Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: B = A.change_ring(CDF)
sage: sv = B.singular_values()
sage: sv[0:2]
[10.1973039..., 0.487045871...]
sage: sv[2] < 1e-14
True
A matrix of rank 3 over the complex numbers.
sage: A = matrix(CDF, [[46*I - 28, -47*I - 50, 21*I + 51, -62*I - 782, 13*I + 22],
... [35*I - 20, -32*I - 46, 18*I + 43, -57*I - 670, 7*I + 3],
... [22*I - 13, -23*I - 23, 9*I + 24, -26*I - 347, 7*I + 13],
... [-44*I + 23, 41*I + 57, -19*I - 54, 60*I + 757, -11*I - 9],
... [30*I - 18, -30*I - 34, 14*I + 34, -42*I - 522, 8*I + 12]])
sage: sv = A.singular_values()
sage: sv[0:3]
[1440.733666, 18.4044034134, 6.83970779714]
sage: (10^-15 < sv[3]) and (sv[3] < 10^-13)
True
sage: (10^-16 < sv[4]) and (sv[4] < 10^-14)
True
A full-rank matrix that is ill-conditioned. We use this to illustrate ways of using the various possibilities for
eps, including one that is ill-advised. Notice that the automatically computed cutoff gets this (difficult)
example slightly wrong. This illustrates the impossibility of any automated process always getting this
right. Use with caution and judgement.
sage: entries = [1/(i+j+1) for i in range(12) for j in range(12)]
sage: B = matrix(QQ, 12, 12, entries)
sage: B.rank()
12
sage: A = B.change_ring(RDF)
sage: A.condition() > 1.6e16 or A.condition()
True
sage: A.singular_values(eps=None) # abs tol 1e-16
[1.79537205956, 0.380275245955, 0.0447385487522, 0.00372231223789, 0.000233089089022, 1.1163
sage: A.singular_values(eps=’auto’) # abs tol 1e-16
[1.79537205956, 0.380275245955, 0.0447385487522, 0.00372231223789, 0.000233089089022, 1.1163
sage: A.singular_values(eps=1e-4)
[1.79537205956, 0.380275245955, 0.0447385487522, 0.00372231223789, 0.000233089089022, 0.0, 0
With Sage’s “verbose” facility, you can compactly see the cutoff at work. In any application of this routine,
or those that build upon it, it would be a good idea to conduct this exercise on samples. We also test here
that all the values are returned in RDF since singular values are always real.
sage: A = matrix(CDF, 4, range(16))
sage: set_verbose(1)
sage: sv = A.singular_values(eps=’auto’); sv
verbose 1 () singular values,
smallest-non-zero:cutoff:largest-zero,
2.2766...:6.2421...e-14:...
[35.139963659, 2.27661020871, 0.0, 0.0]
sage: set_verbose(0)
sage: all([s in RDF for s in sv])
True
388 Chapter 19. Dense matrices using a NumPy backend.