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DOC: clarification on the `which` parameter of ARPACK. #388

Merged
merged 1 commit into from

2 participants

@fabianp
Collaborator

I've had a hard time understanding the which parameter of ARPACK, so
I'm submitting this patch with the hope that it becomes clearer.

Changes:

  • Eigenvectors have all norm 1, thus the parameter which is
    applied to eigenvalues and not eigenvectors.

  • Clarification on what magnitude means in the context of
    complex eigenvalues.

  • Clarification on what 'LA', 'SA' means. ARPACK docs refer to them
    as Largest/Smallest Algebraic value: those are real numbers with
    its respective sign (not in absolute value). I could not understand
    what was meant before as Largest Amplitude.

@fabianp fabianp DOC: clarification on the `which` parameter of ARPACK.
I've had a hard time understanding the `which` parameter of ARPACK, so
I'm submitting this patch with the hope that it becomes clearer.

Changes:

   * Eigenvectors have all norm 1, thus the parameter `which` is
     applied to eigenvalues and not eigenvectors.

   * Clarification on what magnitude means in the context of
     complex eigenvalues.

   * Clarification on what 'LA', 'SA' means. ARPACK docs refer to them
     as Largest/Smallest Algebraic value: those are real numbers with
     its respective sign (not in absolute value). I could not understand
     what was meant before as `Largest Amplitude`.
2d9a5f0
@jakevdp
Collaborator

Nice - I think that is much clearer - Thanks! I'm fine to merge this - I'll give it a couple days to see if there are other comments.

@fabianp
Collaborator

OK glad you liked it!

@jakevdp
Collaborator

Merging - thanks Fabian

@jakevdp jakevdp merged commit a7c0e1e into scipy:master
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Commits on Dec 18, 2012
  1. @fabianp

    DOC: clarification on the `which` parameter of ARPACK.

    fabianp authored
    I've had a hard time understanding the `which` parameter of ARPACK, so
    I'm submitting this patch with the hope that it becomes clearer.
    
    Changes:
    
       * Eigenvectors have all norm 1, thus the parameter `which` is
         applied to eigenvalues and not eigenvectors.
    
       * Clarification on what magnitude means in the context of
         complex eigenvalues.
    
       * Clarification on what 'LA', 'SA' means. ARPACK docs refer to them
         as Largest/Smallest Algebraic value: those are real numbers with
         its respective sign (not in absolute value). I could not understand
         what was meant before as `Largest Amplitude`.
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Showing with 13 additions and 9 deletions.
  1. +13 −9 doc/source/tutorial/arpack.rst
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22 doc/source/tutorial/arpack.rst
@@ -40,15 +40,19 @@ The power of ARPACK is that it can compute only a specified subset of
eigenvalue/eigenvector pairs. This is accomplished through the keyword
``which``. The following values of ``which`` are available:
-* ``which = 'LM'`` : Eigenvectors with largest magnitude (``eigs``, ``eigsh``)
-* ``which = 'SM'`` : Eigenvectors with smallest magnitude (``eigs``, ``eigsh``)
-* ``which = 'LR'`` : Eigenvectors with largest real part (``eigs``)
-* ``which = 'SR'`` : Eigenvectors with smallest real part (``eigs``)
-* ``which = 'LI'`` : Eigenvectors with largest imaginary part (``eigs``)
-* ``which = 'SI'`` : Eigenvectors with smallest imaginary part (``eigs``)
-* ``which = 'LA'`` : Eigenvectors with largest amplitude (``eigsh``)
-* ``which = 'SA'`` : Eigenvectors with smallest amplitude (``eigsh``)
-* ``which = 'BE'`` : Eigenvectors from both ends of the spectrum (``eigsh``)
+* ``which = 'LM'`` : Eigenvalues with largest magnitude (``eigs``, ``eigsh``),
+ that is, largest eigenvalues in the euclidean norm of complex numbers.
+* ``which = 'SM'`` : Eigenvalues with smallest magnitude (``eigs``, ``eigsh``),
+ that is, smallest eigenvalues in the euclidean norm of complex numbers.
+* ``which = 'LR'`` : Eigenvalues with largest real part (``eigs``)
+* ``which = 'SR'`` : Eigenvalues with smallest real part (``eigs``)
+* ``which = 'LI'`` : Eigenvalues with largest imaginary part (``eigs``)
+* ``which = 'SI'`` : Eigenvalues with smallest imaginary part (``eigs``)
+* ``which = 'LA'`` : Eigenvalues with largest algebraic value (``eigsh``),
+ that is, largest eigenvalues inclusive of any negative sign.
+* ``which = 'SA'`` : Eigenvalues with smallest algebraic value (``eigsh``),
+ that is, smallest eigenvalues inclusive of any negative sign.
+* ``which = 'BE'`` : Eigenvalues from both ends of the spectrum (``eigsh``)
Note that ARPACK is generally better at finding extremal eigenvalues: that
is, eigenvalues with large magnitudes. In particular, using ``which = 'SM'``
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