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METHODS FOR NON-LINEAR LEAST SQUARES PROBLEMS - …

IMMMETHODS FORNON-LINEAR LEASTSQUARES PROBLEMS2nd Edition, April 2004K. Madsen, Nielsen, O. TingleffInformatics and Mathematical ModellingTechnical University of DenmarkCONTENTS1. INTRODUCTION The steepest Descent method .. Newton s Method .. Line Search .. Trust Region and Damped METHODS ..113. The Gauss Newton Method .. The Levenberg Marquardt Method .. Powell s Dog Leg Method .. A Hybrid Method: L M and Quasi Newton .. A Secant Version of the L M Method .. A Secant Version of the Dog Leg Method .. Final INTRODUCTION ANDDEFINITIONSIn this booklet we consider the following problem,Definition LEAST SQUARES ProblemFindx , a local minimizer for1)F(x)=12mXi=1(fi(x))2;wherefi:IRn7!I R;i=1;:::;mare given functions, andm important source of LEAST SQUARES PROBLEMS isdata consider thedata points(t1;y1);:::;(tm;ym)shown belowtyFigure pointsf(ti;yi)g(marked by+)and modelM(x;t)(marked by full line.

The Steepest Descent method From (2.5) we see that when we perform a step fi hwith positive fi, then the relative gain in function value satisfies lim fi!0 F(x) ¡F(x+fih) fikhk = ¡ 1 khk h>F0(x)=¡kF0(x)kcosµ; where µis the angle between the vectors h and F0(x). This shows that we get the greatest gain rate if µ=…, ie if we use the ...

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