Opened 11 years ago
Last modified 10 years ago
#10791 closed defect
Fix and upgrade Gram-Schmidt — at Version 1
Reported by: | rbeezer | Owned by: | jason, was |
---|---|---|---|
Priority: | major | Milestone: | sage-4.8 |
Component: | linear algebra | Keywords: | sd32 |
Cc: | Merged in: | ||
Authors: | Rob Beezer | Reviewers: | |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
If a set of vectors has a linearly dependent subset, followed by a vector outside the span of that subset, then the presence of a zero vector in the orthogonal set causes division by a zero dot product.
There are various other improvements that can be made in this routine.
sage: A=matrix(QQ, [[1,2],[2,4],[1,1]]) sage: A.gram_schmidt() --------------------------------------------------------------------------- ZeroDivisionError Traceback (most recent call last) /sage/sage-4.6.2.alpha4/<ipython console> in <module>() /sage/sage-4.6.2.alpha4/local/lib/python2.6/site-packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.gram_schmidt (sage/matrix/matrix2.c:31979)() /sage/sage-4.6.2.alpha4/local/lib/python2.6/site-packages/sage/modules/misc.pyc in gram_schmidt(B) 55 for i in range(1,n): 56 for j in range(i): ---> 57 mu[i,j] = B[i].dot_product(Bstar[j]) / (Bstar[j].dot_product(Bstar[j])) 58 Bstar.append(B[i] - sum(mu[i,j]*Bstar[j] for j in range(i))) 59 return Bstar, mu /sage/sage-4.6.2.alpha4/local/lib/python2.6/site-packages/sage/structure/element.so in sage.structure.element.RingElement.__div__ (sage/structure/element.c:11981)() /sage/sage-4.6.2.alpha4/local/lib/python2.6/site-packages/sage/rings/rational.so in sage.rings.rational.Rational._div_ (sage/rings/rational.c:15177)() ZeroDivisionError: Rational division by zero
Patch summary:
- Original Gram-Schmidt has been adjusted to raise an error on linearly dependent input.
- Free module version of original Gram-Schmidt was used just one place, and it has been referenced directly, rather than through the matrix version.
- Matrix version is totally rewritten, building on QR factorizations as possible to get orthonormal bases. Linearly dependent subsets should be handled properly now. Return format has, by necessity, changed.
- The "noscale" helper method is reminiscent of the old code, but written in a column-oriented QR style.
Change History (2)
Changed 11 years ago by
comment:1 Changed 11 years ago by
- Description modified (diff)
- Status changed from new to needs_review
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