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.
- Since:
- 2.0 (changed to concrete class in 3.0)
- See Also:
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Nested Class Summary
Nested ClassesModifier and TypeClassDescriptionprivate static class
Specialized solver. -
Field Summary
FieldsModifier and TypeFieldDescriptionprivate RealMatrix
Cached value of S (diagonal) matrix.private final RealMatrix
Cached value of U matrix.private RealMatrix
Cached value of transposed U matrix.private final RealMatrix
Cached value of V matrix.private RealMatrix
Cached value of transposed V matrix.private static final double
Relative threshold for small singular values.private final int
max(row dimension, column dimension).private final int
min(row dimension, column dimension).private final double[]
Computed singular values.private static final double
Absolute threshold for small singular values.private final double
Tolerance value for small singular values, calculated once we have populated "singularValues".private final boolean
Indicator for transposed matrix. -
Constructor Summary
ConstructorsConstructorDescriptionSingularValueDecomposition
(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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Field Details
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EPS
private static final double EPSRelative threshold for small singular values.- See Also:
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TINY
private static final double TINYAbsolute threshold for small singular values.- See Also:
-
singularValues
private final double[] singularValuesComputed singular values. -
m
private final int mmax(row dimension, column dimension). -
n
private final int nmin(row dimension, column dimension). -
transposed
private final boolean transposedIndicator for transposed matrix. -
cachedU
Cached value of U matrix. -
cachedUt
Cached value of transposed U matrix. -
cachedS
Cached value of S (diagonal) matrix. -
cachedV
Cached value of V matrix. -
cachedVt
Cached value of transposed V matrix. -
tol
private final double tolTolerance value for small singular values, calculated once we have populated "singularValues".
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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:
-
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:
-
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:
-
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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