# nlxb: Nash variant of Marquardt nonlinear least squares solution... In nlmrt: Functions for Nonlinear Least Squares Solutions

## 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

 ```1 2``` ``` nlxb(formula, start, trace=FALSE, data, lower=-Inf, upper=Inf, masked=NULL, control, ...) ```

## Arguments

 `formula` This is a modeling formula of the form (as in `nls`) lhsvar ~ rhsexpression for example, y ~ b1/(1+b2*exp(-b3*tt)) You may also give this as a string. `start` A named parameter vector. For our example, we could use start=c(b1=1, b2=2.345, b3=0.123) `trace` Logical TRUE if we want intermediate progress to be reported. Default is FALSE. `data` A data frame containing the data of the variables in the formula. This data may, however, be supplied directly in the parent frame. `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. `masked` Character vector of quoted parameter names. These parameters will NOT be altered by the algorithm. `control` A list of controls for the algorithm. These are: `watch`Monitor progress if TRUE. Default is FALSE. `phi`Default is phi=1, which adds phi*Identity to Jacobian inner product. `lamda`Initial Marquardt adjustment (Default 0.0001). Odd spelling is deliberate. `offset`Shift to test for floating-point equality. Default is 100. `laminc`Factor to use to increase lamda. Default is 10. `lamdec`Factor to use to decrease lamda is lamdec/laminc. Default lamdec=4. `femax`Maximum function (sum of squares) evaluations. Default is 10000, which is extremely aggressive. `jemax`Maximum number of Jacobian evaluations. Default is 5000. `rofftest`Default is TRUE. Use a termination test of the relative offset orthogonality type. Useful for nonlinear regression problems. `smallsstest`Default is TRUE. Exit the function if the sum of squares falls below (100 * .Machine\$double.eps)^4 times the initial sumsquares. This is a test for a “small” sum of squares, but there are problems which are very extreme for which this control needs to be set FALSE. `...` 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.

## Details

`nlxb` 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.

## Value

A list of the following items

 `coefficients` A named vector giving the parameter values at the supposed solution. `ssquares` The sum of squared residuals at this set of parameters. `resid` The residual vector at the returned parameters. `jacobian` The jacobian matrix (partial derivatives of residuals w.r.t. the parameters) at the returned parameters. `feval` The number of residual evaluations (sum of squares computations) used. `jeval` The number of Jacobian evaluations used.

## Note

Special notes, if any, will appear here.

## Author(s)

John C Nash <nashjc@uottawa.ca>

## References

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

others!!

Function `nls()`, packages `optim` and `optimx`.
 `1` ```cat("See examples in nlmrt-package.Rd\n") ```