naginterfaces.library.opt.lsq_​uncon_​mod_​func_​comp

naginterfaces.library.opt.lsq_uncon_mod_func_comp(m, lsqfun, x, lsqmon=None, iprint=1, maxcal=None, eta=None, xtol=0.0, stepmx=100000.0, data=None)

lsq_uncon_mod_func_comp is a comprehensive algorithm for finding an unconstrained minimum of a sum of squares of $m$ nonlinear functions in $n$ variables $(m\ge n)$. No derivatives are required.

The function is intended for functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

For full information please refer to the NAG Library document for e04fc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e04/e04fcf.html

Parameters

mint

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

lsqfuncallable (iflag, fvec) = lsqfun(iflag, m, xc, data=None)

$lsqfun$ must calculate the vector of values $f_i\left(x\right)$ at any point $x$. (However, if you do not wish to calculate the residuals at a particular $x$, there is the option of setting an argument to cause lsq_uncon_mod_func_comp to terminate immediately.)

Parameters

iflagint

Has a non-negative value.

mint

$m$, the numbers of residuals.

xcfloat, ndarray, shape $\left(n\right)$

The point $x$ at which the values of the $f_i$ are required.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

iflagint

If $lsqfun$ resets $iflag$ to some negative number, lsq_uncon_mod_func_comp will terminate immediately, with $\text{errno}$ set to your setting of $iflag$.

fvecfloat, array-like, shape $\left(m\right)$

Unless $iflag$ is reset to a negative number, $fvec[\text{i}-1]$ must contain the value of $f_{\text{i}}$ at the point $x$, for $\text{i}=1,2,\dots ,m$.

xfloat, array-like, shape $\left(n\right)$

$x[\text{j}-1]$ must be set to a guess at the $\text{j}$th component of the position of the minimum, for $\text{j}=1,2,\dots ,n$.

lsqmonNone or callable lsqmon(xc, fvec, fjac, s, igrade, niter, nf, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

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

Parameters

xcfloat, ndarray, shape $\left(n\right)$

The coordinates of the current point $x$.

fvecfloat, ndarray, shape $\left(m\right)$

The values of the residuals $f_i$ at the current point $x$.

fjacfloat, ndarray, shape $(m,n)$

$fjac[\text{i}-1,\text{j}-1]$ contains the value of $\frac{\partial f_{\text{i}}}{\partial x_{\text{j}}}$ at the current point $x$, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,m$.

sfloat, ndarray, shape $\left(n\right)$

The singular values of the current approximation to the Jacobian matrix. Thus $s$ may be useful as information about the structure of your problem.

igradeint

lsq_uncon_mod_func_comp estimates the dimension of the subspace for which the Jacobian matrix can be used as a valid approximation to the curvature (see Gill and Murray (1978)). This estimate is called the grade of the Jacobian matrix, and $igrade$ gives its current value.

niterint

The number of iterations which have been performed in lsq_uncon_mod_func_comp.

nfint

The number of times that $lsqfun$ has been called so far. (However, for intermediate calls of $lsqmon$, $nf$ is calculated on the assumption that the latest linear search has been successful. If this is not the case, the $n$ evaluations allowed for approximating the Jacobian at the new point will not in fact have been made. $nf$ will be accurate at the final call of $lsqmon$.)

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

iprintint, optional

The frequency with which $lsqmon$ is to be called.

If $iprint>0$, $lsqmon$ is called once every $iprint$ iterations and just before exit from lsq_uncon_mod_func_comp.

If $iprint=0$, $lsqmon$ is just called at the final point.

If $iprint<0$, $lsqmon$ is not called at all.

$iprint$ should normally be set to a small positive number.

maxcalNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: $400\times n$.

The limit you set on the number of times that $lsqfun$ may be called by lsq_uncon_mod_func_comp. There will be an error exit (see Exceptions) after $maxcal$ calls of $lsqfun$.

etaNone or float, optional

Note: if this argument is None then a default value will be used, determined as follows: if $n=1$: $0.0$; otherwise: $0.5$.

Every iteration of lsq_uncon_mod_func_comp involves a linear minimization, i.e., minimization of $F(x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)})$ with respect to ${\alpha }^{\left(k\right)}$.

Specifies how accurately the 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 lsq_uncon_mod_func_comp, 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.

xtolfloat, optional

The accuracy in $x$ to which the solution is required.

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

$$\parallel x_{sol}-x_{true}\parallel <xtol\times (1.0+\parallel x_{true}\parallel )\text{,}$$

where $\parallel y\parallel =\sqrt{{\sum }_{j=1}^n{y}_j^2}$.

For example, if the elements of $x_{sol}$ are not much larger than $1.0$ in modulus and if $xtol=1.0e-5$, then $x_{sol}$ is usually accurate to about five decimal places. (For further details see Accuracy.)

stepmxfloat, optional

An estimate of the Euclidean distance between the solution and the starting point supplied by you. (For maximum efficiency, a slight overestimate is preferable.) lsq_uncon_mod_func_comp will ensure that, for each iteration,

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

where $k$ is the iteration number. Thus, if the problem has more than one solution, lsq_uncon_mod_func_comp is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence $x^{\left(k\right)}$ 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.

dataarbitrary, optional

User-communication data for callback functions.

Returns

xfloat, ndarray, shape $\left(n\right)$

The final point $x^{\left(k\right)}$. Thus, if no exception or warning is raised on exit, $x[j-1]$ is the $j$th component of the estimated position of the minimum.

fsumsqfloat

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 $x$.

fvecfloat, ndarray, shape $\left(m\right)$

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

fjacfloat, ndarray, shape $(m,n)$

The estimate of the first derivative $\frac{\partial f_{\text{i}}}{\partial x_{\text{j}}}$ at the final point given in $x$, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,m$.

sfloat, ndarray, shape $\left(n\right)$

The singular values of the estimated Jacobian matrix at the final point. Thus $s$ may be useful as information about the structure of your problem.

vfloat, ndarray, shape $(n,n)$

The matrix $V$ associated with the singular value decomposition

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

of the estimated 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 $J^{T}J$.

niterint

The number of iterations which have been performed in lsq_uncon_mod_func_comp.

nfint

The number of times that the residuals have been evaluated (i.e., number of calls of $lsqfun$).

Raises

NagValueError

(errno $1$)

On entry, $stepmx=⟨value⟩$ and $xtol=⟨value⟩$.

Constraint: $stepmx\ge xtol$.

(errno $1$)

On entry, $xtol=⟨value⟩$.

Constraint: $xtol\ge 0.0$.

(errno $1$)

On entry, $eta=⟨value⟩$.

Constraint: $0.0\le eta<1.0$.

(errno $1$)

On entry, $maxcal=⟨value⟩$.

Constraint: $maxcal\ge 1$.

(errno $1$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $m\ge n$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Warns

NagAlgorithmicWarning

(errno $i<0$)

User requested termination by setting $iflag$ negative in $lsqfun$.

(errno $2$)

There have been $maxcal=⟨value⟩$ calls to $lsqfun$.

(errno $3$)

The conditions for a minimum have not all been satisfied, but a lower point could not be found.

(errno $4$)

Failure in computing SVD of estimated Jacobian matrix.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

lsq_uncon_mod_func_comp is essentially identical to the function LSQNDN in the NPL Algorithms Library. It is applicable to problems of the form

$$Minimize\left(F\right)\left(x\right)=\sum _{i=1}^m{\left[f_i\left(x\right)\right]}^2$$

where $x={(x_1,x_2,\dots ,x_n)}^T$ and $m\ge n$. (The functions $f_i\left(x\right)$ are often referred to as ‘residuals’.)

You must supply $lsqfun$ to calculate the values of the $f_i\left(x\right)$ at any point $x$.

From a starting point $x^{\left(1\right)}$ supplied by you, the function generates a sequence of points $x^{\left(2\right)},x^{\left(3\right)},\dots$, which is intended to converge to a local minimum of $F\left(x\right)$. The sequence of points is given by

$$x^{(k+1)}=x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)}$$

where the vector $p^{\left(k\right)}$ is a direction of search, and ${\alpha }^{\left(k\right)}$ is chosen such that $F(x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)})$ is approximately a minimum with respect to ${\alpha }^{\left(k\right)}$.

The vector $p^{\left(k\right)}$ used depends upon the reduction in the sum of squares obtained during the last iteration. If the sum of squares was sufficiently reduced, then $p^{\left(k\right)}$ is an approximation to the Gauss–Newton direction; otherwise additional function evaluations are made so as to enable $p^{\left(k\right)}$ to be a more accurate approximation to the Newton direction.

The method is designed to ensure that steady progress is made whatever the starting point, and to have the rapid ultimate convergence of Newton’s method.

References

Gill, P E and Murray, W, 1978, Algorithms for the solution of the nonlinear least squares problem, SIAM J. Numer. Anal. (15), 977–992

See also

naginterfaces.library.examples.opt.lsq_uncon_mod_func_comp_ex.main()