nlfb: Nash variant of Marquardt nonlinear least squares solution via qr linear solver.

Description

Given a nonlinear model expressed as an expression of the form lhs ~ formula_for_rhs and a start vector where parameters used in the model formula are named, attempts to find the minimum of the residual sum of squares using the Nash variant (Nash, 1979) of the Marquardt algorithm, where the linear sub-problem is solved by a qr method.

Usage

nlfb(start, resfn, jacfn=NULL, trace=FALSE, lower=-Inf, upper=Inf, maskidx=NULL, control, ...)

Arguments

resfn

A function that evaluates the residual vector for computing the elements of the sum of squares function at the set of parameters start.

jacfn

A function that evaluates the Jacobian of the sum of squares function, that is, the matrix of partial derivatives of the residuals with respect to each of the parameters. If NULL (default), uses an approximation. ?? put in character form as

start

A named parameter vector. For our example, we could use start=c(b1=1, b2=2.345, b3=0.123) nls() takes a list, and that is permitted here also.

trace

Logical TRUE if we want intermediate progress to be reported. Default is FALSE.

lower

Lower bounds on the parameters. If a single number, this will be applied to all parameters. Default -Inf.

upper

Upper bounds on the parameters. If a single number, this will be applied to all parameters. Default Inf.

maskidx

Vector if indices of the parameters to be masked. These parameters will NOT be altered by the algorithm. Note that the mechanism here is different from that in nlxb which uses the names of the parameters.

control

A list of controls for the algorithm. These are:

...

Any data needed for computation of the residual vector from the expression rhsexpression - lhsvar. Note that this is the negative of the usual residual, but the sum of squares is the same. It is not clear how the dot variables should be u

Value

A list of the following items
coeffsA named vector giving the parameter values at the supposed solution.
ssquaresThe sum of squared residuals at this set of parameters.
residThe residual vector at the returned parameters.
jacobianThe jacobian matrix (partial derivatives of residuals w.r.t. the parameters) at the returned parameters.
fevalThe number of residual evaluations (sum of squares computations) used.
jevalThe number of Jacobian evaluations used.

Details

nlfb attempts to solve the nonlinear sum of squares problem by using a variant of Marquardt's approach to stabilizing the Gauss-Newton method using the Levenberg-Marquardt adjustment. This is explained in Nash (1979 or 1990) in the sections that discuss Algorithm 23. (?? do we want a vignette. Yes, because folk don't have access to book easily, but finding time.)

In this code, we solve the (adjusted) Marquardt equations by use of the qr.solve(). Rather than forming the J'J + lambda*D matrix, we augment the J matrix with extra rows and the y vector with null elements.

References

Nash, J. C. (1979, 1990) _Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation._ Adam Hilger./Institute of Physics Publications

others!!

Examples

Run this code

cat("See examples in nlmrt-package.Rd
")

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