iconEuler Reference

Singular Values

Singular value decomposition tools.

The singular value decomposition of Euler is based on the builtin function svd. It is used to compute an orthogonal basis of the kernel and the image of a matrix, the condition of a matrix, or the pseudo-inverse.

Other linear algebra routines are contained in the file Linear Algebra.

function svdkernel (A:real)

  Computes the kernel of the quadratic matrix A
  
  This function is using the singular value decomposition, and works
  for real matrices only.
  
  Returns an orthogonal basis of the kernel as columns of a matrix.
  
  See: 
kernel (Linear Algebra)
function svdimage (A:real)

  Computes the image of the quadratic matrix A
  
  This function is using the singular value decomposition, and works
  for real matrices only.
  
  Returns an orthogonal basis of the image as columns of a matrix.
  
  See: 
kernel (Linear Algebra),
image (Linear Algebra),
image (Maxima Documentation),
svdkernel (Linear Algebra),
svdkernel (Singular Values)
function svdcondition (A:real)

  Condition number based on a singular value decompostion of A
  
  Returns the qutient of the largest singular value divided by the
  smallest. 0 means singularity.
  
  A : real matrix
function svddet (A:real)

  Determinant based on a singular value decomposition of A
  
  A : real matrix
  
  See: 
det (Linear Algebra)
function svdeigenvalues (A:real)

  Eigenvalues or singular values of A
  
  For a symmetric A, this returns the eigenvalues of A For a
  non-symmetric A, the singular values.
  
  A : real matrix
  
  See: 
eigenvalues (Linear Algebra),
eigenvalues (Maxima Documentation)
function svdeigenspace (A:real,l:real)

  Returns the eigenspace of A to the eigenvaue l
  
  See: 
eigenspace (Linear Algebra)
function svdsolve (A:real,b:real)

  Minimize |A.x-b| with smallest norm for x
  
  The singular value decomposition is one way to solve the fit
  problem. It has the advantage, that the result will be the result
  with smallest norm, if there is more than one solution.
  
  A : real matrix
  
  See: 
fit (Linear Algebra)
function svdinv (A:real)

  The pseudo-inverse of A.
  
  The pseudo-inverse B of a matrix solves the fit problem to minimize
  |Ax-b| by x=B.b. It is computed in this function using an svd
  decomposition.
  
function pinv (A:real)

  The pseudo-inverse of A.
  
  Alias to svdinv
  
  See: 
svdinv (Linear Algebra),
svdinv (Singular Values)
function ginv (A:real)

  Gilbert inverse of a matrix A
  
  This inverse has the property A.G.A=A

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