Interface Matrix

  • All Superinterfaces:
    java.io.Serializable
    All Known Subinterfaces:
    DenseMatrix
    All Known Implementing Classes:
    JMatrix

    public interface Matrix
    extends java.io.Serializable
    An abstract interface of matrix. The most important method is the matrix vector multiplication, which is the only operation needed in many iterative matrix algorithms, e.g. biconjugate gradient method for solving linear equations and power iteration and Lanczos algorithm for eigen decomposition, which are usually very efficient for very large and sparse matrices.

    A matrix is a rectangular array of numbers. An item in a matrix is called an entry or an element. Entries are often denoted by a variable with two subscripts. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general.

    Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations and matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.

    There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix transformation or matrix decomposition techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.

    The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The QR decomposition factors matrices as a product of an orthogonal (Q) and a right triangular matrix (R). QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm. Singular value decomposition expresses any matrix A as a product UDV', where U and V are unitary matrices and D is a diagonal matrix. The eigendecomposition or diagonalization expresses A as a product VDV-1, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable.

    Author:
    Haifeng Li
    • Method Summary

      All Methods Static Methods Instance Methods Abstract Methods Default Methods 
      Modifier and Type Method Description
      Matrix aat()
      Returns A * A'
      default double apply​(int i, int j)
      Returns the entry value at row i and column j.
      Matrix ata()
      Returns A' * A
      double[] atx​(double[] x, double[] y)
      y = A' * x
      double[] atxpy​(double[] x, double[] y)
      y = A' * x + y
      double[] atxpy​(double[] x, double[] y, double b)
      y = A' * x + b * y
      double[] ax​(double[] x, double[] y)
      y = A * x
      double[] axpy​(double[] x, double[] y)
      y = A * x + y
      double[] axpy​(double[] x, double[] y, double b)
      y = A * x + b * y
      default double[] diag()
      Returns the diagonal elements.
      static DenseMatrix diag​(double[] A)
      Returns a square diagonal matrix with the elements of vector diag on the main diagonal.
      default EVD eigen​(int k)
      Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
      default EVD eigen​(int k, double kappa, int maxIter)
      Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
      static DenseMatrix eye​(int n)
      Returns an n-by-n identity matrix.
      static DenseMatrix eye​(int m, int n)
      Returns an m-by-n identity matrix.
      double get​(int i, int j)
      Returns the entry value at row i and column j.
      default boolean isSymmetric()
      Returns true if the matrix is symmetric.
      int ncols()
      Returns the number of columns.
      static DenseMatrix newInstance​(double[] A)
      Returns a column vector/matrix initialized by given one-dimensional array.
      static DenseMatrix newInstance​(double[][] A)
      Returns an matrix initialized by given two-dimensional array.
      static DenseMatrix newInstance​(int rows, int cols, double value)
      Creates a matrix filled with given value.
      int nrows()
      Returns the number of rows.
      static DenseMatrix ones​(int rows, int cols)
      Return an all-one matrix.
      static DenseMatrix randn​(int rows, int cols)
      Returns a random matrix of standard normal distributed values with given mean and standard dev.
      static DenseMatrix randn​(int rows, int cols, double mu, double sigma)
      Returns a random matrix of normal distributed values with given mean and standard dev.
      default void setSymmetric​(boolean symmetric)
      Sets if the matrix is symmetric.
      default SVD svd​(int k)
      Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
      default SVD svd​(int k, double kappa, int maxIter)
      Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
      default java.lang.String toString​(boolean full)
      Returns the string representation of matrix.
      default double trace()
      Returns the matrix trace.
      Matrix transpose()
      Returns the matrix transpose.
      static DenseMatrix zeros​(int rows, int cols)
      Returns all-zero matrix.
    • Method Detail

      • newInstance

        static DenseMatrix newInstance​(double[][] A)
        Returns an matrix initialized by given two-dimensional array.
      • newInstance

        static DenseMatrix newInstance​(double[] A)
        Returns a column vector/matrix initialized by given one-dimensional array.
      • newInstance

        static DenseMatrix newInstance​(int rows,
                                       int cols,
                                       double value)
        Creates a matrix filled with given value.
      • zeros

        static DenseMatrix zeros​(int rows,
                                 int cols)
        Returns all-zero matrix.
      • ones

        static DenseMatrix ones​(int rows,
                                int cols)
        Return an all-one matrix.
      • eye

        static DenseMatrix eye​(int n)
        Returns an n-by-n identity matrix.
      • eye

        static DenseMatrix eye​(int m,
                               int n)
        Returns an m-by-n identity matrix.
      • diag

        static DenseMatrix diag​(double[] A)
        Returns a square diagonal matrix with the elements of vector diag on the main diagonal.
        Parameters:
        A - the array of diagonal elements.
      • randn

        static DenseMatrix randn​(int rows,
                                 int cols)
        Returns a random matrix of standard normal distributed values with given mean and standard dev.
      • randn

        static DenseMatrix randn​(int rows,
                                 int cols,
                                 double mu,
                                 double sigma)
        Returns a random matrix of normal distributed values with given mean and standard dev.
      • toString

        default java.lang.String toString​(boolean full)
        Returns the string representation of matrix.
        Parameters:
        full - Print the full matrix if true. Otherwise only print top left 7 x 7 submatrix.
      • isSymmetric

        default boolean isSymmetric()
        Returns true if the matrix is symmetric.
      • setSymmetric

        default void setSymmetric​(boolean symmetric)
        Sets if the matrix is symmetric. It is the caller's responability to make sure if the matrix symmetric. Also the matrix won't update this property if the matrix values are changed.
      • nrows

        int nrows()
        Returns the number of rows.
      • ncols

        int ncols()
        Returns the number of columns.
      • transpose

        Matrix transpose()
        Returns the matrix transpose.
      • get

        double get​(int i,
                   int j)
        Returns the entry value at row i and column j.
      • apply

        default double apply​(int i,
                             int j)
        Returns the entry value at row i and column j. For Scala users.
      • diag

        default double[] diag()
        Returns the diagonal elements.
      • trace

        default double trace()
        Returns the matrix trace. The sum of the diagonal elements.
      • ata

        Matrix ata()
        Returns A' * A
      • aat

        Matrix aat()
        Returns A * A'
      • ax

        double[] ax​(double[] x,
                    double[] y)
        y = A * x
        Returns:
        y
      • axpy

        double[] axpy​(double[] x,
                      double[] y)
        y = A * x + y
        Returns:
        y
      • axpy

        double[] axpy​(double[] x,
                      double[] y,
                      double b)
        y = A * x + b * y
        Returns:
        y
      • atx

        double[] atx​(double[] x,
                     double[] y)
        y = A' * x
        Returns:
        y
      • atxpy

        double[] atxpy​(double[] x,
                       double[] y)
        y = A' * x + y
        Returns:
        y
      • atxpy

        double[] atxpy​(double[] x,
                       double[] y,
                       double b)
        y = A' * x + b * y
        Returns:
        y
      • eigen

        default EVD eigen​(int k)
        Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
        Parameters:
        k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.
      • eigen

        default EVD eigen​(int k,
                          double kappa,
                          int maxIter)
        Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
        Parameters:
        k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.
        kappa - relative accuracy of ritz values acceptable as eigenvalues.
        maxIter - Maximum number of iterations.
      • svd

        default SVD svd​(int k)
        Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
        Parameters:
        k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.
      • svd

        default SVD svd​(int k,
                        double kappa,
                        int maxIter)
        Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
        Parameters:
        k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.
        kappa - relative accuracy of ritz values acceptable as singular values.
        maxIter - Maximum number of iterations.