DenseFloatAlgebra
cern.colt.matrix.tfloat.algo

Class DenseFloatAlgebra

    • Field Detail

      • DEFAULT

        public static final DenseFloatAlgebra DEFAULT
        A default Algebra object; has FloatProperty.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 DenseFloatAlgebra ZERO
        A default Algebra object; has FloatProperty.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

      • DenseFloatAlgebra

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

        public DenseFloatAlgebra(float 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 float cond(FloatMatrix2D A)
        Returns the condition of matrix A, which is the ratio of largest to smallest singular value.
      • det

        public float det(FloatMatrix2D A)
        Returns the determinant of matrix A.
        Returns:
        the determinant.
      • hypot

        public static float hypot(float a,          float b)
        Returns sqrt(a^2 + b^2) without under/overflow.
      • hypotFunction

        public static FloatFloatFunction hypotFunction()
        Returns sqrt(a^2 + b^2) without under/overflow.
      • inverse

        public FloatMatrix2D inverse(FloatMatrix2D 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 float mult(FloatMatrix1D x,         FloatMatrix1D 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 FloatMatrix1D mult(FloatMatrix2D A,                 FloatMatrix1D 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 FloatMatrix2D mult(FloatMatrix2D A,                 FloatMatrix2D 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 FloatMatrix2D multOuter(FloatMatrix1D x,                      FloatMatrix1D y,                      FloatMatrix2D 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 float norm1(FloatMatrix1D x)
        Returns the one-norm of vector x, which is Sum(abs(x[i])).
      • norm1

        public float norm1(FloatMatrix2D A)
        Returns the one-norm of matrix A, which is the maximum absolute column sum.
      • norm2

        public float norm2(FloatMatrix1D x)
        Returns the two-norm (aka euclidean norm) of vector x; equivalent to Sqrt(mult(x,x)).
      • vectorNorm2

        public float vectorNorm2(FloatMatrix2D X)
        Returns the two-norm (aka euclidean norm) of vector X.vectorize();
      • vectorNorm2

        public float vectorNorm2(FloatMatrix3D X)
        Returns the two-norm (aka euclidean norm) of vector X.vectorize();
      • norm2

        public float norm2(FloatMatrix2D A)
        Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.
      • normF

        public float normF(FloatMatrix2D A)
        Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).
      • normF

        public float normF(FloatMatrix1D A)
        Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i]2)).
      • normInfinity

        public float normInfinity(FloatMatrix1D x)
        Returns the infinity norm of vector x, which is Max(abs(x[i])).
      • normInfinity

        public float normInfinity(FloatMatrix2D A)
        Returns the infinity norm of matrix A, which is the maximum absolute row sum.
      • permute

        public FloatMatrix1D permute(FloatMatrix1D A,                    int[] indexes,                    float[] 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 FloatMatrix2D permute(FloatMatrix2D A,                    int[] rowIndexes,                    int[] columnIndexes)
        Constructs and returns a new row and column permuted selection view of matrix A; equivalent to FloatMatrix2D.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 FloatMatrix2D permuteColumns(FloatMatrix2D 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 FloatMatrix2D permuteRows(FloatMatrix2D 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 FloatMatrix2D pow(FloatMatrix2D 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).
      • rank

        public int rank(FloatMatrix2D A)
        Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
      • setProperty

        public void setProperty(FloatProperty property)
        Attaches the given property object to this Algebra, defining tolerance.
        Parameters:
        property - 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
      • backwardSolve

        public FloatMatrix1D backwardSolve(FloatMatrix2D U,                          FloatMatrix1D b)
        Solves the upper triangular system U*x=b;
        Parameters:
        U - upper triangular matrix
        b - right-hand side
        Returns:
        x, a new independent matrix;
      • forwardSolve

        public FloatMatrix1D forwardSolve(FloatMatrix2D L,                         FloatMatrix1D b)
        Solves the lower triangular system U*x=b;
        Parameters:
        L - lower triangular matrix
        b - right-hand side
        Returns:
        x, a new independent matrix;
      • solveTranspose

        public FloatMatrix2D solveTranspose(FloatMatrix2D A,                           FloatMatrix2D 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 FloatMatrix2D subMatrix(FloatMatrix2D A,                      int[] rowIndexes,                      int columnFrom,                      int columnTo)
        Copies the columns of the indicated rows into a new sub matrix. sub[0..rowIndexes.length-1,0..columnTo-columnFrom] = A[rowIndexes(:),columnFrom..columnTo] ; The returned matrix is not backed by this matrix, so changes in the returned matrix are not reflected in this matrix, and vice-versa.
        Parameters:
        A - the source matrix to copy from.
        rowIndexes - the indexes of the rows to copy. May be unsorted.
        columnFrom - the index of the first column to copy (inclusive).
        columnTo - the index of the last column to copy (inclusive).
        Returns:
        a new sub matrix; with sub.rows()==rowIndexes.length; sub.columns()==columnTo-columnFrom+1 .
        Throws:
        IndexOutOfBoundsException - if columnFrom<0 || columnTo-columnFrom+1<0 || columnTo+1>matrix.columns() || for any row=rowIndexes[i]: row < 0 || row >= matrix.rows() .
      • subMatrix

        public FloatMatrix2D subMatrix(FloatMatrix2D A,                      int rowFrom,                      int rowTo,                      int[] columnIndexes)
        Copies the rows of the indicated columns into a new sub matrix. sub[0..rowTo-rowFrom,0..columnIndexes.length-1] = A[rowFrom..rowTo,columnIndexes(:)] ; The returned matrix is not backed by this matrix, so changes in the returned matrix are not reflected in this matrix, and vice-versa.
        Parameters:
        A - the source matrix to copy from.
        rowFrom - the index of the first row to copy (inclusive).
        rowTo - the index of the last row to copy (inclusive).
        columnIndexes - the indexes of the columns to copy. May be unsorted.
        Returns:
        a new sub matrix; with sub.rows()==rowTo-rowFrom+1; sub.columns()==columnIndexes.length .
        Throws:
        IndexOutOfBoundsException - if rowFrom<0 || rowTo-rowFrom+1<0 || rowTo+1>matrix.rows() || for any col=columnIndexes[i]: col < 0 || col >= matrix.columns() .
      • subMatrix

        public FloatMatrix2D subMatrix(FloatMatrix2D 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(FloatMatrix2D matrix)
        Returns a String with (propertyName, propertyValue) pairs. Useful for debugging or to quickly get the rough picture. For example,
                 cond          : 14.073264490042144         det           : Illegal operation or error: Matrix must be square.         norm1         : 0.9620244354009628         norm2         : 3.0         normF         : 1.304841791648992         normInfinity  : 1.5406551198102534         rank          : 3         trace         : 0  
      • toVerboseString

        public String toVerboseString(FloatMatrix2D 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 matrix         249  66  68         104 214 108         144 146 293          cond         : 3.931600417472078         det          : 9638870.0         norm1        : 497.0         norm2        : 473.34508217011404         normF        : 516.873292016525         normInfinity : 583.0         rank         : 3         trace        : 756.0          density                      : 1.0         isDiagonal                   : false         isDiagonallyDominantByColumn : true         isDiagonallyDominantByRow    : true         isIdentity                   : false         isLowerBidiagonal            : false         isLowerTriangular            : false         isNonNegative                : true         isOrthogonal                 : false         isPositive                   : true         isSingular                   : false         isSkewSymmetric              : false         isSquare                     : true         isStrictlyLowerTriangular    : false         isStrictlyTriangular         : false         isStrictlyUpperTriangular    : false         isSymmetric                  : false         isTriangular                 : false         isTridiagonal                : false         isUnitTriangular             : false         isUpperBidiagonal            : false         isUpperTriangular            : false         isZero                       : false         lowerBandwidth               : 2         semiBandwidth                : 3         upperBandwidth               : 2          -----------------------------------------------------------------------------         LUDecompositionQuick(A) --> isNonSingular(A), det(A), pivot, L, U, inverse(A)         -----------------------------------------------------------------------------         isNonSingular = true         det = 9638870.0         pivot = [0, 1, 2]          L = 3 x 3 matrix         1        0       0         0.417671 1       0         0.578313 0.57839 1          U = 3 x 3 matrix         249  66         68                0 186.433735  79.598394         0   0        207.635819          inverse(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 = true          H = 3 x 3 matrix         1.814086 0        0         0.34002  1.903675 0         0.470797 0.428218 2          Q = 3 x 3 matrix         -0.814086  0.508871  0.279845         -0.34002  -0.808296  0.48067          -0.470797 -0.296154 -0.831049          R = 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 = false          L = 3 x 3 matrix         15.779734  0         0                6.590732 13.059948  0                9.125629  6.573948 12.903724          inverse(A) = Illegal operation or error: Matrix is not symmetric positive definite.          ---------------------------------------------------------------------         EigenvalueDecomposition(A) --> D, V, realEigenvalues, imagEigenvalues         ---------------------------------------------------------------------         realEigenvalues = 1 x 3 matrix         462.796507 172.382058 120.821435         imagEigenvalues = 1 x 3 matrix         0 0 0          D = 3 x 3 matrix         462.796507   0          0                0        172.382058   0                0          0        120.821435          V = 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 = 3.931600417472078         rank = 3         norm2 = 473.34508217011404          U = 3 x 3 matrix         0.46657  -0.877519  0.110777         0.50486   0.161382 -0.847982         0.726243  0.45157   0.51832           S = 3 x 3 matrix         473.345082   0          0                0        169.137441   0                0          0        120.395013          V = 3 x 3 matrix         0.577296 -0.808174  0.116546         0.517308  0.251562 -0.817991         0.631761  0.532513  0.563301  
      • trace

        public float trace(FloatMatrix2D A)
        Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).
      • transpose

        public FloatMatrix2D transpose(FloatMatrix2D 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.
      • trapezoidalLower

        public FloatMatrix2D trapezoidalLower(FloatMatrix2D A)
        Modifies the matrix to be a lower trapezoidal matrix.
        Returns:
        A (for convenience only).
      • xmultOuter

        public FloatMatrix2D xmultOuter(FloatMatrix1D x,                       FloatMatrix1D y)
        Outer product of two vectors; Returns a matrix with A[i,j] = x[i] * y[j].
        Parameters:
        x - the first source vector.
        y - the second source vector.
        Returns:
        the outer product A.
      • xpowSlow

        public FloatMatrix2D xpowSlow(FloatMatrix2D A,                     int k)
        Linear algebraic matrix power; B = Ak <==> B = A*A*...*A.
        Parameters:
        A - the source matrix; must be square.
        k - the exponent, can be any number.
        Returns:
        a new result matrix.
        Throws:
        IllegalArgumentException - if !Testing.isSquare(A).

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