Algebra
cern.colt.matrix.linalg

Class Algebra

    • Field Detail

      • DEFAULT

        public static final Algebra DEFAULT
        A default Algebra object; has Property.DEFAULT attached for tolerance. Allows ommiting to construct an Algebra object time and again. Note that this Algebra object is immutable. Any attempt to assign a new Property object to it (via method setProperty), or to alter the tolerance of its property object (via property().setTolerance(...)) will throw an exception.
      • ZERO

        public static final Algebra ZERO
        A default Algebra object; has Property.ZERO attached for tolerance. Allows ommiting to construct an Algebra object time and again. Note that this Algebra object is immutable. Any attempt to assign a new Property object to it (via method setProperty), or to alter the tolerance of its property object (via property().setTolerance(...)) will throw an exception.
    • Constructor Detail

      • Algebra

        public Algebra()
        Constructs a new instance with an equality tolerance given by Property.DEFAULT.tolerance().
      • Algebra

        public Algebra(double tolerance)
        Constructs a new instance with the given equality tolerance.
        Parameters:
        tolerance - the tolerance to be used for equality operations.
    • Method Detail

      • clone

        public Object clone()
        Returns a copy of the receiver. The attached property object is also copied. Hence, the property object of the copy is mutable.
        Overrides:
        clone in class PersistentObject
        Returns:
        a copy of the receiver.
      • cond

        public double cond(DoubleMatrix2D A)
        Returns the condition of matrix A, which is the ratio of largest to smallest singular value.
      • det

        public double det(DoubleMatrix2D A)
        Returns the determinant of matrix A.
        Returns:
        the determinant.
      • inverse

        public DoubleMatrix2D inverse(DoubleMatrix2D A)
        Returns the inverse or pseudo-inverse of matrix A.
        Returns:
        a new independent matrix; inverse(matrix) if the matrix is square, pseudoinverse otherwise.
      • mult

        public double mult(DoubleMatrix1D x,          DoubleMatrix1D y)
        Inner product of two vectors; Sum(x[i] * y[i]). Also known as dot product.
        Equivalent to x.zDotProduct(y).
        Parameters:
        x - the first source vector.
        y - the second source matrix.
        Returns:
        the inner product.
        Throws:
        IllegalArgumentException - if x.size() != y.size().
      • mult

        public DoubleMatrix1D mult(DoubleMatrix2D A,                  DoubleMatrix1D y)
        Linear algebraic matrix-vector multiplication; z = A * y. z[i] = Sum(A[i,j] * y[j]), i=0..A.rows()-1, j=0..y.size()-1.
        Parameters:
        A - the source matrix.
        y - the source vector.
        Returns:
        z; a new vector with z.size()==A.rows().
        Throws:
        IllegalArgumentException - if A.columns() != y.size().
      • mult

        public DoubleMatrix2D mult(DoubleMatrix2D A,                  DoubleMatrix2D B)
        Linear algebraic matrix-matrix multiplication; C = A x B. C[i,j] = Sum(A[i,k] * B[k,j]), k=0..n-1.
        Matrix shapes: A(m x n), B(n x p), C(m x p).
        Parameters:
        A - the first source matrix.
        B - the second source matrix.
        Returns:
        C; a new matrix holding the results, with C.rows()=A.rows(), C.columns()==B.columns().
        Throws:
        IllegalArgumentException - if B.rows() != A.columns().
      • multOuter

        public DoubleMatrix2D multOuter(DoubleMatrix1D x,                       DoubleMatrix1D y,                       DoubleMatrix2D A)
        Outer product of two vectors; Sets A[i,j] = x[i] * y[j].
        Parameters:
        x - the first source vector.
        y - the second source vector.
        A - the matrix to hold the results. Set this parameter to null to indicate that a new result matrix shall be constructed.
        Returns:
        A (for convenience only).
        Throws:
        IllegalArgumentException - if A.rows() != x.size() || A.columns() != y.size().
      • norm1

        public double norm1(DoubleMatrix1D x)
        Returns the one-norm of vector x, which is Sum(abs(x[i])).
      • norm1

        public double norm1(DoubleMatrix2D A)
        Returns the one-norm of matrix A, which is the maximum absolute column sum.
      • norm2

        public double norm2(DoubleMatrix1D x)
        Returns the two-norm (aka euclidean norm) of vector x; equivalent to mult(x,x).
      • norm2

        public double norm2(DoubleMatrix2D A)
        Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.
      • normF

        public double normF(DoubleMatrix2D A)
        Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).
      • normInfinity

        public double normInfinity(DoubleMatrix1D x)
        Returns the infinity norm of vector x, which is Max(abs(x[i])).
      • normInfinity

        public double normInfinity(DoubleMatrix2D A)
        Returns the infinity norm of matrix A, which is the maximum absolute row sum.
      • permute

        public DoubleMatrix1D permute(DoubleMatrix1D A,                     int[] indexes,                     double[] work)
        Modifies the given vector A such that it is permuted as specified; Useful for pivoting.Cell A[i] will go into cell A[indexes[i]].

        Example:

        Reordering[A,B,C,D,E] with indexes [0,4,2,3,1] yields [A,E,C,D,B]In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[2], A[3]<--A[3], A[4]<--A[1].Reordering[A,B,C,D,E] with indexes [0,4,1,2,3] yields [A,E,B,C,D]In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[1], A[3]<--A[2], A[4]<--A[3].
        Parameters:
        A - the vector to permute.
        indexes - the permutation indexes, must satisfy indexes.length==A.size() && indexes[i] >= 0 && indexes[i] < A.size();
        work - the working storage, must satisfy work.length >= A.size(); set work==null if you don't care about performance.
        Returns:
        the modified A (for convenience only).
        Throws:
        IndexOutOfBoundsException - if indexes.length != A.size().
      • permute

        public DoubleMatrix2D permute(DoubleMatrix2D A,                     int[] rowIndexes,                     int[] columnIndexes)
        Constructs and returns a new row and column permuted selection view of matrix A; equivalent to DoubleMatrix2D.viewSelection(int[],int[]).The returned matrix is backed by this matrix, so changes in the returned matrix are reflected in this matrix, and vice-versa.Use idioms like result = permute(...).copy() to generate an independent sub matrix.
        Returns:
        the new permuted selection view.
      • permuteColumns

        public DoubleMatrix2D permuteColumns(DoubleMatrix2D A,                            int[] indexes,                            int[] work)
        Modifies the given matrix A such that it's columns are permuted as specified; Useful for pivoting.Column A[i] will go into column A[indexes[i]].Equivalent to permuteRows(transpose(A), indexes, work).
        Parameters:
        A - the matrix to permute.
        indexes - the permutation indexes, must satisfy indexes.length==A.columns() && indexes[i] >= 0 && indexes[i] < A.columns();
        work - the working storage, must satisfy work.length >= A.columns(); set work==null if you don't care about performance.
        Returns:
        the modified A (for convenience only).
        Throws:
        IndexOutOfBoundsException - if indexes.length != A.columns().
      • permuteRows

        public DoubleMatrix2D permuteRows(DoubleMatrix2D A,                         int[] indexes,                         int[] work)
        Modifies the given matrix A such that it's rows are permuted as specified; Useful for pivoting.Row A[i] will go into row A[indexes[i]].

        Example:

        Reordering[A,B,C,D,E] with indexes [0,4,2,3,1] yields [A,E,C,D,B]In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[2], A[3]<--A[3], A[4]<--A[1].Reordering[A,B,C,D,E] with indexes [0,4,1,2,3] yields [A,E,B,C,D]In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[1], A[3]<--A[2], A[4]<--A[3].
        Parameters:
        A - the matrix to permute.
        indexes - the permutation indexes, must satisfy indexes.length==A.rows() && indexes[i] >= 0 && indexes[i] < A.rows();
        work - the working storage, must satisfy work.length >= A.rows(); set work==null if you don't care about performance.
        Returns:
        the modified A (for convenience only).
        Throws:
        IndexOutOfBoundsException - if indexes.length != A.rows().
      • pow

        public DoubleMatrix2D pow(DoubleMatrix2D A,                 int p)
        Linear algebraic matrix power; B = Ak <==> B = A*A*...*A.
        • p >= 1: B = A*A*...*A.
        • p == 0: B = identity matrix.
        • p < 0: B = pow(inverse(A),-p).
        Implementation: Based on logarithms of 2, memory usage minimized.
        Parameters:
        A - the source matrix; must be square; stays unaffected by this operation.
        p - the exponent, can be any number.
        Returns:
        B, a newly constructed result matrix; storage-independent of A.
        Throws:
        IllegalArgumentException - if !property().isSquare(A).
      • property

        public Property property()
        Returns the property object attached to this Algebra, defining tolerance.
        Returns:
        the Property object.
        See Also:
        setProperty(Property)
      • rank

        public int rank(DoubleMatrix2D A)
        Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
      • setProperty

        public void setProperty(Property property)
        Attaches the given property object to this Algebra, defining tolerance.
        Parameters:
        the - Property object to be attached.
        Throws:
        UnsupportedOperationException - if this==DEFAULT && property!=this.property() - The DEFAULT Algebra object is immutable.
        UnsupportedOperationException - if this==ZERO && property!=this.property() - The ZERO Algebra object is immutable.
        See Also:
        property
      • solveTranspose

        public DoubleMatrix2D solveTranspose(DoubleMatrix2D A,                            DoubleMatrix2D B)
        Solves X*A = B, which is also A'*X' = B'.
        Returns:
        X; a new independent matrix; solution if A is square, least squares solution otherwise.
      • subMatrix

        public DoubleMatrix2D subMatrix(DoubleMatrix2D A,                       int fromRow,                       int toRow,                       int fromColumn,                       int toColumn)
        Constructs and returns a new sub-range view which is the sub matrix A[fromRow..toRow,fromColumn..toColumn].The returned matrix is backed by this matrix, so changes in the returned matrix are reflected in this matrix, and vice-versa.Use idioms like result = subMatrix(...).copy() to generate an independent sub matrix.
        Parameters:
        A - the source matrix.
        fromRow - The index of the first row (inclusive).
        toRow - The index of the last row (inclusive).
        fromColumn - The index of the first column (inclusive).
        toColumn - The index of the last column (inclusive).
        Returns:
        a new sub-range view.
        Throws:
        IndexOutOfBoundsException - if fromColumn<0 || toColumn-fromColumn+1<0 || toColumn>=A.columns() || fromRow<0 || toRow-fromRow+1<0 || toRow>=A.rows()
      • toString

        public String toString(DoubleMatrix2D matrix)
        Returns a String with (propertyName, propertyValue) pairs.Useful for debugging or to quickly get the rough picture.For example,
        cond          : 14.073264490042144det           : Illegal operation or error: Matrix must be square.norm1         : 0.9620244354009628norm2         : 3.0normF         : 1.304841791648992normInfinity  : 1.5406551198102534rank          : 3trace         : 0
      • toVerboseString

        public String toVerboseString(DoubleMatrix2D matrix)
        Returns the results of toString(A) and additionally the results of all sorts of decompositions applied to the given matrix.Useful for debugging or to quickly get the rough picture.For example,
        A = 3 x 3 matrix249  66  68104 214 108144 146 293cond         : 3.931600417472078det          : 9638870.0norm1        : 497.0norm2        : 473.34508217011404normF        : 516.873292016525normInfinity : 583.0rank         : 3trace        : 756.0density                      : 1.0isDiagonal                   : falseisDiagonallyDominantByColumn : trueisDiagonallyDominantByRow    : trueisIdentity                   : falseisLowerBidiagonal            : falseisLowerTriangular            : falseisNonNegative                : trueisOrthogonal                 : falseisPositive                   : trueisSingular                   : falseisSkewSymmetric              : falseisSquare                     : trueisStrictlyLowerTriangular    : falseisStrictlyTriangular         : falseisStrictlyUpperTriangular    : falseisSymmetric                  : falseisTriangular                 : falseisTridiagonal                : falseisUnitTriangular             : falseisUpperBidiagonal            : falseisUpperTriangular            : falseisZero                       : falselowerBandwidth               : 2semiBandwidth                : 3upperBandwidth               : 2-----------------------------------------------------------------------------LUDecompositionQuick(A) --> isNonSingular(A), det(A), pivot, L, U, inverse(A)-----------------------------------------------------------------------------isNonSingular = truedet = 9638870.0pivot = [0, 1, 2]L = 3 x 3 matrix1        0       00.417671 1       00.578313 0.57839 1U = 3 x 3 matrix249  66         68         0 186.433735  79.598394  0   0        207.635819inverse(A) = 3 x 3 matrix 0.004869 -0.000976 -0.00077 -0.001548  0.006553 -0.002056-0.001622 -0.002786  0.004816-----------------------------------------------------------------QRDecomposition(A) --> hasFullRank(A), H, Q, R, pseudo inverse(A)-----------------------------------------------------------------hasFullRank = trueH = 3 x 3 matrix1.814086 0        00.34002  1.903675 00.470797 0.428218 2Q = 3 x 3 matrix-0.814086  0.508871  0.279845-0.34002  -0.808296  0.48067 -0.470797 -0.296154 -0.831049R = 3 x 3 matrix-305.864349 -195.230337 -230.023539   0        -182.628353  467.703164   0           0        -309.13388 pseudo inverse(A) = 3 x 3 matrix 0.006601  0.001998 -0.005912-0.005105  0.000444  0.008506-0.000905 -0.001555  0.002688--------------------------------------------------------------------------CholeskyDecomposition(A) --> isSymmetricPositiveDefinite(A), L, inverse(A)--------------------------------------------------------------------------isSymmetricPositiveDefinite = falseL = 3 x 3 matrix15.779734  0         0        6.590732 13.059948  0        9.125629  6.573948 12.903724inverse(A) = Illegal operation or error: Matrix is not symmetric positive definite.---------------------------------------------------------------------EigenvalueDecomposition(A) --> D, V, realEigenvalues, imagEigenvalues---------------------------------------------------------------------realEigenvalues = 1 x 3 matrix462.796507 172.382058 120.821435imagEigenvalues = 1 x 3 matrix0 0 0D = 3 x 3 matrix462.796507   0          0         0        172.382058   0         0          0        120.821435V = 3 x 3 matrix-0.398877 -0.778282  0.094294-0.500327  0.217793 -0.806319-0.768485  0.66553   0.604862---------------------------------------------------------------------SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V---------------------------------------------------------------------cond = 3.931600417472078rank = 3norm2 = 473.34508217011404U = 3 x 3 matrix0.46657  -0.877519  0.1107770.50486   0.161382 -0.8479820.726243  0.45157   0.51832 S = 3 x 3 matrix473.345082   0          0         0        169.137441   0         0          0        120.395013V = 3 x 3 matrix0.577296 -0.808174  0.1165460.517308  0.251562 -0.8179910.631761  0.532513  0.563301
      • trace

        public double trace(DoubleMatrix2D A)
        Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).
      • transpose

        public DoubleMatrix2D transpose(DoubleMatrix2D A)
        Constructs and returns a new view which is the transposition of the given matrix A.Equivalent to A.viewDice().This is a zero-copy transposition, taking O(1), i.e. constant time.The returned view is backed by this matrix, so changes in the returned view are reflected in this matrix, and vice-versa. Use idioms like result = transpose(A).copy() to generate an independent matrix.

        Example:

        2 x 3 matrix:
        1, 2, 3
        4, 5, 6
        transpose ==> 3 x 2 matrix:
        1, 4
        2, 5
        3, 6
        transpose ==> 2 x 3 matrix:
        1, 2, 3
        4, 5, 6
        Returns:
        a new transposed view.

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