matrix is found singular over CC, nonsingular over CDF

edit

Dealing with rounding errors

Why is this happening and is there another way I can avoid large rounding errors when taking the inverse?

This is happening because CC and CDF use floating-point numbers, and computations are thus inexact and subject to rounding errors.

Instead of working with CC or CDF, one could

• use matrices over GaussianIntegers() or ZZ[i] or QQ[i] or QQbar for exact computations

• use matrices over CBF ("complex ball field") for floating-point computations with rigorous error bounds

Providing a large example

I'm not really sure how to provide an example of this error occurring since the smallest example of the array I am working with is (52x52).

Communicating a 52 x 52 matrix with entries in {0, 1, -1, i, -i} is doable.

You can encode the matrix and provide decoding instructions rather than the matrix itself.

One possible encoding, since the entries can only take 5 values, take them as 5-ary digits; then each row can be thought of as the 5-ary expansion of a number. If we agree on the digits and communicate the numbers, we're good.

Here is a random matrix a with entries in {0, 1, -1, i, -i}:

sage: G = GaussianIntegers()
sage: U = [0, 1, -1, i, -i]

sage: a = matrix(G, 52, lambda i, j: U[randint(0, 4)])

Encoded matrix:

sage: b = [Integer(tuple(U.index(e) for e in row), 5) for row in a]

sage: print(b)
[1986835196674783292709127771671427698, 372532889760445898415871372893604877,
 29903254311383361818758881193795318, 1358695302115484662298806740046547384,
 1537321601024823802437657316988965568, 807200500103255107975400681439180354,
 567350510853383980331713019521919561, 1502850803304703506587108729499355006,
 846884730762028905888265726139310683, 137381915837271267181515478275643286,
 522105367460276163750943030646864364, 1062521690576328608603843582516522599,
 899547582492594668160772320652245190, 767732247768012897653127687752209191,
 629354811960707798077793065391805161, 922202162939977177914455168534168765,
 1729869497057901779087386639626774141, 1868018985439944039499425518652161649,
 1296299082686624643142119402841062894, 1064799841987832209268578137251968874,
 2155066789175869415783929309121456698, 267366791364966820259349410583047599,
 1632002494019334195780113340724191796, 2192661273536615612340019587291366905,
 1762917404578120599684457836032844498, 228986171419406644876547615831863389,
 2101920423365829833128536344495944345, 660985714605962040665526677335410210,
 873334831095240711598788236281716427, 1744121894438415469560277438117935725,
 2210059965689974187926035573546483126, 1594508474721458635473949546648185838,
 518211385383214049244941993118895991, 231206763245123017363282654207827645,
 1748059359372987512622900717899928645, 259988738192311572545249238464600548,
 2007721547194093983826996655201647232, 1240843285105582949996674635390136465,
 1736982326629685244276013105887136315, 435682393610221223713098672542297123,
 66566788503415822985489945441756435, 1141561970732274264554486688206105649,
 2044822779971527183978088623745613537, 17299608296692077102235770236630252,
 780853597789891512314851615527578148, 427428566224687826328803693567479441,
 1922351158930802787176924223394683886, 2104373328835002410314401979576425296,
 1317300586469460326137557530572476230, 1736284365530465805899187629996562866,
 2132757211837100962505268985637520292, 147031314086172436786678135727403627]

To decode:

sage: aa = matrix(G, 52, [[U[d] for d in r.digits(5, padto=52)] for r in b])
sage: aa == a
True

Here is a function that takes a matrix as an argument, and returns "short enough code" to recreate that matrix.

def matrix_code(a):
    r"""
    Return code for a matrix with entries in {0, 1, -1, i, -i}.

    EXAMPLE::

        sage: G = GaussianIntegers()
        sage: U = [0, 1, -1, i, -i]
        sage: a = matrix(G, 52, lambda i, j: U[randint(0, 4)])
        sage: matrix_code(a)
        ...
    """
    G = GaussianIntegers()
    U = [0, 1, -1, i, -i]
    m = a.nrows()
    n = a.ncols()
    b = [Integer(tuple(U.index(e) for e in row), 5) for row in a]
    print(f"""
    G = GaussianIntegers()
    U = [0, 1, -1, i, -i]
    b = {b}
    a = matrix(G, {m}, {n}, [[U[d] for d in r.digits(5, padto={n})] for r in b])
    """)

Run it on your matrix. Edit your question, pasting the output. Then we can all play with the same matrix.

Exploring an example

(Added after the question was updated to provide a concrete example.)

Thanks for the concrete example. It seems to uncover several bugs!

Define the gaussian integers and their fraction field:

sage: Zi = GaussianIntegers()
sage: Qi = Zi.fraction_field()

The "digits" for decoding the encoded matrix in the updated question:

sage: U = [0, 1, -1, i, -i]

The decoded matrix:

sage: a = matrix(Zi, 52, [[U[d] for d in r.digits(5, padto=52)] for r in b])

Several variations of the same matrix, varying the base ring:

sage: a_Zi = a
sage: a_Qi = a.change_ring(Qi)
sage: a_CC = a.change_ring(CC)
sage: a_CDF = a.change_ring(CDF)
sage: a_QQbar = a_Qi.change_ring(QQbar)

First bug: changing ring directly from the gaussian integers to QQbar fails:

sage: a.change_ring(QQbar)
Traceback (most recent call last)
...
AttributeError: 'AbsoluteOrder_with_category' object has no attribute 'polynomial'

Second bug: all our matrices don't compare equal:

sage: aa = [a_Zi, a_Qi, a_QQbar, a_CC, a_CDF]
sage: print('\n'.join(' '.join(str(int(a == b)) for b in aa) for a in aa))
1 1 0 1 1
1 1 0 1 1
0 0 1 1 1
1 1 1 1 1
1 1 1 1 1

Third bug: the determinant computation over CC fails:

sage: ringdets = [('ZZ[i]', a_Zi.det()),
....:             ('QQ[i]', a_Qi.det()),
....:             ('QQbar', a_QQbar.det()),
....:             ('CC', a_CC.det()),
....:             ('CDF', a_CDF.det())]

sage: print('\n'.join('%5s %18s' % ringdet for ringdet in ringdets))
ZZ[i]                  2
QQ[i]                  2
QQbar                  2
   CC 512.000000000000*I
  CDF                2.0

Taking the inverse

I don't observe problems when taking the inverse:

sage: b_Zi = a_Zi.inverse()
sage: b_Qi = a_Qi.inverse()
sage: b_QQbar = a_QQbar.inverse()
sage: b_CDF = a_CDF.inverse()
sage: b_CC = a_CC.inverse()

and the inverse seems to be computed correctly; the result for CC is rather more accurate than for CDF.

sage: print(b_Zi[:8, :4])
[  -1    1    0    0]
[  -2    1   -1   -1]
[   0    0   -1   -1]
[   0    0    0  1/2]
[   0    0    0  1/2]
[   0    1   -1   -1]
[  -1    1   -1   -1]
[   1    0   -1 -1/2]
sage: print(b_Qi[:8, :4])
[  -1    1    0    0]
[  -2    1   -1   -1]
[   0    0   -1   -1]
[   0    0    0  1/2]
[   0    0    0  1/2]
[   0    1   -1   -1]
[  -1    1   -1   -1]
[   1    0   -1 -1/2]
sage: print(b_QQbar[:8, :4])
[  -1    1    0    0]
[  -2    1   -1   -1]
[   0    0   -1   -1]
[   0    0    0  1/2]
[   0    0    0  1/2]
[   0    1   -1   -1]
[  -1    1   -1   -1]
[   1    0   -1 -1/2]
sage: print(b_CDF[:8, :4])
[    -0.9999999999999998      1.0000000000000004  -7.771561172376096e-16  -2.220446049250313e-16]
[    -1.9999999999999996      1.0000000000000009                    -1.0     -1.0000000000000004]
[ 1.1102230246251565e-16   5.551115123125783e-17                    -1.0     -0.9999999999999999]
[ -6.106226635438361e-16  2.7755575615628914e-16  2.7755575615628914e-16      0.5000000000000002]
[-1.1102230246251568e-16 -1.1102230246251563e-16                     0.0      0.4999999999999999]
[  5.551115123125783e-16      0.9999999999999997     -1.0000000000000004     -1.0000000000000004]
[    -1.0000000000000004      1.0000000000000009     -1.0000000000000004                    -1.0]
[     0.9999999999999998  1.1102230246251565e-16     -1.0000000000000004     -0.5000000000000002]
sage: print(b_CC[:8, :4])
[ -1.00000000000000   1.00000000000000  0.000000000000000  0.000000000000000]
[ -2.00000000000000   1.00000000000000  -1.00000000000000  -1.00000000000000]
[ 0.000000000000000  0.000000000000000  -1.00000000000000  -1.00000000000000]
[ 0.000000000000000  0.000000000000000  0.000000000000000  0.500000000000000]
[ 0.000000000000000  0.000000000000000  0.000000000000000  0.500000000000000]
[ 0.000000000000000   1.00000000000000  -1.00000000000000  -1.00000000000000]