Packages

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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Instance Constructors

  1. new SVD(a: MatT)

    a

    the m-by-n matrix to factor/decompose (requires m >= n)

Type Members

  1. 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
  2. type FactorTypeFull = (MatriD, MatriD, MatriD)
    Definition Classes
    SVDecomp

Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  2. final def ##: Int
    Definition Classes
    AnyRef → Any
  3. final def ==(arg0: Any): Boolean
    Definition Classes
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  4. final def asInstanceOf[T0]: T0
    Definition Classes
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  5. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  6. 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
  7. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  8. def equals(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef → Any
  9. 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
    SVDecompFactorization
  10. 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
  11. 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
  12. 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'.

    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'.

    Definition Classes
    SVDSVDecomp
  13. 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
  14. val factored: Boolean

    Flag indicating whether the matrix has been factored

    Flag indicating whether the matrix has been factored

    Attributes
    protected
    Definition Classes
    Factorization
  15. def factors: (MatriD, MatriD)

    Return the two factored matrices.

    Return the two factored matrices.

    Definition Classes
    SVDecompFactorization
  16. 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
  17. 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
  18. 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

    Definition Classes
    SVDecomp
  19. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  20. def hashCode(): Int
    Definition Classes
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    @native() @HotSpotIntrinsicCandidate()
  21. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  22. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  23. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  24. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  25. 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

  26. 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
  27. 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
    SVDSVDecompFactorization
  28. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  29. 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

  30. def toString(): String
    Definition Classes
    AnyRef → Any
  31. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  32. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  33. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])

Deprecated Value Members

  1. def finalize(): Unit
    Attributes
    protected[lang]
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    @throws(classOf[java.lang.Throwable]) @Deprecated
    Deprecated

Inherited from Error

Inherited from SVDecomp

Inherited from Factorization

Inherited from AnyRef

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