Numerical Analysis

In order to explain our analysis and to provide an exposition of the
PDA method we now commence a careful description of the implementation
of the PDA solutions.
We will discuss both exact sets which include the
minimal exact solution of the previous section and solve the generating equation and other approximate sets which also include the minimal exact solution.
We begin by describing one set in detail and then repeat the procedure for
the other sets without giving all the details.
The first of the non-minimal exact sets
to be used in
solving the PDAs is given in Table 1. This set has been selected to be
fairly small for ease of explanation
but not so small that characteristic features are lost.
The table uses the
graphical notation developed by Fisher [7]
and used by Styer
in his published program [8].
In this graphical notation one gives each
two-variable polynomial
a
matrix of X-es and blanks, an X in
the (*i*,*j*)th element means that *a*_{ij} can take values other than 0,
while a blank in the (*i*,*j*)th element means *a*_{ij}=0.
This graphical notation aids in visualizing the shape of the polynomials
in the two variable space.

The set J of coefficients of V |
The set L
of coefficients of P |

XX | |

X | |

X | |

The set M of coefficients of Q |
The set N of
coefficients of R |

XXX | XX |

XX | XX |

XX | X |

X | |

The polynomials themselves (which solve the generating equation in our case) are:

where

As we can see directly from (4.1) *V* is zero so all rows and columns have
blanks in the *J* set.
The expansion of (4.1) is

where

For the numerical analysis we used 15-35 terms of the series, making analyses with many different generalizations of the minimal approximant. Finally, we added ``noise'' to mimic series with numerical uncertainties. In solving the generating equation numerically we used a Fortran program, based on Styer's PDA subroutine library [8]. We tested our version of the program by repeating the bicritical analysis of [7,11,9], and wish to note that this validation was complicated by problems with the published tables of the bicritical series of [13]. Details of this will be given in [12]. Part of our validation was based on material from the PhD thesis of D. Styer [9]. We have developed a graphical interface for our version of the PDA routine and would be happy to provide this on request.

After validation was complete we first used our
program to approximate *f*(*T*,*U*)
with the approximant of Table 1 and different matching sets *K*
of |*K*|=16 (see Table 2) coefficients
from the series (2.7) (to answer the requirement for unconstrained
approximants
|*K*|=|*J*|+|*L*|+|*M*|+|*N*|-1) with the normalization condition *p*_{00}=1,
and then with non-minimal sets (also unconstrained approximants).
For example set #1 (from Table 2) corresponds to matching the coefficients *c*_{ij} (
with
,
,
,
and
)
of *F*(*T*,*U*) with the corresponding coefficients of *f*(*t*,*u*).
All the approximants from Tables 1 and 2 gave the exact result of
*T*_{0}=*U*_{0}=1,
,
,
,
and *e*_{2}=0.

#1 | #2 | #3 | #4 | #5 |

XXXXX | XXXXX | XXXX | XXXXX | XXXX |

XXXX | XXXXX | XXXX | XXXXX | XXXX |

XXX | XXX | XXXX | XXXX | XXXX |

XX X | XX | XXXX | XX | XX |

X | X | XX | ||

#6 | #7 | #8 | #9 | #10 |

XXX | XXXXX | XXXXXX | XXXXXX | XXXXXX |

XXX | XXX | XXX | XXXX | XX |

XXX | XXX | XX | XX | XX |

XXX | XXX | XX | X | XX |

XX | X | X | X | XX |

XX | X | X | X | X |

X | X | X | X |

As well as the above described case
we solved the
generating equation for a second set of other non-minimal approximants
(Table 3). Similarly
to the case of the first approximants all other approximants gave correct results,
but one approximant (#1 from Table 3) gave the following results:
*T*_{0}=1.0216,
*U*_{0}=1.168,
,
and
*e*_{1}=*e*_{2}=0, for which we have no explanation.

To examine the stability of the approximants and to better mimic series that are not
expansions of exact solutions we inserted increasing amounts of noise into the series.
The noise was introduced by changing randomly the coefficients of the series in
the following way:
;
where *N* is the noise amplitude taking
the values 0.0001, 0.00001, 0.000001, 0.0000001, and 0.00000001, while *R* is a
random number in the interval [-1,1].

# | J |
L |
M |
N |
K |
# | J |
L |
M |
N |
K |

1 | XX | X | XXX | XX | XXXXX | 2 | X | XX | XXX | XXX | XXXXXX |

X | X | XX | X | XXXX | XX | XXX | XX | XXXXXX | |||

X | X | X | XXX | X | XX | X | XXXX | ||||

X | XX X | X | XXX | ||||||||

X | X | ||||||||||

3 | XXX | XX | XXX | XX | XXXXX | 4 | XXXX | XXX | XX | XXXXXX | |

XX | XX | XX | XX | XXXXX | X | XXX | XX | XXXXX | |||

X | XX | X | X | XXXXX | X | XX | XXXX | ||||

XXXXX | X | XX | |||||||||

XX | X | ||||||||||

5 | X | XXX | XXX | XXXXX | 6 | XXXX | XXX | XXXX | XXXX | XXXXXXXX | |

X | XXX | XX | XXXXX | XXX | XX | XXX | XXX | XXXXXXX | |||

X | XXX | X | XXXX | XX | X | XX | XX | XXXXXX | |||

X | XX | X | X | X | XXXXX | ||||||

XX | XXXX | ||||||||||

XXX | |||||||||||

XX | |||||||||||

7 | XX | XX | XXX | XX | XXXXX | 8 | XXX | XXX | XX | XXXXXX | |

X | XX | XX | XXX | XXX | XXX | XX | XXXXX | ||||

X | XXX | XXX | XXX | XX | XXXX | ||||||

X | XXX | XX | XXXX | ||||||||

X | XXX | ||||||||||

XX | |||||||||||

X | |||||||||||

9 | XXX | XXXX | XXX | XXXXXXX | 10 | XXXX | XX | XXX | XXXXXX | ||

XXX | XXXX | XXX | XXXXX | X | XX | XX | XXXX | ||||

X | XX | XX | XXXX | X | XX | XX | XXX | ||||

X | XXX | X | XXX | ||||||||

XXX | XX | ||||||||||

XX | X | ||||||||||

X |

For the first approximants one can see in Fig. 1 and Fig. 2 that as the
noise in the coefficients is reduced, the value of the approximated *T*_{c} and *U*_{c}
converges to 1 as expected, except for approximant #2 which approaches to
*T*_{0}=0.9993
and
*U*_{0}=0.906.

Similarly in Fig. 3 and Fig. 4 one can see that the values of the exponents
and
converge to -0.5 and 2 correspondingly; except for approximant #2 which converges
to
and
.
Notice that the approximant #2 gives bad results for both
the tricritical point and the critical exponents, though in the case of critical exponents the
errors were considerably larger.

FIG.1.

The value of

FIG.2.

The value of

FIG.3.

The value of approximated as a function of the noise for the first approximants

FIG.4.

The value of approximated as a function of the noise for the first approximants

For the second approximants the results were also very encouraging as one can see from
Fig. 5 and Fig. 6, which show the values of the approximated *T*_{c}
and *U*_{c}, correspondingly, as a function of the noise. We can see that except for
approximant #1, which converges to
*T*_{0}=1.004 and
*U*_{0}=1.042, all approximations
converge to 1.

FIG.5.

The value of

FIG.6.

The value of

FIG.7.

The value of approximated as a function of the noise for the second approximants

FIG.8.

The value of approximated as a function of the noise for the second approximants

Similarly the approximated values of the exponents and converge to -0.5 and 2 correspondingly, as can be seen in Fig. 7 and Fig. 8, except for approximant #1, which converges to and . Note that approximant #1 does not yield exact results for either the tricritical point nor the critical exponents, with and without noise. We are not certain what causes this deviation; however it is very clear that in each sample nine of the ten give exact results and so the defective one can be discarded. This is a higher typical success rate than occurs in many Padé type analyses.