Derive Modified Gram Schmidt QR Decomposition Algorithm from Gram Schmidt Orthogonalisation (part 2)
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From eq. 10 in part 1 we can find the Classical Gram Schmidt Algorithm, which is numerically unstable or sensitive to perturbation:
1 2 3 4 5 6 7 8 9 | Q=zeros((m,n)) R=zeros((n,n)) for k= 1 to n tmpk=cAk for i= 1 to k-1 R[i][k]=cAk^T*cQi tmpk=tmpk-R[i][k]*cQi R[k][k]=||tmpk|| cQk=tmpk /R [k][k] |
Q calculated by Classical Gram Schmidt algorithm is not very orthogonal due to rounding errors. An example is provided by X. Jiao[1]
The error introduced in q1 also cause errors in q2 and q3. We will discuss how to avoid this in the next part.
[1] http://www.ams.sunysb.edu/~jiao/teaching/ams526_fall11/lectures/lecture06.pdf
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