# Arbitrary precision complex ball matrices using Arb¶

AUTHORS:

• Clemens Heuberger (2014-10-25): Initial version.

This is a rudimentary binding to the Arb library; it may be useful to refer to its documentation for more details.

class sage.matrix.matrix_complex_ball_dense.Matrix_complex_ball_dense

Matrix over a complex ball field. Implemented using the acb_mat type of the Arb library.

EXAMPLES:

sage: MatrixSpace(CBF, 3)(2)
[2.000000000000000                 0                 0]
[                0 2.000000000000000                 0]
[                0                 0 2.000000000000000]
sage: matrix(CBF, 1, 3, [1, 2, -3])
[ 1.000000000000000  2.000000000000000 -3.000000000000000]

charpoly(var='x', algorithm=None)

Compute the characteristic polynomial of this matrix.

EXAMPLES:

sage: from sage.matrix.benchmark import hilbert_matrix
sage: mat = hilbert_matrix(5).change_ring(ComplexBallField(10))
sage: mat.charpoly()
x^5 + ([-1.8 +/- 0.0258])*x^4 + ([0.3 +/- 0.05...)*x^3 +
([+/- 0.0...])*x^2 + ([+/- 0.0...])*x + [+/- 0.0...]

contains(other)

Test if the set of complex matrices represented by self is contained in that represented by other.

EXAMPLES:

sage: b = CBF(0, RBF(0, rad=.1r)); b
[+/- 0.101]*I
sage: matrix(CBF, [0, b]).contains(matrix(CBF, [0, 0]))
True
sage: matrix(CBF, [0, b]).contains(matrix(CBF, [b, 0]))
False
sage: matrix(CBF, [b, b]).contains(matrix(CBF, [b, 0]))
True

determinant()

Compute the determinant of this matrix.

EXAMPLES:

sage: matrix(CBF, [[1/2, 1/3], [1, 1]]).determinant()
[0.1666666666666667 +/- ...e-17]
sage: matrix(CBF, [[1/2, 1/3], [1, 1]]).det()
[0.1666666666666667 +/- ...e-17]
sage: matrix(CBF, [[1/2, 1/3]]).determinant()
Traceback (most recent call last):
...
ValueError: self must be a square matrix

exp()

Compute the exponential of this matrix.

EXAMPLES:

sage: matrix(CBF, [[i*pi, 1], [0, i*pi]]).exp()
[[-1.00000000000000 +/- ...e-16] + [+/- ...e-16]*I [-1.00000000000000 +/- ...e-16] + [+/- ...e-16]*I]
[                                                0 [-1.00000000000000 +/- ...e-16] + [+/- ...e-16]*I]
sage: matrix(CBF, [[1/2, 1/3]]).exp()
Traceback (most recent call last):
...
ValueError: self must be a square matrix

identical(other)

Test if the corresponding entries of two complex ball matrices represent the same balls.

EXAMPLES:

sage: a = matrix(CBF, [[1/3,2],[3,4]])
sage: b = matrix(CBF, [[1/3,2],[3,4]])
sage: a == b
False
sage: a.identical(b)
True

overlaps(other)

Test if two matrices with complex ball entries represent overlapping sets of complex matrices.

EXAMPLES:

sage: b = CBF(0, RBF(0, rad=0.1r)); b
[+/- 0.101]*I
sage: matrix(CBF, [0, b]).overlaps(matrix(CBF, [b, 0]))
True
sage: matrix(CBF, [1, 0]).overlaps(matrix(CBF, [b, 0]))
False

trace()

Compute the trace of this matrix.

EXAMPLES:

sage: matrix(CBF, [[1/3, 1/3], [1, 1]]).trace()
[1.333333333333333 +/- ...e-16]
sage: matrix(CBF, [[1/2, 1/3]]).trace()
Traceback (most recent call last):
...
ValueError: self must be a square matrix