class SVD[MatT <: MatriD] extends SVDecomp with Error
The SVD class is used to compute the Singular Value Decomposition 'SVD' of
matrix 'a' using the Golub-Kahan-Reinsch Algorithm.
Factor/decompose matrix 'a' into the product of three matrices:
a = u * q * v.t where u is an m-by-n matrix of orthogonal eigenvectors of 'a * a.t' (LEFT SINGULAR VECTORS) q is an n-by-n diagonal matrix of square roots of eigenvalues of 'a.t * a' & 'a * a.t' (SINGULAR VALUES) v is an n-by-n matrix of orthogonal eigenvectors of 'a.t * a' (RIGHT SINGULAR VECTORS)
The SVD algorithm implemented is based on plane rotations. It involves transforming the input matrix to its Bidiagonal form using the Householder transformation procedure. The Bidiagonal matrix is then used to obtain singular values using the QR algorithm. ------------------------------------------------------------------------------
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new
SVD(a: MatT)
- a
the m-by-n matrix to factor/decompose (requires m >= n)
Type Members
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type
FactorType = (MatriD, VectoD, MatriD)
Factor type contains 'u, s, v' which are the left orthogonal matrix, the diagonal matrix/vector containing singular values and the right orthogonal matrix.
Factor type contains 'u, s, v' which are the left orthogonal matrix, the diagonal matrix/vector containing singular values and the right orthogonal matrix.
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- SVDecomp
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type
FactorTypeFull = (MatriD, MatriD, MatriD)
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- SVDecomp
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def
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def
conditionNum: Double
Compute the condition number of 'this' matrix, i.e., the ratio of the largest singular value to the smallest.
Compute the condition number of 'this' matrix, i.e., the ratio of the largest singular value to the smallest. Note, if not of full rank, it will be infinity.
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- SVDecomp
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final
def
eq(arg0: AnyRef): Boolean
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def
equals(arg0: Any): Boolean
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def
factor(): SVDecomp
Factor/deflate the matrix by iteratively turning elements not in the main diagonal to zero.
Factor/deflate the matrix by iteratively turning elements not in the main diagonal to zero.
- Definition Classes
- SVDecomp → Factorization
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def
factor1(): MatriD
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the first matrix.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the first matrix.
- Definition Classes
- Factorization
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def
factor12(): (MatriD, MatriD)
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t' or a = q * r, returning both the first and second matrices.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t' or a = q * r, returning both the first and second matrices.
- Definition Classes
- Factorization
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def
factor123(): FactorType
Factor matrix 'a' into the product of a matrix of left singular vectors 'u', a vector of singular values 'q' and a matrix of right singular vectors 'v' such that 'a = u ** q * v.t'.
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def
factor2(): MatriD
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the second matrix.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the second matrix.
- Definition Classes
- Factorization
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val
factored: Boolean
Flag indicating whether the matrix has been factored
Flag indicating whether the matrix has been factored
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- Factorization
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def
factors: (MatriD, MatriD)
Return the two factored matrices.
Return the two factored matrices.
- Definition Classes
- SVDecomp → Factorization
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final
def
flaw(method: String, message: String): Unit
Show the flaw by printing the error message.
Show the flaw by printing the error message.
- method
the method where the error occurred
- message
the error message
- Definition Classes
- Error
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def
flip(u: MatriD, v: MatriD): Unit
Flip negative main diagonal elements in the singular vectors to positive.
Flip negative main diagonal elements in the singular vectors to positive.
- u
the left orthongonal matrix
- v
the right orthongonal matrix
- Definition Classes
- SVDecomp
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def
flip(u: MatriD, s: VectoD): Unit
Flip negative singular values to positive and set singular values close to zero to zero.
Flip negative singular values to positive and set singular values close to zero to zero.
- u
the left orthongonal matrix
- s
the vector of singular values
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final
def
getClass(): Class[_]
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hashCode(): Int
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notify(): Unit
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notifyAll(): Unit
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def
recip(d: VectoD): VectoD
Calculate the reciprocal of the given vector avoiding divide by zero.
Calculate the reciprocal of the given vector avoiding divide by zero. To handle the problem of a singular matrix, the singular values which are zero are left zero. (This function is similar to 'VectorD.recip ()' except for the zeros part.)
- d
the vector of singular values
- See also
www.cs.princeton.edu/courses/archive/fall11/cos323/notes/cos323_f11_lecture09_svd.pdf
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def
reorder(ft: FactorType): Unit
Reorder the singular values to be in non-increasing order.
Reorder the singular values to be in non-increasing order. Must swap singular vectors in lock step with singular values. To minimize the number of swaps, selection sort is used.
- ft
the factored matrix (u, s, v)
- Definition Classes
- SVDecomp
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def
solve(b: VectoD): VectoD
Solve for 'x' in 'a^t*a*x = b' using
SVD.Solve for 'x' in 'a^t*a*x = b' using
SVD.- b
the constant vector
- Definition Classes
- SVD → SVDecomp → Factorization
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final
def
synchronized[T0](arg0: ⇒ T0): T0
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def
testFSplitting(k: Int, e: VectoD, q: VectoD): Unit
Test super-diagonal element 'e(l)' and main diagonal element 'q(l-1)' to set the lower index 'l'.
Test super-diagonal element 'e(l)' and main diagonal element 'q(l-1)' to set the lower index 'l'.
- k
the upper index
- e
the super-diagonal
- q
the main diagonal
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def
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