Convergence analysis of the general Gauss-Newton algorithm

SummaryThe convergence of the Gauss-Newton algorithm for solving discrete nonlinear approximation problems is analyzed for general norms and families of functions. Aquantitative global convergence theorem and several theorems on the rate of local convergence are derived. A general stepsize control procedure and two regularization principles are incorporated. Examples indicate the limits of the convergence theorems.

[1]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

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

[3]  W. Rheinboldt.  A unified convergence theory for a class of iterative processes. , 1968 .

[4]  P. Deuflhard.  A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting , 1974 .

[5]  P. Deuflhard.  A relaxation stratery for the modified Newton method , 1975 .

[6]  K. Madsen.  An Algorithm for Minimax Solution of Overdetermined Systems of Non-linear Equations , 1975 .

[7]  Ludwig J. Cromme,et al.  Eine Klasse von Verfahren zur Ermittlung bester nichtlinearer Tschebyscheff-Approximationen , 1975 .

[8]  R. Schaback.  Globale Konvergene von Verfahren Zur Nichtlinearen Approximation , 1976 .

[9]  M. R. Osborne.  Nonlinear least squares — the Levenberg algorithm revisited , 1976, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[10]  Jorge J. Moré,et al.  The Levenberg-Marquardt algo-rithm: Implementation and theory , 1977 .

[11]  L. Cromme.  Strong uniqueness , 1978 .

[12]  P. Gill,et al.  Algorithms for the Solution of the Nonlinear Least-Squares Problem , 1978 .

[13]  P. Deuflhard,et al.  Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods , 1979 .

[14]  Axel Ruhe.  Accelerated Gauss-Newton algorithms for nonlinear least squares problems , 1979 .

[15]  Peter Deuflhard,et al.  On rank-deficient pseudoinverses , 1980 .

[16]  Larry Nazareth,et al.  Some Recent Approaches to Solving Large Residual Nonlinear Least Squares Problems , 1980 .