Non-Evaluation of Numerical Expression in Maxima

Question

I start with a simple Maxima question, the answer to which may provide the answer to the actual problem I'm grappling with.

Related Simple Question: How can I get maxima to calculate: bfloat((1+%i)^0.3); Might there be an option variable that can be set so that this evaluates to a complex number?

Actual Question: In evaluating approximations for numerical time integration for finite element methods, for this purpose I'm using spectral analysis, which requires the calculation of the eigenvalues of a 4 x 4 matrix. This matrix "cav" is also calculated within maxima, using some of the algebra capabilities of maxima, but sustituting numerical values, so that matrix is entirely numerical, i.e. containing no variables. I've calculated the eigenvalues with Mathematica and it returns 4 real eigenvalues. However Maxima calculates horrenduously complicated expressions for this case, which apparently it does not "know" how to simplify, even numerically as "bigfloat". Perhaps this problem arises because Maxima first approximates the matrix "cac" by rational numbers (i.e. fractions) and then tries to solve the problem fully exactly, instead of simply using numerical "bigfloat" computations throughout. Is there I way I can change this?

Note that if you only change the input value of gzv to say 0.5 it works fine, and returns numerical values of complex eigenvalues.

I include the code below. Note that all of the code up until "cav:subst(vs,ca)$" is just for the definition of the matrix cav and seems to work fine. It is in the few statements thereafter that it fails to calculate numerical values for the eigenvalues.

v1:v0+ (1-gg)*a0+gg*a1$
d1:d0+v0+(1/2-gb)*a0+gb*a1$
obf:a1+(1+ga)*(w^2*d1 + 2*gz*w*(d1-d0)) -
          ga *(w^2*d0 + 2*gz*w*(d0-g0))$
obf:expand(obf)$
cd:subst([a1=1,d0=0,v0=0,a0=0,g0=0],obf)$
fd:subst([a1=0,d0=1,v0=0,a0=0,g0=0],obf)$
fv:subst([a1=0,d0=0,v0=1,a0=0,g0=0],obf)$
fa:subst([a1=0,d0=0,v0=0,a0=1,g0=0],obf)$
fg:subst([a1=0,d0=0,v0=0,a0=0,g0=1],obf)$
f:[fd,fv,fa,fg]$
cad1:expand(cd*[1,1,1/2-gb,0] - gb*f)$
cad2:expand(cd*[0,1,1-gg,0]   - gg*f)$
cad3:expand(-f)$
cad4:[cd,0,0,0]$
cad:matrix(cad1,cad2,cad3,cad4)$
gav:-0.05$
ggv:1/2-gav$
gbv:(ggv+1/2)^2/4$
gzv:1.1$
dt:0.01$
wv:bfloat(dt*2*%pi)$
vs:[ga=gav,gg=ggv,gb=gbv,gz=gzv,w=wv]$
cav:subst(vs,ca)$
cav:bfloat(cav)$
evam:eigenvalues(cav)$
evam:bfloat(evam)$
eva:evam[1]$

Answer 1

The main problem here is that Maxima tries pretty hard to make computations exact, and it's hard to tell it to ease up and allow inexact results.

Is there a mistake in the code you posted above? You have cav:subst(vs,ca) but ca is not defined. Is that supposed to be cav:subst(vs,cad) ?

For the short problem, usually rectform can simplify complex expressions to something more usable:

(%i58) rectform (bfloat((1+%i)^0.3));
`rat' replaced 1.0B0 by 1/1 = 1.0B0
(%o58)       2.59023849130283b-1 %i + 1.078911979230303b0

About the long problem, if fixed-precision (i.e. ordinary floats, not bigfloats) is acceptable to you, then you can use the LAPACK function dgeev to compute eigenvalues and/or eigenvectors.

(%i51) load (lapack);
<bunch of messages here>
(%o51)    /usr/share/maxima/5.39.0/share/lapack/lapack.mac
(%i52) dgeev (cav);
(%o52)    [[- 0.02759949957202372, 0.06804641655485913, 0.997993508502892, 0.928429191717788], false, false]

If you really need variable precision, I don't know what to try. In principle it's possible to rework the LAPACK code to work with variable-precision floats, but that's a substantial task and I'm not sure about the details.