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svd_math.nb
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Computing the Singular Value Decomposition of 3 x3 Matrices with Minimal \
Branching and Elementary Floating Point Operations\
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Short math writeup for implementing the paper.\
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The Problem\
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Step 1 : Iterative Jacobi Conjugation to diagonalize matrix\
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First thing we need to do is diagonalize the normal equations matrix S, which \
is a symmetric matrix S that we build by multiplying our matrix M by its \
transpose. This is done via iterative, cyclic Jacobi conjugation.
In each loop of the conjugation we repeatedly perform Q\[CloseCurlyQuote] * S \
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There is a sequence of chaining 3 quaternions together. TODO : compute the \
analytic form of the final chained matrix?
the first one is (ch,0,0,sh)
the second one is (ch,0,-sh,0)
the third one is (ch,sh,0,0)
\
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Step 4 : Returning Polar Decomposition\
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The form of polar decomposition is also conducive to symmetric matrix \
multiplication.
Furthermore since the order of givens conjugations is fixed we can be smart \
about skipping computation for the non-(p,q) entries of those matrices and \
avoid converting to a quaternion data structure explicitly.\
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Here is a routine that converts a quaternion to a 3 x3 givens rotation matrix.\
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