Welcome to Pade
Test Data – generated as described in gleg\Welcome.doc
Figure 1 Test data 3073 points between -6 and 0
The points shown in figure 1 have relative errors less than 10-15.
 1
Ln  AiGau  x    Ln 
 
x

 exp  t  dt 
2
(1)

Probability integral.docx

exp( Ln  ALiGau  x  
2
exp

t
dt



 

 1  exp( Ln  ALiGau   x  
1
x
 
Pade Approximate
Npade
cx
Poly ( x, c ) 
1
i 1
Npade 1

i 1
i 1
i
(1)
ci  Npade x
The total number of c’s is
Ncon  2 Npade  1 (2)
This is coded into
PolyDA.for(3)
i
x  0
x0
(2)
Pade approximates are notorious for zeroes in the denominator that can cause the approximating
function to have spikes between data points. One solution is to use the absolute values of ci+Npade. This
rather severely limits the denominator.
A second problem is a tendency for the numerator and the denominator to become very large.
Penalty
A penalty form is coded into PolyDA.for which at least prevents the very large coefficients and is
relatively successful in eliminating zeroes when data is plentiful.
 f  x , c   dati 
 c     A i

erri
i 1 

Ndat
2
2 Npow
 Nv

    ci2 Npen 
 i 3

At the beginning of the minimization  should be large enough that the penalty accounts for on the
order of 10% of the total chi. The difference is available after the last step when nlfit calculates chit
which does not include the penalty.
Npow
gleg\Welcome.doc describes the generation of the test data. Below -3, the data was generated by
integrating from -∞ to z using Gauss Laguerre quadrature. This made each point an independent.
(4)
Above -3, the points were integrated by adding the difference from the previous point making the entire
set share the error.
pade.dir
, OUTPUT FILE NAME
Laigau.out, DATA SET NAME
1,8 dimension of X in FA(X), NPOW in ((f-fA)/err)**(2*NPOW)
C,1E-34,0,1E-15 'R' read data point,err, 'C' err=(a+bf)^.5 +c
19 total number of constants (ones with vary 0 are not being f
2000, maximum number of minimization steps
0.9999,Pen,2 2e0,abs, initial fr, Penalty-if all caps-,Npen Ala
1 VARY O FIX FITTED CONS ERROR IN CONS
1 -6.9314718055994506E-01 +- 1.243E-13
1 2.4248558638803122E+00 +- 15.8
1 -3.8596778751581624E+00 +- 46.9
1 3.6689470530103137E+00 +- 62.4
1 -2.2927494647421449E+00 +- 48.8
1 9.7372893169815855E-01 +- 24.4
1 -2.7912983009006404E-01 +- 7.91
1 5.1295970022983153E-02 +- 1.59
1 -5.2810630653471174E-03 +- 0.170
1 2.0483089058487465E-04 +- 5.820E-03
1 -1.8704205010795294E+00 +- 22.9
1 1.6050192612265761E+00 +- 30.4
1 -8.1419573615274221E-01 +- 19.6
1 2.5844644415456330E-01 +- 7.24
1 -5.0213969382383122E-02 +- 1.55
1 5.2757905896337618E-03 +- 0.170
1 -2.0491315246831215E-04 +- 5.820E-03
1 -1.3797084627869877E-09 +- 3.158E-08
1 -1.4116380869588198E-11 +- 5.803E-10
ended because abs(chb-chl)/chl<1e-9
CHI,CHIPEN= 1.5105E-03 0.000 FOR 3073 DATA POINTS
CHI USED IN CALCULATING ERRORS IS 3054.
SUM ((FA-F)/EP)**2 = 88.5263721791144094876991883243923
NDAT =
3073
It was necessary to start with Npow = 2 to get the last two parameters away from 0. The minimization
was with nlfitqmp.
Range
The data in figure 1 has 3073 data points from x=-6 to x=0.
The code PlotRes.wpj allowsextrapolation to any value.
Watfor
This FORTRAN interpreter checks for uninitialized variables. It is the most complete debugger
that I have ever found. It also stores things in a nonstandard format and has slightly less accurate
double precision than most FORTRAN compilers. I tested both a double precision code and a
double/multiple precision code.
More details are in ..\W4\Welcome.docx.
Watcom – nlfit/nlfitmp Intel - nlfitq
The double precision code is nlfit.wpj. Quadruple precision, nlfitq.vfproj, differs by the presence
of real*16, including the first derivatives, and the use of the Intel compiler. Nlfitmp uses multiple
precision to invert the matrices (250 digits) , but uses double precision for the derivatives. Nlfitqmp
uses quadruple precision where appropriate and multiple precision to invert the matrices. Nlfitmpmp
uses multiple precision for both the derivatives and the matrix inversion. The codes are listed in order
of the time required for them to run. The most successful in this example is Nlfitqmp.
Figure 2 Residuals as in (4) Npow = 6.
pade.dir
, OUTPUT FILE NAME
Laigau.out, DATA SET NAME
1,5 dimension of X in FA(X), NPOW in ((f-fA)/err)**(2*NPOW)
C,1E-4,0,1E-20 'R' read data point,err, 'C' err=(a+bf)^.5 +cf
5 total number of constants (ones with vary 0 are not being f
2000, maximum number of minimization steps
0.9999,PEN,2 1e1,abs, initial fr, Penalty-if all caps-,Npen Ala
1 VARY O FIX FITTED CONS ERROR IN CONS
1 -7.0296649242537557E-01 +- 5.440E-04
1 1.0179908657537597E+00 +- 1.311E-03
1 -7.2084655816792254E-01 +- 1.711E-03
1 4.1276799623572773E-02 +- 5.050E-04
1 2.3987903246443639E-03 +- 3.829E-05
ended because abs(chb-chl)/chl<1e-9
CHI,CHIPEN= 38.44 13.44 FOR 3073 DATA POINTS
CHI USED IN CALCULATING ERRORS IS 3068.
SUM ((FA-F)/EP)**2 = 728.3170098400192956 NDAT =
3073
Nlfitmp lowered the 38.44 to 38.42 after a few hundred steps when I stopped it.
Nlfitq lowered this to 38.41 in relatively few steps.
Bob*** 8/6/2015
The double precision code eventually fails. The inverted array does not correct with large values of .
The code returns the values to this point. The intel quadruple precision code caries it further. Then
nlfitmp.wpj with multiple precision carries it further yet, though the improvement is very small.
Figure 3 The difference between the data and the fit is less than 0.003.
pade2.dir
, OUTPUT FILE NAME
Laigau.out, DATA SET NAME
1,6 dimension of X in FA(X), NPOW in ((f-fA)/err)**(2*NPOW)
C,1e-8,0,1E-20 'R' read data point,err, 'C' err=(a+bf)^.5 +cf
15 total number of constants (ones with vary 0 are not being f
2000, maximum number of minimization steps
0.9999,Pen,1e5,abs, initial fr, Penalty-if all caps-,lambda, ab
1 VARY O FIX FITTED CONS ERROR IN CONS
1 -6.932790915727651E-01 +- 1.830E-06
1 1.082256637822175E+00 +- 9.20
1 -5.789703219287241E-01 +- 17.7
1 6.557079075029873E-02 +- 13.4
1 1.762611667084673E-02 +- 4.42
1 3.260599094603882E-03 +- 0.372
1 3.780360755424656E-04 +- 5.556E-02
1 1.808451488260390E-05 +- 1.797E-03
1 6.320201829048598E-02 +- 13.3
1 9.520415409816399E-03 +- 6.18
1 2.430484480365099E-04 +- 0.116
1 -1.703774166936710E-04 +- 5.785E-02
1 -2.316481624453219E-05 +- 2.393E-02
1 -1.263327602876655E-06 +- 3.202E-03
1 -3.901269297096165E-08 +- 1.591E-04
ended because abs(chb-chl)/chl<1e-9
CHI= 594.5 FOR 3073 DATA POINTS
CHI USED IN CALCULATING ERRORS IS 3058.
SUM ((FA-F)/EP)**2 = 1283.3741324289612749 NDAT =
3073
Minimize numerator and denominator separately
The above file with 15 constants is the first for which the numerator and denominator can be
minimized separately. Below 15, such minimizations followed by a complete minimization always lead
to noticeably lower minima. This one stays the same for both nlfitq and nlfitmp minimizations.
Separate numerator and denominator minimizationsare much faster than the combined minimization.
pade2.dir
, OUTPUT FILE NAME
Laigau.out, DATA SET NAME
1,6 dimension of X in FA(X), NPOW in ((f-fA)/err)**(2*NPOW)
C,1e-12,0,1E-6 'R' read data point,err, 'C' err=(a+bf)^.5 +cf
21 total number of constants (ones with vary 1 are not being f
2000, maximum number of minimization steps
0.9999,Pen,1e5,abs, initial fr, Penalty-if all caps-,lambda, ab
1 VARY O FIX FITTED CONS ERROR IN CONS
1 -6.931462809758711E-01 +- 4.186E-06
1 1.084620407832966E+00 +- 1.929E+07
1 -5.714716776144769E-01 +- 2.802E+07
1 7.464461953405718E-02 +- 1.292E+07
1 2.273120438423467E-02 +- 8.675E+05
1 4.527216313088876E-03 +- 6.203E+05
1 3.827639996659974E-04 +- 2.035E+05
1 -5.347508789098885E-05 +- 3.910E+04
1 -1.667601702385410E-05 +- 4.378E+03
1 -1.614346303156150E-06 +- 276.
1 -5.912633759139942E-08 +- 8.02
1 6.319970956417620E-02 +- 2.783E+07
1 9.525414012300504E-03 +- 3.132E+06
1 2.704498484726000E-04 +- 5.760E+05
1 -1.331806990347220E-04 +- 8.463E+04
1 1.701260991175848E-06 +- 1.204E+04
1 8.290546980453564E-06 +- 1.770E+03
1 2.175308297642102E-06 +- 238.
1 3.063719509989111E-07 +- 26.3
1 2.333281314912452E-08 +- 1.99
1 7.533626441023104E-10 +- 6.922E-02
ended because abs(chb-chl)/chl<1e-9
CHI=0.5714E-02 FOR 3073 DATA POINTS
CHI USED IN CALCULATING ERRORS IS 3052.
SUM ((FA-F)/EP)**2 = 170.122405915483724810218893015300
3073
NDAT =
Taming the Pade (making a denominator with no zeroes)
Write
Den  x, b, c   1  bx  cx 2

b
b 2  4c 
b
b 2  4c 
x 
 x  




2
2
2
2



 b
b 2  4c b
b 2  4c   b 2 b 2  4c 
 x2  x   
 
 

 2
  4
2
2
2
4 


 x 2  bx  c
(5)
c  b / 4 , there are no zeroes on the real axis. Thus a denominator with no zeroes can be written as
c 

Den  x, c4 , c5   1  c4 x   c52  4  x 2
4

(6)
This is overly restrictive for functions over finite or even semi infinite ranges, but for the test function
above, allows all desirable variation. The approximation in Error! Reference source not found. however
has ln(-x) values from - to  for which this is appropriate.
Write
N

c

Den  x, c4 , c5 ,.., c2 N , c2 N 1    1  c2i x   c22i 1  2i
4

i 2 
 2
 x  (7)
 
N

c

ln Den  x, c4 , c5 ,.., c2 N , c2 N 1    ln 1  c2i x   c22i 1  2i
4

i 2

 ln Den  x, c4 , c5 ,.., c2 N , c2 N 1 
x  x2 / 4

c 
c2i

1  c2i x   c22i 1  2i  x 2
4 

 ln Den  x, c4 , c5 ,.., c2 N , c2 N 1 
2c2i 1 x 2

c 
c2i 1

1  c2i x   c22i 1  2i  x 2
4 

 2
x 
 
More Constants
The form in Error! Reference source not found. assumes that as xinfinity, Pade constant
2N
Pade  x, c  
c x
  x  c
i 0
Nd
i 1
i
i
2 i 1 2 N

2

 c22i  2 N

(8)

 Nd

 2N

2
   ci x i  exp   ln  x  c2i 1 2 N   c22i 1 2 N 
 i 0

 i 1

Equation (8) has 2N+1 terms in the numerator and 2Nd terms in the denominator. The denominator for
large x  x2Nd, while the numerator  x2N.