class SVD3 extends SVDecomp
The SVD3 class is used to solve Singular Value Decomposition for bidiagonal matrices.
It computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition 'SVD' of a real n-by-n (upper) bidiagonal matrix B using the implicit zero-shift 'QR' algorithm. The 'SVD' of B has the form
B = Q * S * P.t
where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P.t * VT instead of P.T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by DGEBRD, then
A = (U*Q) * S * (P.t*VT)
is the 'SVD' of the general matrix A. A positive tolerance 'TOL' gives relative accuracy; for absolute accuracy negate it.
- See also
LAPACK SUBROUTINE DBDSQR (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)
fortranwiki.org/fortran/show/svd
Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992
"Accurate singular values and differential qd algorithms," B. Parlett and V. Fernando,
J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. 11:5, pp. 873-912, Sept 1990)
"Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy,"
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Instance Constructors
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new
SVD3(b: BidMatrixD, vt: MatrixD = new MatrixD (0, 0), u: MatrixD = new MatrixD (0, 0))
- vt
the right orthogonal matrix from b = bidiagonalize (a)
- u
the left orthogonal matrix from b = bidiagonalize (a)
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.
- Definition Classes
- SVDecomp
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type
FactorTypeFull = (MatriD, MatriD, MatriD)
- Definition Classes
- SVDecomp
Value Members
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final
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!=(arg0: Any): Boolean
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def
asInstanceOf[T0]: T0
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def
calcThreshold(): Double
Calculate the desired threshold for setting elements to zero.
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def
clone(): AnyRef
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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.
- Definition Classes
- SVDecomp
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def
countNonzeroElements(): Int
Count the number of nonzero elements in the super-diagonal.
Count the number of nonzero elements in the super-diagonal. Call if the maximum number of iterations exceeded, failure to converge
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def
deflate(): VectorD
Deflate the bidiagonal matrix by iteratively turning super-diagonal elements to zero.
Deflate the bidiagonal matrix by iteratively turning super-diagonal elements to zero. Then return the vector of singular values (i.e., the main diagonal).
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def
deflateV(): (VectorD, MatrixD, MatrixD)
Deflate the bidiagonal matrix by iteratively turning super-diagonal elements to zero.
Deflate the bidiagonal matrix by iteratively turning super-diagonal elements to zero. Then return the vector of singular values and the matrices of singular vectors.
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def
deflate_2by2(ll: Int): Unit
Deflate 2 by 2 block, handle separately.
Deflate 2 by 2 block, handle separately.
- ll
the lower index
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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(): Unit
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' forming a diagonal matrix consisting of singular values and return the singular values in a vector.
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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.
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- 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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def
factors: (MatriD, MatriD)
Return the two factored matrices.
Return the two factored matrices.
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def
finalize(): Unit
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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
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final
def
notify(): Unit
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def
notifyAll(): Unit
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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
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synchronized[T0](arg0: ⇒ T0): T0
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