   Next: Conclusions Up: Series Analysis of Tricritical Previous: Analytic Solution by PDA

# 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  and used by Styer in his published program . 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 aij can take values other than 0, while a blank in the (i,j)th element means aij=0. This graphical notation aids in visualizing the shape of the polynomials in the two variable space.

Table 1: The sets J,L,M, and N, corresponding to the approximant of equation (4.1).
 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 p is a constant parameter.

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 pij, qij, and rij are such that i denotes columns and j denotes rows in Table 1. In the present model p00=4p ; p10=-3p ; p01=-2p ; p02=p etc. it is because of the exact solution that we are able to specify the coefficients, When no exact solution is avaliable a polynomial of the same shape would still be indicated by two X-es in the first row corresponding to the non-zero constant term p00 and T coefficient p10, and those in the first column of the second and third rows, correspond to the coefficients p01 and p02 for U and U2, correspondingly. Similarly for the coefficients of Q and R; we can see that the three X-es in the first row of the set M correspond to the constant, T and T2 terms. For this first non-minimal set we checked that it is an analytic solution and that it gives results identical to the minimal exact approximant.

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 . 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 . Details of this will be given in . Part of our validation was based on material from the PhD thesis of D. Styer . 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 p00=1, and then with non-minimal sets (also unconstrained approximants). For example set #1 (from Table 2) corresponds to matching the coefficients cij ( 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 T0=U0=1, , , , and e2=0.

Table 2: The sets K used with the first approximant, all with |K|=16
 #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: T0=1.0216, U0=1.168, , and e1=e2=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].

Table 3: The other approximant sets J, L, M, N, and K.
 # 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 Tc and Uc converges to 1 as expected, except for approximant #2 which approaches to T0=0.9993 and U0=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
Tc approximated as a function of the noise for the first approximants FIG.2.
The value of
Uc approximated as a function of the noise for the first approximants 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 Tc and Uc, correspondingly, as a function of the noise. We can see that except for approximant #1, which converges to T0=1.004 and U0=1.042, all approximations converge to 1. FIG.5.
The value of
Tc approximated as a function of the noise for the second approximants FIG.6.
The value of
Uc approximated as a function of the noise for the second approximants 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.   Next: Conclusions Up: Series Analysis of Tricritical Previous: Analytic Solution by PDA

1998-12-30