Matrix
org.jscience.mathematics.vector

Class Matrix<F extends Field<F>>

  • All Implemented Interfaces:
    javolution.lang.Immutable, javolution.lang.Realtime, javolution.lang.ValueType, GroupAdditive<Matrix<F>>, Ring<Matrix<F>>, Structure<Matrix<F>>, VectorSpace<Matrix<F>,F>
    Direct Known Subclasses:
    ComplexMatrix, DenseMatrix, Float64Matrix, SparseMatrix


    public abstract class Matrix<F extends Field<F>>extends Objectimplements VectorSpace<Matrix<F>,F>, Ring<Matrix<F>>, javolution.lang.ValueType, javolution.lang.Realtime

    This class represents a rectangular table of elements of a ring-like algebraic structure.

    Instances of this class can be used to resolve system of linear equations involving any kind of Field elements (e.g. Real, Complex, Amount<?>, Function, etc). For example:

            // Creates a dense matrix (2x2) of Rational numbers.        DenseMatrix<Rational> M = DenseMatrix.valueOf(            { Rational.valueOf(23, 45), Rational.valueOf(33, 75) },            { Rational.valueOf(15, 31), Rational.valueOf(-20, 45)});                    // Creates a sparse matrix (16x2) of Real numbers.        SparseMatrix<Real> M = SparseMatrix.valueOf(            SparseVector.valueOf(16, Real.ZERO, 0, Real.valueOf(5)),            SparseVector.valueOf(16, Real.ZERO, 15, Real.valueOf(-3)));                    // Creates a floating-point (64 bits) matrix (3x2).        Float64Matrix M = Float64Matrix.valueOf(           {{ 1.0, 2.0, 3.0}, { 4.0, 5.0, 6.0}});                    // Creates a complex single column matrix (1x2).        ComplexMatrix M = ComplexMatrix.valueOf(           {{ Complex.valueOf(1.0, 2.0), Complex.valueOf(4.0, 5.0)}}).transpose();                    // Creates an identity matrix (2x2) for modulo integer.        SparseMatrix<ModuloInteger> IDENTITY = SparseMatrix.valueOf(           DenseVector.valueOf(ModuloInteger.ONE, ModuloInteger.ONE), ModuloInteger.ZERO);     

    Non-commutative field multiplication is supported. Invertible square matrices may form a non-commutative field (also called a division ring). In which case this class may be used to resolve system of linear equations with matrix coefficients.

    Implementation Note: Matrices may use StackContext and ConcurrentContext in order to minimize heap allocation and accelerate calculations on multi-core systems.

    See Also:
    Wikipedia: Matrix (mathematics)
    • Method Summary

      Methods 
      Modifier and TypeMethod and Description
      abstract Matrix<F>adjoint()
      Returns the adjoint of this matrix.
      abstract Fcofactor(int i, int j)
      Returns the cofactor of an element in this matrix.
      abstract Matrix<F>copy()
      Returns a copy of this matrix allocated by the calling thread (possibly on the stack).
      abstract Fdeterminant()
      Returns the determinant of this matrix.
      Matrix<F>divide(Matrix<F> that)
      Returns this matrix divided by the one specified.
      booleanequals(Matrix<F> that, Comparator<F> cmp)
      Indicates if this matrix can be considered equals to the one specified using the specified comparator when testing for element equality.
      booleanequals(Object that)
      Indicates if this matrix is strictly equal to the object specified.
      abstract Fget(int i, int j)
      Returns a single element from this matrix.
      abstract Vector<F>getColumn(int j)
      Returns the column identified by the specified index in this matrix.
      abstract Vector<F>getDiagonal()
      Returns the diagonal vector.
      abstract intgetNumberOfColumns()
      Returns the number of columns n for this matrix.
      abstract intgetNumberOfRows()
      Returns the number of rows m for this matrix.
      abstract Vector<F>getRow(int i)
      Returns the row identified by the specified index in this matrix.
      inthashCode()
      Returns a hash code value for this matrix.
      abstract Matrix<F>inverse()
      Returns the inverse of this matrix (must be square).
      booleanisSquare()
      Indicates if this matrix is square.
      Matrix<F>minus(Matrix<F> that)
      Returns the difference between this matrix and the one specified.
      abstract Matrix<F>opposite()
      Returns the negation of this matrix.
      abstract Matrix<F>plus(Matrix<F> that)
      Returns the sum of this matrix with the one specified.
      Matrix<F>pow(int exp)
      Returns this matrix raised at the specified exponent.
      Matrix<F>pseudoInverse()
      Returns the inverse or pseudo-inverse if this matrix if not square.
      Matrix<F>solve(Matrix<F> y)
      Solves this matrix for the specified matrix (returns x such as this \xc2\xb7 x = y).
      Vector<F>solve(Vector<F> y)
      Solves this matrix for the specified vector (returns x such as this \xc2\xb7 x = y).
      abstract Matrix<F>tensor(Matrix<F> that)
      Returns the linear algebraic matrix tensor product of this matrix and another (Kronecker product).
      abstract Matrix<F>times(F k)
      Returns the product of this matrix by the specified factor.
      abstract Matrix<F>times(Matrix<F> that)
      Returns the product of this matrix with the one specified.
      abstract Vector<F>times(Vector<F> v)
      Returns the product of this matrix by the specified vector.
      StringtoString()
      Returns the text representation of this matrix as a java.lang.String.
      javolution.text.TexttoText()
      Returns the text representation of this matrix.
      Ftrace()
      Returns the trace of this matrix.
      abstract Matrix<F>transpose()
      Returns the transpose of this matrix.
      abstract Vector<F>vectorization()
      Returns the vectorization of this matrix.
    • Method Detail

      • getNumberOfRows

        public abstract int getNumberOfRows()
        Returns the number of rows m for this matrix.
        Returns:
        m, the number of rows.
      • getNumberOfColumns

        public abstract int getNumberOfColumns()
        Returns the number of columns n for this matrix.
        Returns:
        n, the number of columns.
      • get

        public abstract F get(int i,    int j)
        Returns a single element from this matrix.
        Parameters:
        i - the row index (range [0..m[).
        j - the column index (range [0..n[).
        Returns:
        the element read at [i,j].
        Throws:
        IndexOutOfBoundsException - ((i < 0) || (i >= m)) || ((j < 0) || (j >= n))
      • getRow

        public abstract Vector<F> getRow(int i)
        Returns the row identified by the specified index in this matrix.
        Parameters:
        i - the row index (range [0..m[).
        Returns:
        the vector holding the specified row.
        Throws:
        IndexOutOfBoundsException - (i < 0) || (i >= m)
      • getColumn

        public abstract Vector<F> getColumn(int j)
        Returns the column identified by the specified index in this matrix.
        Parameters:
        j - the column index (range [0..n[).
        Returns:
        the vector holding the specified column.
        Throws:
        IndexOutOfBoundsException - (j < 0) || (j >= n)
      • getDiagonal

        public abstract Vector<F> getDiagonal()
        Returns the diagonal vector.
        Returns:
        the vector holding the diagonal elements.
      • minus

        public Matrix<F> minus(Matrix<F> that)
        Returns the difference between this matrix and the one specified.
        Parameters:
        that - the matrix to be subtracted.
        Returns:
        this - that.
        Throws:
        DimensionException - matrices\'s dimensions are different.
      • times

        public abstract Matrix<F> times(F k)
        Returns the product of this matrix by the specified factor.
        Specified by:
        times in interface VectorSpace<Matrix<F extends Field<F>>,F extends Field<F>>
        Parameters:
        k - the coefficient multiplier.
        Returns:
        this \xc2\xb7 k
      • times

        public abstract Vector<F> times(Vector<F> v)
        Returns the product of this matrix by the specified vector.
        Parameters:
        v - the vector.
        Returns:
        this \xc2\xb7 v
        Throws:
        DimensionException - if v.getDimension() != this.getNumberOfColumns()
      • times

        public abstract Matrix<F> times(Matrix<F> that)
        Returns the product of this matrix with the one specified.
        Specified by:
        times in interface Ring<Matrix<F extends Field<F>>>
        Parameters:
        that - the matrix multiplier.
        Returns:
        this \xc2\xb7 that.
        Throws:
        DimensionException - if this.getNumberOfColumns() != that.getNumberOfRows().
      • inverse

        public abstract Matrix<F> inverse()
        Returns the inverse of this matrix (must be square).
        Returns:
        1 / this
        Throws:
        DimensionException - if this matrix is not square.
      • divide

        public Matrix<F> divide(Matrix<F> that)
        Returns this matrix divided by the one specified.
        Parameters:
        that - the matrix divisor.
        Returns:
        this / that.
        Throws:
        DimensionException - if that matrix is not square or dimensions do not match.
      • pseudoInverse

        public Matrix<F> pseudoInverse()
        Returns the inverse or pseudo-inverse if this matrix if not square.

        Note: To resolve the equation A * X = B, it is usually faster to calculate A.lu().solve(B) rather than A.inverse().times(B).

        Returns:
        the inverse or pseudo-inverse of this matrix.
      • determinant

        public abstract F determinant()
        Returns the determinant of this matrix.
        Returns:
        this matrix determinant.
        Throws:
        DimensionException - if this matrix is not square.
      • transpose

        public abstract Matrix<F> transpose()
        Returns the transpose of this matrix.
        Returns:
        A\'.
      • cofactor

        public abstract F cofactor(int i,         int j)
        Returns the cofactor of an element in this matrix. It is the value obtained by evaluating the determinant formed by the elements not in that particular row or column.
        Parameters:
        i - the row index.
        j - the column index.
        Returns:
        the cofactor of THIS[i,j].
        Throws:
        DimensionException - matrix is not square or its dimension is less than 2.
      • adjoint

        public abstract Matrix<F> adjoint()
        Returns the adjoint of this matrix. It is obtained by replacing each element in this matrix with its cofactor and applying a + or - sign according (-1)**(i+j), and then finding the transpose of the resulting matrix.
        Returns:
        the adjoint of this matrix.
        Throws:
        DimensionException - if this matrix is not square or if its dimension is less than 2.
      • isSquare

        public boolean isSquare()
        Indicates if this matrix is square.
        Returns:
        getNumberOfRows() == getNumberOfColumns()
      • solve

        public Vector<F> solve(Vector<F> y)
        Solves this matrix for the specified vector (returns x such as this \xc2\xb7 x = y).
        Parameters:
        y - the vector for which the solution is calculated.
        Returns:
        x such as this \xc2\xb7 x = y
        Throws:
        DimensionException - if that matrix is not square or dimensions do not match.
      • solve

        public Matrix<F> solve(Matrix<F> y)
        Solves this matrix for the specified matrix (returns x such as this \xc2\xb7 x = y).
        Parameters:
        y - the matrix for which the solution is calculated.
        Returns:
        x such as this \xc2\xb7 x = y
        Throws:
        DimensionException - if that matrix is not square or dimensions do not match.
      • pow

        public Matrix<F> pow(int exp)
        Returns this matrix raised at the specified exponent.
        Parameters:
        exp - the exponent.
        Returns:
        thisexp
        Throws:
        DimensionException - if this matrix is not square.
      • trace

        public F trace()
        Returns the trace of this matrix.
        Returns:
        the sum of the diagonal elements.
      • tensor

        public abstract Matrix<F> tensor(Matrix<F> that)
        Returns the linear algebraic matrix tensor product of this matrix and another (Kronecker product). The default implementation returns a DenseMatrix.
        Parameters:
        that - the second matrix.
        Returns:
        this ⊗ that
        See Also:
        Wikipedia: Kronecker Product
      • vectorization

        public abstract Vector<F> vectorization()
        Returns the vectorization of this matrix. The vectorization of a matrix is the column vector obtain by stacking the columns of the matrix on top of one another. The default implementation returns a DenseVector.
        Returns:
        the vectorization of this matrix.
        See Also:
        Wikipedia: Vectorization.
      • toText

        public javolution.text.Text toText()
        Returns the text representation of this matrix.
        Specified by:
        toText in interface javolution.lang.Realtime
        Returns:
        the text representation of this matrix.
      • toString

        public final String toString()
        Returns the text representation of this matrix as a java.lang.String.
        Overrides:
        toString in class Object
        Returns:
        toText().toString()
      • equals

        public boolean equals(Matrix<F> that,             Comparator<F> cmp)
        Indicates if this matrix can be considered equals to the one specified using the specified comparator when testing for element equality. The specified comparator may allow for some tolerance in the difference between the matrix elements.
        Parameters:
        that - the matrix to compare for equality.
        cmp - the comparator to use when testing for element equality.
        Returns:
        true if this matrix and the specified matrix are both matrices with equal elements according to the specified comparator; false otherwise.
      • equals

        public boolean equals(Object that)
        Indicates if this matrix is strictly equal to the object specified.
        Overrides:
        equals in class Object
        Parameters:
        that - the object to compare for equality.
        Returns:
        true if this matrix and the specified object are both matrices with equal elements; false otherwise.
        See Also:
        equals(Matrix, Comparator)
      • copy

        public abstract Matrix<F> copy()
        Returns a copy of this matrix allocated by the calling thread (possibly on the stack).
        Specified by:
        copy in interface javolution.lang.ValueType
        Returns:
        an identical and independant copy of this matrix.

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