@@ -788,27 +788,28 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
788788 ...
789789 ValueError: polynomial with interval coefficients, use multiplicities=False
790790
791- sage: ( x^ 4 - 1/3) . roots( multiplicities=False) # indirect doctest
792- [ [-0.759835685651593 +/- ...e -16 ] + [+/- ...e -16 ]* I,
793- [0.759835685651593 +/- ...e -16 ] + [+/- ...e -16 ]* I,
794- [+/- ...e -16 ] + [0.759835685651593 +/- ...e -16 ]* I,
795- [+/- ...e -16 ] + [-0.759835685651593 +/- ...e -16 ]* I]
791+ sage: set (( x^ 4 - 1/3) . roots( multiplicities=False)) # indirect doctest
792+ {[ +/- 1.27e -16 ] + [-0.759835685651593 +/- 5.90e -16 ]* I,
793+ [+/- 1.27e -16 ] + [0.759835685651593 +/- 5.90e -16 ]* I,
794+ [-0.759835685651593 +/- 5.90e -16 ] + [+/- 1.27e -16 ]* I,
795+ [0.759835685651593 +/- 5.90e -16 ] + [+/- 1.27e -16 ]* I}
796796
797- sage: ( x^ 4 - 1/3) . roots( RBF, multiplicities=False)
798- [ [ -0.759835685651593 +/- ...e -16 ], [0.759835685651593 +/- ...e -16 ]]
797+ sage: set (( x^ 4 - 1/3) . roots( RBF, multiplicities=False) )
798+ {[ -0.759835685651593 +/- 5.90e -16 ], [0.759835685651593 +/- 5.90e -16 ]}
799799
800- sage: ( x^ 4 - 3) . roots( RealBallField( 100) , multiplicities=False)
801- [ [ -1.316074012952492460819218901797 +/- ...e -34 ],
802- [1.316074012952492460819218901797 +/- ...e -34 ]]
800+ sage: set (( x^ 4 - 3) . roots( RealBallField( 100) , multiplicities=False) )
801+ {[ -1.316074012952492460819218901797 +/- 9.7e -34 ],
802+ [1.316074012952492460819218901797 +/- 9.7e -34 ]}
803803
804- sage: ( x^ 4 - 3) . roots( ComplexIntervalField( 100) , multiplicities=False)
804+ sage: sorted(( x^ 4 - 3) . roots( ComplexIntervalField( 100) ,
805+ .... : multiplicities=False) , key=repr)
805806 [-1.31607401295249246081921890180? + 0.?e-37*I,
806- 1.31607401295249246081921890180? + 0.?e-37*I,
807807 0.?e-37 + 1.31607401295249246081921890180?*I,
808- 0.?e-37 - 1.31607401295249246081921890180?*I ]
808+ 0.?e-37 - 1.31607401295249246081921890180?*I,
809+ 1.31607401295249246081921890180? + 0.?e-37*I ]
809810
810- sage: ( x^ 2 - i/3) . roots( ComplexBallField( 2) , multiplicities=False)
811- [ [ +/- 0.409 ] + [+/- 0.409 ]* I, [+/- 0.409 ] + [+/- 0.409 ]* I]
811+ sage: set (( x^ 2 - i/3) . roots( ComplexBallField( 2) , multiplicities=False) )
812+ {[ +/- 0.409 ] + [+/- 0.409 ]* I, [+/- 0.409 ] + [+/- 0.409 ]* I}
812813
813814 sage: (( x - 1) ^ 2) . roots( multiplicities=False)
814815 Traceback ( most recent call last) :
@@ -818,8 +819,8 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
818819 sage: (( x - 1) ^ 2) . roots( multiplicities=False, proof=False)
819820 doctest:...
820821 UserWarning: roots may have been lost...
821- [[1.00000000000 +/- ...e-12 ] + [+/- ...e-11 ]* I,
822- [1.0000000000 +/- ...e-12 ] + [+/- ...e-12 ]* I]
822+ [[1.000000000... +/- ... ] + [+/- ... ]* I,
823+ [1.000000000... +/- ... ] + [+/- ... ]* I]
823824
824825 sage: pol = x^ 7 - 2* ( 1000* x - 1) ^ 2 # Mignotte polynomial
825826 sage: pol. roots( multiplicities=False)
@@ -844,7 +845,7 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
844845 sage: (( x - 1) ^ 2 + 2^ ( -70) * i/3) . roots( RBF, multiplicities=False)
845846 Traceback ( most recent call last) :
846847 ...
847- ValueError: unable to determine which roots are real
848+ ValueError: ...
848849
849850 TESTS::
850851
@@ -4261,14 +4262,14 @@ cdef class ComplexBall(RingElement):
42614262
42624263 EXAMPLES::
42634264
4264- sage: CBF(1, 1).Ei()
4265- [1.76462598556385 +/- ...e -15] + [2.38776985151052 +/- ...e -15]*I
4265+ sage: CBF(1, 1).Ei() # abs tol 6e-15
4266+ [1.76462598556385 +/- 6.03e -15] + [2.38776985151052 +/- 4.23e -15]*I
42664267 sage: CBF(0).Ei()
4267- nan
4268+ nan...
42684269
42694270 TESTS:
42704271
4271- sage: CBF(Ei(I)) # abs tol 1e-16 # needs sage.symbolic
4272+ sage: CBF(Ei(I)) # abs tol 2e-15 # needs sage.symbolic
42724273 [0.337403922900968 +/- 3.76e-16] + [2.51687939716208 +/- 2.01e-15]*I
42734274 """
42744275 cdef ComplexBall result = self ._new()
@@ -4283,14 +4284,14 @@ cdef class ComplexBall(RingElement):
42834284
42844285 EXAMPLES::
42854286
4286- sage: CBF(1, 1).Si()
4287- [1.10422265823558 +/- ...e -15] + [0.88245380500792 +/- ...e -15]*I
4287+ sage: CBF(1, 1).Si() # abs tol 3e-15
4288+ [1.10422265823558 +/- 2.48e -15] + [0.88245380500792 +/- 3.36e -15]*I
42884289 sage: CBF(0).Si()
42894290 0
42904291
42914292 TESTS:
42924293
4293- sage: CBF(Si(I)) # needs sage.symbolic
4294+ sage: CBF(Si(I)) # abs tol 3e-15 # needs sage.symbolic
42944295 [1.05725087537573 +/- 2.77e-15]*I
42954296 """
42964297 cdef ComplexBall result = self ._new()
@@ -4307,14 +4308,14 @@ cdef class ComplexBall(RingElement):
43074308
43084309 EXAMPLES::
43094310
4310- sage: CBF(1, 1).Ci()
4311- [0.882172180555936 +/- ...e -16] + [0.287249133519956 +/- ...e -16]*I
4311+ sage: CBF(1, 1).Ci() # abs tol 5e-16
4312+ [0.882172180555936 +/- 5.89e -16] + [0.287249133519956 +/- 3.37e -16]*I
43124313 sage: CBF(0).Ci()
43134314 nan + nan*I
43144315
43154316 TESTS:
43164317
4317- sage: CBF(Ci(I)) # abs tol 1e-17 # needs sage.symbolic
4318+ sage: CBF(Ci(I)) # abs tol 5e-16 # needs sage.symbolic
43184319 [0.837866940980208 +/- 4.72e-16] + [1.570796326794897 +/- 5.54e-16]*I
43194320 """
43204321 cdef ComplexBall result = self ._new()
@@ -4331,8 +4332,8 @@ cdef class ComplexBall(RingElement):
43314332
43324333 EXAMPLES::
43334334
4334- sage: CBF(1, 1).Shi()
4335- [0.88245380500792 +/- ...e -15] + [1.10422265823558 +/- ...e -15]*I
4335+ sage: CBF(1, 1).Shi() # abs tol 3e-15
4336+ [0.88245380500792 +/- 3.36e -15] + [1.10422265823558 +/- 2.48e -15]*I
43364337 sage: CBF(0).Shi()
43374338 0
43384339
@@ -4355,14 +4356,14 @@ cdef class ComplexBall(RingElement):
43554356
43564357 EXAMPLES::
43574358
4358- sage: CBF(1, 1).Chi()
4359- [0.882172180555936 +/- ...e -16] + [1.28354719327494 +/- ...e -15]*I
4359+ sage: CBF(1, 1).Chi() # abs tol 1e-15
4360+ [0.882172180555936 +/- 5.89e -16] + [1.28354719327494 +/- 1.01e -15]*I
43604361 sage: CBF(0).Chi()
43614362 nan + nan*I
43624363
43634364 TESTS:
43644365
4365- sage: CBF(Chi(I)) # abs tol 1e -16 # needs sage.symbolic
4366+ sage: CBF(Chi(I)) # abs tol 5e -16 # needs sage.symbolic
43664367 [0.337403922900968 +/- 3.25e-16] + [1.570796326794897 +/- 5.54e-16]*I
43674368 """
43684369 cdef ComplexBall result = self ._new()
@@ -4381,8 +4382,8 @@ cdef class ComplexBall(RingElement):
43814382
43824383 EXAMPLES::
43834384
4384- sage: CBF(1, 1).li()
4385- [0.61391166922120 +/- ...e -15] + [2.05958421419258 +/- ...e -15]*I
4385+ sage: CBF(1, 1).li() # abs tol 6e-15
4386+ [0.61391166922120 +/- 6.23e -15] + [2.05958421419258 +/- 5.59e -15]*I
43864387 sage: CBF(0).li()
43874388 0
43884389 sage: CBF(0).li(offset=True)
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