A method for fitting multiequation models to data sets of finite precision is proposed. This is based on the Gauss–Newton algorithm devised by Britt and Luecke (1973); the inclusion of several equations of condition to be satisfied at each data point results in a block diagonal form for the effective weighting matrix. This method allows generalized nonlinear least-squares fitting of functions that are more easily represented in the parametric form (x(t),y(t)) than as an explicit functional relationship of the form y=f(x). The Aitken (1935) formulas appropriate to multiequation weighted nonlinear least squares are recovered in the limiting case where the variances and covariances of the independent variables are zero. Practical considerations relevant to the performance of such calculations, such as the evaluation of the required partial derivatives and matrix products, are discussed in detail, and the operation of the algorithm is illustrated by applying it to the fit of complex permittivity data to the Debye equation.

1.
John E. Freund, Mathematical Statistics, 2nd ed. (Prentice–Hall, Englewood Cliffs, N.J. 1972), pp. 358–392.
2.
P. G. Hoel, Introduction to Mathematical Statistics, 4th ed. (Wiley, New York, 1971), pp. 142–189.
3.
Douglas M. Bates and Donald G. Watts, Nonlinear Regression Analysis and its Applications (Wiley, New York, 1988).
4.
C. Daniel, F. S. Wood, and J. W. Gorman, editors, Fitting Equations to Data (Wiley-Interscience, New York, 1971), pp. 8 and 54.
5.
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing 2nd ed. (Cambridge University Press, Cambridge, U.K., 1992), pp. 675–694.
6.
G. A. F. Seber and C. J. Wild, Nonlinear Regression (Wiley, New York, 1989).
7.
K.
Pearson
, “
On lines and planes of closest fit to points in space
,”
Philos. Mag.
2
,
559
572
(
1901
).
8.
W. E. Deming, Statistical Adjustment of Data (Wiley, New York, 1943).
9.
J. R.
Macdonald
and
W. J.
Thompson
, “
Least-squares fitting when both variables contain errors: pitfalls and possibilities
,”
Am. J. Phys.
60
,
66
73
(
1992
).
10.
D. R.
Powell
and
J. R.
Macdonald
, “
A rapidly convergent iterative method for the solution of the generalized nonlinear least squares problem
,”
Comput. Phys.
15
,
148
155
(
1972
).
11.
H. I.
Britt
and
R. H.
Luecke
, “
The estimation of parameters in nonlinear, implicit models
,”
Technometrics
15
,
233
247
(
1973
).
12.
W. H.
Jefferys
, “
On the method of least squares
,”
Astrophys. J.
85
,
177
181
(
1980
).
13.
M.
Lybanon
, “
A better least-squares method when both variables have uncertainties
,”
Am. J. Phys.
52
,
22
26
(
1984
).
14.
M.
Lybanon
, “
A simple generalized least-squares algorithm
,”
Comput. Geosci.
11
,
501
508
(
1985
).
15.
W. H.
Sachs
, “
Implicit multifunctional nonlinear regression analysis
,”
Technometrics
18
,
161
173
(
1976
).
16.
T. F.
Anderson
,
D. S.
Abrams
, and
E. A.
Grens
II
, “
Evaluation of parameters for nonlinear thermodynamic models
,”
AIChE J.
24
,
20
29
(
1978
).
17.
H.
Schwetlick
and
V.
Tiller
, “
Numerical methods for estimating parameters in nonlinear models with errors in the variables
,”
Technometrics
27
,
17
24
(
1985
).
18.
Reference 6, p. 502.
19.
T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).
20.
Einar Hille and Saturnino L. Salas, Calculus: One and Several Variables (Xerox College Publishing, Waltham, MA, 1971), p. 647.
21.
Yonathan Bard, Nonlinear Parameter Estimation (Academic, New York, 1974), pp. 20 and 62.
22.
A. C.
Aitken
, “
On least squares and combinations of observations
,”
Proc. R. Soc. Edinburgh
55
,
42
48
(
1935
).
23.
Reference 3, Chap. 4.
24.
Reference 5, pp. 675–683.
25.
Reference 3, p. 40.
26.
M.
Clutton-Brock
, “
Likelihood distributions for estimating functions when both variables are subject to error
,”
Technometrics
9
,
261
269
(
1967
).
27.
M. J.
Box
, “
Improved parameter estimation
,”
Technometrics
12
,
219
229
(
1970
).
28.
Y.
Bard
and
L.
Lapidus
, “
Kinetic analysis by digital parameter estimation
,”
Catal. Rev.
2
,
67
75
(
1968
).
29.
S.
Kemény
,
J.
Manczinger
,
S.
Skjold-Jørgensen
, and
M.
Tóth
, “
Reduction of thermodynamic data by means of the multiresponse maximum likelihood principle
,”
AIChE J.
28
,
20
30
(
1982
).
30.
M.
Lybanon
, “
Comment on ‘Least squares when both variables have uncertainties,’ 
Am. J. Phys.
52
,
276
277
(
1984
).
31.
A.
Gingle
and
T. M.
Knasel
, “
Undergraduate laboratory investigation of the dielectric constant of ice
,”
Am. J. Phys.
43
,
161
167
(
1975
).
32.
W. R.
Jupin
, “
Measurement of dielectric constants and capacitor dissipation using resonant circuits
,”
Am. J. Phys.
45
,
663
666
(
1977
).
33.
J.
Mwanje
, “
Dielectric loss measurements on raw materials
,”
Am. J. Phys.
48
,
837
839
(
1979
).
34.
J. R.
Macdonald
, “
Comparison and application of two methods for the least-squares analysis of immittance data
,”
Solid State Ionics
58
,
97
107
(
1992
).
35.
J. R.
Macdonald
, “
Exact and approximate nonlinear least-squares inversion of dielectric relaxation spectra
,”
J. Chem. Phys.
102
,
6241
6250
(
1995
).
36.
J. R.
Macdonald
and
J. A.
Garber
, “
Analysis of impedance and admittance data for solids and liquids
,”
J. Electrochem. Soc.
124
,
1022
1030
(
1977
).
37.
J. R.
Macdonald
,
J.
Schoonman
, and
A. P.
Lehnen
, “
The applicability and power of complex nonlinear least squares for the analysis of impedance and admittance data
,”
J. Electroanal. Chem. Interfacial Electrochem.
131
,
77
95
(
1982
).
38.
S. H.
Chung
and
J. R.
Stevens
, “
Time-dependent correlation and the evaluation of the stretched exponential or Kohlrausch–Williams–Watts function
,”
Am. J. Phys.
59
,
1024
1030
(
1991
).
39.
A. K.
Lyashchenko
,
V. S.
Khar’kin
,
A. S.
Lileev
, and
P. V.
Efremov
, “
Complex permittivity and relaxation in aqueous solutions of methyl ethyl ketone
,”
Russ. J. Phys. Chem.
75
,
195
201
(
2001
).
40.
D.
York
, “
Least-squares fitting of a straight line
,”
Can. J. Phys.
44
,
1079
1086
(
1966
).
41.
George E. P.
Box
and
Norman R.
Draper
, “
The Bayesian estimation of common parameters from several responses
,”
Biometrika
52
,
355
365
(
1965
).
42.
Reference 21, pp. 63–66.
43.
Reference 21, pp. 123–127.
44.
Douglas M.
Bates
and
Donald G.
Watts
, “
A generalized Gauss-Newton procedure for multi-response parameter estimation
,”
SIAM J. Sci. Stat. Comput.
8
,
49
55
(
1987
).
45.
Reference 3, pp. 141–146.
46.
R.
Fletcher
and
C. M.
Reeves
, “
Function minimization by conjugate gradients
,”
Comput. J.
7
,
149
154
(
1964
).
47.
David A. Wismer and R. Chattergy, Introduction to Nonlinear Optimization: A Problem Solving Approach, North–Holland Series in System Science and Engineering (North–Holland, New York, 1978), pp. 226–229.
48.
Arnold
Zellner
, “
An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias
,”
J. Am. Stat. Assoc.
57
,
348
368
(
1962
).
49.
John J.
Beauchamp
and
Richard G.
Cornell
, “
Simultaneous nonlinear estimation
,”
Technometrics
8
,
319
326
(
1966
).
50.
Reference 6, pp. 531–536.
51.
J. M.
Varah
, “
A spline least squares method for numerical parameter estimation in differential equations
,”
SIAM J. Sci. Stat. Comput.
3
,
28
46
(
1982
).
52.
Reference 3, p. 147.
53.
J.
Lekner
, “
Parametric solution of the van der Waals liquid–vapor coexistence curve
,”
Am. J. Phys.
50
,
161
163
(
1982
).
54.
J. R.
Macdonald
, “
Review of some experimental and analytical equations of state
,”
Rev. Mod. Phys.
41
(
3
),
316
349
(
1969
).
55.
U.
Setzmann
and
W.
Wagner
, “
A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 625 K at pressures up to 1000 MPa
,”
J. Phys. Chem. Ref. Data
20
,
1061
1155
(
1991
).
56.
R.
Span
and
W.
Wagner
, “
A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa
,”
J. Phys. Chem. Ref. Data
25
,
1509
1596
(
1996
).
57.
C.
Tegeler
,
R.
Span
, and
W.
Wagner
, “
A new equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressure up to 1000 MPa
,”
J. Phys. Chem. Ref. Data
28
,
779
850
(
1999
).
58.
W.
Wagner
and
A.
Pruß
, “
The IAPWS Formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use
,”
J. Phys. Chem. Ref. Data
31
,
387
536
(
2002
).
59.
Reference 5, pp. 182–184.
60.
Reference 3, p. 82.
61.
Reference 5, pp. 881–886.
62.
G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Calculations (Prentice–Hall, Englewood Cliffs, N.J., 1977), pp. 13–14.
63.
Reference 5, pp. 684–687.
64.
Reference 5, pp. 271–280.
65.
Reference 5, p. 674.
66.
Reference 5, pp. 57–58 and 670–673.
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.