Abstract
In the trapezoidal method, linear interpolation between data points tends to overestimate or underestimate the area, depending on the concavity of the curve. In some instances, area estimates can be obtained by linear interpolation of logarithmically transformed data. Two alternative algorithms based on known interpolating functions have been implemented for area calculations. In the Lagrange method, the linear interpolations are replaced by cubic polynomial interpolations. In the spline method, the cubic functions are further modified so that the fitted curves are completely smooth. This report describes their computing procedures with numerical examples.
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References
B. Carnahan, H. A. Luther, and J. O. Wilkes.Applied Numerical Methods, Wiley, New York, 1969, pp. 71–72.
D. D. McCracken and W. S. Dorn.Numerical Methods and FORTRAN Programming, Wiley, New York, 1964, pp. 161–163.
J. H. Ahlberg, E. N. Nilson, and J. L. Walsh.The Theory of Splines and Their Applications, Academic Press, New York, 1967.
S. Wold. Spline functions in data analysis.Tecknometrics 16:1–11 (1974).
B. Carnahan, H. A. Luther, and J. O. Wilkes.Applied Numerical Methods, Wiley, New York, pp. 27–34.
F. Scheid.Schaum's Outline of Theory and Problems of Numerical Analysis, McGraw-Hill, New York, 1968, p. 80.
T. N. E. Greville. Spline functions, interpolation, and numerical quadrature. In A. Ralston and H. W. Wilf (eds.),Mathematical Methods for Digital Computers, Vol. II, Wiley, New York, 1967, pp. 156–168.
L. G. Dunfield and J. F. Read. Determination of reaction rates by the use of cubic spline interpolation.J. Phys. Chem. 57:2178–2183 (1972).
B. Carnahan, H. A. Luther, and J. O. Wilkes.Applied Numerical Methods, Wiley, New York, p. 361.
User's Guide: Numerical Analysis Routines, General Electric, Bethesda, Md., 1971, p. 358.
K. C. Yen, Kinetic parameter estimation by numerical algorithms and multiple linear regression: theoretical.J. Pharm. Sci. 66:1688–1691 (1977).
HP-67 Math Pac. Hewlett-Packard, Cupertino, Calif., 1976.
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Yeh, K.C., Kwan, K.C. A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation. Journal of Pharmacokinetics and Biopharmaceutics 6, 79–98 (1978). https://doi.org/10.1007/BF01066064
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DOI: https://doi.org/10.1007/BF01066064