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Algorithm 6 . 1 procedure COMPACT _ SVD ( A ) , Vlarreig ( A H A ) l a r r 2 Calculate the
Algorithm procedure COMPACTSVD Vlarreig Calculate the eigenvalues and eigenvectors of Calculate the singular values of Sort the singular values from greatest to least. Vlarrsort Sort the eigenvectors the same way as in the previous step. rlarrcount Count the number of nonzero singular values the rank of larr Keep only the positive singular values. Keep only the corresponding eigenvectors. larrA Construct with array broadcasting. return Problem Write a function that accepts a matrix A and a small error tolerance tol. Use Algorithm to compute the compact SVD of In step compute by counting the number of singular values that are greater than tol. Consider the following tips for implementing the algorithm. The Hermitian can be computed with A conj T In step the way that is sorted needs to be stored so that the columns of can be sorted the same way. Consider using npargsort and fancy indexing to do this, but remember that by default it sorts from least to greatest not greatest to least Step can be done by looping over the columns of but it can be done more easily and efficiently with array broadcasting. Test your function by calculating the compact SVD for random matrices. Verify that and are orthonormal, that and that the number of nonzero singular values is the rank of You may also want to compre your results to SciPy's SVD algorithm.
Algorithm
procedure COMPACTSVD
Vlarreig
Calculate the eigenvalues and eigenvectors of
Calculate the singular values of
Sort the singular values from greatest to least.
Vlarrsort Sort the eigenvectors the same way as in the previous step.
rlarrcount Count the number of nonzero singular values the rank of
larr Keep only the positive singular values.
Keep only the corresponding eigenvectors.
larrA Construct with array broadcasting.
return
Problem Write a function that accepts a matrix A and a small error tolerance tol. Use
Algorithm to compute the compact SVD of In step compute by counting the number
of singular values that are greater than tol.
Consider the following tips for implementing the algorithm.
The Hermitian can be computed with A conj T
In step the way that is sorted needs to be stored so that the columns of can be
sorted the same way. Consider using npargsort and fancy indexing to do this, but
remember that by default it sorts from least to greatest not greatest to least
Step can be done by looping over the columns of but it can be done more easily and
efficiently with array broadcasting.
Test your function by calculating the compact SVD for random matrices. Verify that
and are orthonormal, that and that the number of nonzero singular values is
the rank of You may also want to compre your results to SciPy's SVD algorithm.
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