Class SingularValueDecomposition
The Singular Value Decomposition of matrix A is a set of three matrices: U, Σ and V such that A = U × Σ × VT. Let A be a m × n matrix, then U is a m × p orthogonal matrix, Σ is a p × p diagonal matrix with positive or null elements, V is a p × n orthogonal matrix (hence VT is also orthogonal) where p=min(m,n).
This class is similar to the class with similar name from the JAMA library, with the following changes:
- the
norm2
method which has been renamed asgetNorm
, - the
cond
method which has been renamed asgetConditionNumber
, - the
rank
method which has been renamed asgetRank
, - a
getUT
method has been added, - a
getVT
method has been added, - a
getSolver
method has been added, - a
getCovariance
method has been added.
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Constructor Summary
ConstructorDescriptionSingularValueDecomposition
(RealMatrix matrix) Calculates the compact Singular Value Decomposition of the given matrix. -
Method Summary
Modifier and TypeMethodDescriptiondouble
Return the condition number of the matrix.getCovariance
(double minSingularValue) Returns the n × n covariance matrix.double
Computes the inverse of the condition number.double
getNorm()
Returns the L2 norm of the matrix.int
getRank()
Return the effective numerical matrix rank.getS()
Returns the diagonal matrix Σ of the decomposition.double[]
Returns the diagonal elements of the matrix Σ of the decomposition.Get a solver for finding the A × X = B solution in least square sense.getU()
Returns the matrix U of the decomposition.getUT()
Returns the transpose of the matrix U of the decomposition.getV()
Returns the matrix V of the decomposition.getVT()
Returns the transpose of the matrix V of the decomposition.
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Constructor Details
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SingularValueDecomposition
Calculates the compact Singular Value Decomposition of the given matrix.- Parameters:
matrix
- Matrix to decompose.
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Method Details
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getU
Returns the matrix U of the decomposition.U is an orthogonal matrix, i.e. its transpose is also its inverse.
- Returns:
- the U matrix
- See Also:
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getUT
Returns the transpose of the matrix U of the decomposition.U is an orthogonal matrix, i.e. its transpose is also its inverse.
- Returns:
- the U matrix (or null if decomposed matrix is singular)
- See Also:
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getS
Returns the diagonal matrix Σ of the decomposition.Σ is a diagonal matrix. The singular values are provided in non-increasing order, for compatibility with Jama.
- Returns:
- the Σ matrix
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getSingularValues
public double[] getSingularValues()Returns the diagonal elements of the matrix Σ of the decomposition.The singular values are provided in non-increasing order, for compatibility with Jama.
- Returns:
- the diagonal elements of the Σ matrix
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getV
Returns the matrix V of the decomposition.V is an orthogonal matrix, i.e. its transpose is also its inverse.
- Returns:
- the V matrix (or null if decomposed matrix is singular)
- See Also:
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getVT
Returns the transpose of the matrix V of the decomposition.V is an orthogonal matrix, i.e. its transpose is also its inverse.
- Returns:
- the V matrix (or null if decomposed matrix is singular)
- See Also:
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getCovariance
Returns the n × n covariance matrix.The covariance matrix is V × J × VT where J is the diagonal matrix of the inverse of the squares of the singular values.
- Parameters:
minSingularValue
- value below which singular values are ignored (a 0 or negative value implies all singular value will be used)- Returns:
- covariance matrix
- Throws:
IllegalArgumentException
- if minSingularValue is larger than the largest singular value, meaning all singular values are ignored
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getNorm
public double getNorm()Returns the L2 norm of the matrix.The L2 norm is max(|A × u|2 / |u|2), where |.|2 denotes the vectorial 2-norm (i.e. the traditional euclidian norm).
- Returns:
- norm
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getConditionNumber
public double getConditionNumber()Return the condition number of the matrix.- Returns:
- condition number of the matrix
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getInverseConditionNumber
public double getInverseConditionNumber()Computes the inverse of the condition number. In cases of rank deficiency, thecondition number
will become undefined.- Returns:
- the inverse of the condition number.
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getRank
public int getRank()Return the effective numerical matrix rank.The effective numerical rank is the number of non-negligible singular values. The threshold used to identify non-negligible terms is max(m,n) × ulp(s1) where ulp(s1) is the least significant bit of the largest singular value.
- Returns:
- effective numerical matrix rank
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getSolver
Get a solver for finding the A × X = B solution in least square sense.- Returns:
- a solver
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