

I assume there is a standard approach to this, but I've been having trouble finding it.

You might wonder why I need the full Q matrix. I'm using it to apply a constraint matrix for "natural" splines to a truncated power series basis expansion. I'm doing this in Java, but am looking for a language-independent answer.

Upvotes: 3

Views: 1954

Answers (1)

JeremyKun
JeremyKun

Reputation: 3002

Successively add columns to Q in the following way:

  1. Pick a vector not already in the span of Q
  2. Orthogonalize it with respect to the columns of Q
  3. Add the orthogonalized vector as a new column of Q.
  4. Add a row of zeros to the bottom of R

For reference, see these illustrative albeit mathematical lecture notes

Just in case, the process of "orthogonalization" of a new vector is an old technique called the Gram-Schmidt process, and there is a variant which is numerically stable.

Upvotes: 2

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