Package dk.alexandra.fresco.stat

Interface AdvancedLinearAlgebra

All Known Implementing Classes:
DefaultLinearAlgebra
public interface AdvancedLinearAlgebra
This computation directory contains variuous linear algebra functions. See also FixedLinearAlgebra.

  • Method Summary

    Modifier and Type Method Description
    DRes<ArrayList<DRes<SFixed>>> backSubstitution​(Matrix<DRes<SFixed>> a, ArrayList<DRes<SFixed>> b)
    Use back substitution to compute a vector x such that ax = b where a is an upper triangular square matrix.
    DRes<ArrayList<DRes<SFixed>>> forwardSubstitution​(Matrix<DRes<SFixed>> a, ArrayList<DRes<SFixed>> b)
    Use forward substitution to compute a vector x such that ax = b where a is a lower triangular square matrix.
    DRes<List<ArrayList<DRes<SFixed>>>> gramSchmidt​(List<ArrayList<DRes<SFixed>>> vectors)
    Return a list of mutually orthogonal vectors spanning the same space as the given vectors.
    DRes<Matrix<DRes<SFixed>>> invertLowerTriangularMatrix​(Matrix<DRes<SFixed>> l)
    Compute the inverse of a lower triangular matrix.
    DRes<List<DRes<SFixed>>> iterativeEigenvalues​(Matrix<DRes<SFixed>> a, int iterations)
    Approximate the eigenvalues of a matrix using the QR-algorithm.
    DRes<ArrayList<DRes<SFixed>>> linearInverseProblem​(Matrix<DRes<SFixed>> a, ArrayList<DRes<SFixed>> b)
    Solve a linear inverse problem, eg.
    DRes<Matrix<DRes<SFixed>>> moorePenrosePseudoInverse​(Matrix<DRes<SFixed>> a)
    Compute the Moore-Penrose pseudo-inverse of an m×n-matrix with full column rank.
    DRes<ArrayList<DRes<SFixed>>> normalizeVector​(ArrayList<DRes<SFixed>> u)
    Normalize a non-zero vector.
    DRes<ArrayList<DRes<SFixed>>> projection​(ArrayList<DRes<SFixed>> a, ArrayList<DRes<SFixed>> u)
    Compute the projection of a vector a onto another vector u.
    DRes<Pair<Matrix<DRes<SFixed>>,​Matrix<DRes<SFixed>>>> qrDecomposition​(Matrix<DRes<SFixed>> a)
    Compute the QR-decomposition of an mxn-matrix a with m ≥ n and full column rank.
    static AdvancedLinearAlgebra using​(ProtocolBuilderNumeric builder)  

  • Method Details

    • using

      static AdvancedLinearAlgebra using​(ProtocolBuilderNumeric builder)

    • backSubstitution

      DRes<ArrayList<DRes<SFixed>>> backSubstitution​(Matrix<DRes<SFixed>> a, ArrayList<DRes<SFixed>> b)
      Use back substitution to compute a vector x such that ax = b where a is an upper triangular square matrix.
      Parameters:
      a - An upper triangular matrix.
      b - A vector.
      Returns:
      A vector x such that ax = b.

    • forwardSubstitution

      DRes<ArrayList<DRes<SFixed>>> forwardSubstitution​(Matrix<DRes<SFixed>> a, ArrayList<DRes<SFixed>> b)
      Use forward substitution to compute a vector x such that ax = b where a is a lower triangular square matrix.
      Parameters:
      a - An lower triangular matrix.
      b - A vector.
      Returns:
      A vector x such that ax = b.

    • gramSchmidt

      DRes<List<ArrayList<DRes<SFixed>>>> gramSchmidt​(List<ArrayList<DRes<SFixed>>> vectors)
      Return a list of mutually orthogonal vectors spanning the same space as the given vectors.
      Parameters:
      vectors - A set of linearly independent vectors.
      Returns:
      A set of mutually orthogonal vectors spanning the same space as the input.

    • invertLowerTriangularMatrix

      DRes<Matrix<DRes<SFixed>>> invertLowerTriangularMatrix​(Matrix<DRes<SFixed>> l)
      Compute the inverse of a lower triangular matrix.
      Parameters:
      l - A lower triangular matrix.
      Returns:
      The inverse of l

    • linearInverseProblem

      DRes<ArrayList<DRes<SFixed>>> linearInverseProblem​(Matrix<DRes<SFixed>> a, ArrayList<DRes<SFixed>> b)
      Solve a linear inverse problem, eg. find an x such that ax = b where a is an m×n-matrix and b is an n-dimensional vector. If a system is overdetermined (m ≥ n), the computation will find the x minimising ||ax - b||.
      Parameters:
      a - An m×n-matrix.
      b - An n-dimensional vector.
      Returns:
      A solution to the equation ax = b or a matrix minimising ax - b.

    • moorePenrosePseudoInverse

      DRes<Matrix<DRes<SFixed>>> moorePenrosePseudoInverse​(Matrix<DRes<SFixed>> a)
      Compute the Moore-Penrose pseudo-inverse of an m×n-matrix with full column rank.
      Parameters:
      a - An m×n-matrix with full column rank.
      Returns:
      The Moore-Penrose pseudo-inverse of a.

    • normalizeVector

      DRes<ArrayList<DRes<SFixed>>> normalizeVector​(ArrayList<DRes<SFixed>> u)
      Normalize a non-zero vector.
      Parameters:
      u - A non-zero vector.
      Returns:
      A vector with the same direction as u and magnitude 1.

    • projection

      DRes<ArrayList<DRes<SFixed>>> projection​(ArrayList<DRes<SFixed>> a, ArrayList<DRes<SFixed>> u)
      Compute the projection of a vector a onto another vector u.
      Parameters:
      a - A vector.
      u - A vector.
      Returns:
      a projected onto u.

    • iterativeEigenvalues

      DRes<List<DRes<SFixed>>> iterativeEigenvalues​(Matrix<DRes<SFixed>> a, int iterations)
      Approximate the eigenvalues of a matrix using the QR-algorithm.
      Parameters:
      a - A square matrix.
      iterations - The number of iterations.
      Returns:
      An approximation of the eigenvalues of a.

    • qrDecomposition

      DRes<Pair<Matrix<DRes<SFixed>>,​Matrix<DRes<SFixed>>>> qrDecomposition​(Matrix<DRes<SFixed>> a)
      Compute the QR-decomposition of an mxn-matrix a with m ≥ n and full column rank. The QR-decomposition is a pair of matrices (Q,R) with A = QR and where Q is an mxn-matrix with orthonormal columns and R is an upper-triangular nxn-matrix.
      Parameters:
      a - An mxn-matrix
      Returns:
      A pair of matrices (Q,R) with a = QR and where * Q is an mxn-matrix with orthonormal columns and R is an upper-triangular * nxn-matrix.