Fitting nonlinear models: numerical techniques

SUMMARY Computational methods for fitting nonlinear models have developed considerably in the last decade. Statistical use of these techniques has, as yet, lagged somewhat behind. The present paper gives a critical review of current numerical techniques and relates these to some statistical needs. We emphasize the basic properties of methods of optimization and nonlinear least squares, cite some advantages and difficulties of the methods and suggest a basic library of fitting procedures. A classified bibliography is included.

[1]  R. Outlaw,et al.  The solution of transcendental equations , 1973 .

[2]  R. Barham,et al.  An Algorithm for Least Squares Estimation of Nonlinear Parameters When Some of the Parameters Are Linear , 1972 .

[3]  A. I. Cohen.  Rate of convergence of several conjugate gradient algorithms. , 1972 .

[4]  A General Convergence Result for Unconstrained Minimization Methods , 1972 .

[5]  P. Gill,et al.  Quasi-Newton Methods for Unconstrained Optimization , 1972 .

[6]  J. Greenstadt.  A Quasi-Newton Method with No Derivatives , 1972 .

[7]  R. Bass.  A rank two algorithm for unconstrained minimization , 1972 .

[8]  R. O'Neill.  Algorithm AS 47: Function Minimization Using a Simplex Procedure , 1971 .

[9]  R. Fletcher.  A General Quadratic Programming Algorithm , 1971 .

[10]  F. Smith,et al.  An Improved Marquardt Procedure for Nonlinear Regressions , 1971 .

[11]  G. J. S. Ross,et al.  The Efficient Use of Function Minimization in Non‐Linear Maximum‐Likelihood Estimation , 1970 .

[12]  D. Shanno.  Parameter Selection for Modified Newton Methods for Function Minimization , 1970 .

[13]  D. Shanno,et al.  Optimal conditioning of quasi-Newton methods , 1970 .

[14]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

[15]  Donald W. Marquaridt.  Generalized Inverses, Ridge Regression, Biased Linear Estimation, and Nonlinear Estimation , 1970 .

[16]  The application of third variations to function minimization , 1970 .

[17]  H. Y. Huang.  Unified approach to quadratically convergent algorithms for function minimization , 1970 .

[18]  Yonathan Bard,et al.  Comparison of Gradient Methods for the Solution of Nonlinear Parameter Estimation Problems , 1970 .

[19]  C. G. Broyden.  The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations , 1970 .

[20]  R. Fletcher,et al.  A Class of Methods for Nonlinear Programming II Computational Experience , 1970 .

[21]  M. Powell.  A Survey of Numerical Methods for Unconstrained Optimization , 1970 .

[22]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[23]  A. Jones,et al.  Spiral - A New Algorithm for Non-Linear Parameter Estimation Using Least Squares , 1970, Comput. J..

[24]  R. R. Allran,et al.  An Algorithm for Solving Nonlinear Programming Problems Subject to Nonlinear Inequatlity Constraints , 1970, Comput. J..

[25]  J. Greenstadt.  Variations on Variable-Metric Methods , 1970 .

[26]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[27]  F. J. Gould.  Extensions of Lagrange Multipliers in Nonlinear Programming , 1969 .

[28]  P. Rabinowitz.  New Chebyshev Polynomial Approximations to Mills' Ratio , 1969 .

[29]  M. Powell.  A method for nonlinear constraints in minimization problems , 1969 .

[30]  G. Golub.  MATRIX DECOMPOSITIONS AND STATISTICAL CALCULATIONS , 1969 .

[31]  William C. Davidon,et al.  Variance Algorithm for Minimization , 1968, Comput. J..

[32]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

[33]  J. Greenstadt.  A Ricocheting Gradient Method for Nonlinear Optimization , 1966 .

[34]  R. Fletcher,et al.  Function Minimization Without Evaluating Derivatives - a Review , 1965, Comput. J..

[35]  M. J. Box.  A New Method of Constrained Optimization and a Comparison With Other Methods , 1965, Comput. J..

[36]  W. Fleming.  Functions of Several Variables , 1965 .

[37]  M. J. D. Powell,et al.  An efficient method for finding the minimum of a function of several variables without calculating derivatives , 1964, Comput. J..

[38]  C. M. Reeves,et al.  Function minimization by conjugate gradients , 1964, Comput. J..

[39]  Roger Fletcher,et al.  A Rapidly Convergent Descent Method for Minimization , 1963, Comput. J..

[40]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[41]  G. R. Hext,et al.  Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation , 1962 .

[42]  Charles W. Carroll.  The Created Response Surface Technique for Optimizing Nonlinear, Restrained Systems , 1961 .

[43]  H. H. Rosenbrock,et al.  An Automatic Method for Finding the Greatest or Least Value of a Function , 1960, Comput. J..

[44]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .