Parameters

m

Type: System..::..Int32

On entry: the number $m$ of residuals, $f_i\left(x\right)$, and the number $n$ of variables, $x_j$.

Constraint: $1\le \mathbf{n}\le \mathbf{m}$.

n

Type: System..::..Int32

On entry: the number $m$ of residuals, $f_i\left(x\right)$, and the number $n$ of variables, $x_j$.

Constraint: $1\le \mathbf{n}\le \mathbf{m}$.

lsqfun

Type: NagLibrary..::..E04..::..E04HE_LSQFUN

lsqfun must calculate the vector of values $f_i\left(x\right)$ and Jacobian matrix of first derivatives $\frac{\partial f_i}{\partial x_j}$ at any point $x$. (However, if you do not wish to calculate the residuals or first derivatives at a particular $x$, there is the option of setting a parameter to cause e04he to terminate immediately.)

A delegate of type E04HE_LSQFUN.

Note:  lsqfun should be tested separately before being used in conjunction with e04he.

lsqhes

Type: NagLibrary..::..E04..::..E04HE_LSQHES

lsqhes must calculate the elements of the symmetric matrix

$$B\left(x\right)=\sum _{i=1}^m{f}_i\left(x\right)G_i\left(x\right)\text{,}$$

at any point $x$, where $G_i\left(x\right)$ is the Hessian matrix of $f_i\left(x\right)$. (As with lsqfun, there is the option of causing e04he to terminate immediately.)

A delegate of type E04HE_LSQHES.

Note:  lsqhes should be tested separately before being used in conjunction with e04he.

lsqmon

Type: NagLibrary..::..E04..::..E04HE_LSQMON

If $\mathbf{iprint}\ge 0$, you must supply lsqmon which is suitable for monitoring the minimization process. lsqmon must not change the values of any of its parameters.

If $\mathbf{iprint}<0$, the dummy method E04FDZ can be used as lsqmon.

A delegate of type E04HE_LSQMON.

Note:  you should normally print the sum of squares of residuals, so as to be able to examine the sequence of values of $F\left(x\right)$ mentioned in [Accuracy]. It is usually helpful to also print xc, the gradient of the sum of squares, niter and nf.

iprint

Type: System..::..Int32

On entry: specifies the frequency with which lsqmon is to be called.

$\mathbf{iprint}>0$

lsqmon is called once every iprint iterations and just before exit from e04he.

$\mathbf{iprint}=0$

lsqmon is just called at the final point.

$\mathbf{iprint}<0$

lsqmon is not called at all.

iprint should normally be set to a small positive number.

Suggested value: $\mathbf{iprint}=1$.

maxcal

Type: System..::..Int32

On entry: this parameter is present so as to enable you to limit the number of times that lsqfun is called by e04he. There will be an error exit (see [Error Indicators and Warnings]) after maxcal calls of lsqfun.

Suggested value: $\mathbf{maxcal}=50\times n$.

Constraint: $\mathbf{maxcal}\ge 1$.

eta

Type: System..::..Double

On entry: every iteration of e04he involves a linear minimization (i.e., minimization of $F\left(x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)}\right)$ with respect to ${\alpha }^{\left(k\right)}$). eta must lie in the range $0.0\le \mathbf{eta}<1.0$, and specifies how accurately these linear minimizations are to be performed. The minimum with respect to ${\alpha }^{\left(k\right)}$ will be located more accurately for small values of eta (say, $0.01$) than for large values (say, $0.9$).

Although accurate linear minimizations will generally reduce the number of iterations performed by e04he, they will increase the number of calls of lsqfun made each iteration. On balance it is usually more efficient to perform a low accuracy minimization.

Suggested value: $\mathbf{eta}=0.5$ ($\mathbf{eta}=0.0$ if $\mathbf{n}=1$).

Constraint: $0.0\le \mathbf{eta}<1.0$.

xtol

Type: System..::..Double

On entry: the accuracy in $x$ to which the solution is required.

If $x_{\mathrm{true}}$ is the true value of $x$ at the minimum, then $x_{\mathrm{sol}}$, the estimated position before a normal exit, is such that

$$‖x_{\mathrm{sol}}-x_{\mathrm{true}}‖<\mathbf{xtol}\times \left(1.0+‖x_{\mathrm{true}}‖\right)\text{,}$$

where $‖y‖=\sqrt{{\displaystyle \sum _{j=1}^n}{y}_j^2}$. For example, if the elements of $x_{\mathrm{sol}}$ are not much larger than $1.0$ in modulus and if $\mathbf{xtol}=\text{1.0E−5}$, then $x_{\mathrm{sol}}$ is usually accurate to about five decimal places. (For further details see [Accuracy].)

If $F\left(x\right)$ and the variables are scaled roughly as described in [Further Comments] and $\epsilon$ is the machine precision, then a setting of order $\mathbf{xtol}=\sqrt{\epsilon }$ will usually be appropriate. If xtol is set to $0.0$ or some positive value less than $10\epsilon$, e04he will use $10\epsilon$ instead of xtol, since $10\epsilon$ is probably the smallest reasonable setting.

Constraint: $\mathbf{xtol}\ge 0.0$.

stepmx

Type: System..::..Double

On entry: an estimate of the Euclidean distance between the solution and the starting point supplied by you. (For maximum efficiency, a slight overestimate is preferable.)

e04he will ensure that, for each iteration

$$\sum _{j=1}^n{\left(x_j^{\left(k\right)}-x_j^{\left(k-1\right)}\right)}^2\le {\left(\mathbf{stepmx}\right)}^2\text{,}$$

where $k$ is the iteration number. Thus, if the problem has more than one solution, e04he is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence of $x^{\left(k\right)}$ entering a region where the problem is ill-behaved and can help avoid overflow in the evaluation of $F\left(x\right)$. However, an underestimate of stepmx can lead to inefficiency.

Suggested value: $\mathbf{stepmx}=100000.0$.

Constraint: $\mathbf{stepmx}\ge \mathbf{xtol}$.

x

Type: array<System..::..Double>[]()[][]

An array of size [n]

On entry: $\mathbf{x}\left[\mathit{j}-1\right]$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$.

On exit: the final point $x^{\left(k\right)}$. Thus, if $\mathbf{ifail}=0$ on exit, $\mathbf{x}\left[j-1\right]$ is the $j$th component of the estimated position of the minimum.

fsumsq

Type: System..::..Double%

On exit: the value of $F\left(x\right)$, the sum of squares of the residuals $f_i\left(x\right)$, at the final point given in x.

fvec

Type: array<System..::..Double>[]()[][]

An array of size [m]

On exit: the value of the residual $f_{\mathit{i}}\left(x\right)$ at the final point given in x, for $\mathit{i}=1,2,\dots ,m$.

fjac

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1, n]

Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \mathbf{m}$

On exit: the value of the first derivative $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ evaluated at the final point given in x, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

s

Type: array<System..::..Double>[]()[][]

An array of size [n]

On exit: the singular values of the Jacobian matrix at the final point. Thus s may be useful as information about the structure of your problem.

v

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1, n]

Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \mathbf{n}$

On exit: the matrix $V$ associated with the singular value decomposition

$$J=US{V}^{\mathrm{T}}$$

of the Jacobian matrix at the final point, stored by columns. This matrix may be useful for statistical purposes, since it is the matrix of orthonormalized eigenvectors of $J^{\mathrm{T}}J$.

niter

Type: System..::..Int32%

On exit: the number of iterations which have been performed in e04he.

nf

Type: System..::..Int32%

On exit: the number of times that the residuals and Jacobian matrix have been evaluated (i.e., number of calls of lsqfun).

ifail

Type: System..::..Int32%

On exit: $\mathbf{ifail}=0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).