naginterfaces.library.opt.lsq_​uncon_​mod_​deriv2_​comp

naginterfaces.library.opt.lsq_uncon_mod_deriv2_comp(m, lsqfun, lsqhes, xtol, x, lsqmon=None, iprint=1, maxcal=None, eta=None, stepmx=100000.0, data=None, spiked_sorder='C')

lsq_uncon_mod_deriv2_comp is a comprehensive modified Gauss–Newton algorithm for finding an unconstrained minimum of a sum of squares of $m$ nonlinear functions in $n$ variables $(m\ge n)$. First and second 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 e04he

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e04/e04hef.html

Parameters

mint

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

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

$lsqfun$ 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 an argument to cause lsq_uncon_mod_deriv2_comp to terminate immediately.)

Parameters

iflagint

To $lsqfun$, $iflag$ will be set to $2$.

mint

$m$, the numbers of residuals.

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

The point $x$ at which the values of the $f_i$ and the $\frac{\partial f_i}{\partial x_j}$ are required.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

iflagint

If it is not possible to evaluate the $f_i\left(x\right)$ or their first derivatives at the point given in $xc$ (or if it wished to stop the calculations for any other reason), you should reset $iflag$ to some negative number and return control to lsq_uncon_mod_deriv2_comp. lsq_uncon_mod_deriv2_comp will then 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 $\text{i}=1,2,\dots ,m$.

fjacfloat, array-like, shape $(m,n)$

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

lsqhescallable (iflag, b) = lsqhes(iflag, fvec, xc, lb, data=None)

$lsqhes$ 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 lsq_uncon_mod_deriv2_comp to terminate immediately.)

Parameters

iflagint

Is set to a non-negative number.

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

The value of the residual $f_{\text{i}}$ at the point $x$, for $\text{i}=1,2,\dots ,m$, so that the values of the $f_{\text{i}}$ can be used in the calculation of the elements of $b$.

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

The point $x$ at which the elements of $b$ are to be evaluated.

lbint

The length of the array $b$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

iflagint

If $lsqhes$ resets $iflag$ to some negative number, lsq_uncon_mod_deriv2_comp will terminate immediately, with $\text{errno}$ set to your setting of $iflag$.

bfloat, array-like, shape $\left(lb\right)$

Unless $iflag$ is reset to a negative number, $b$ must contain the lower triangle of the matrix $b[x-1]$, evaluated at the point $x$, stored by rows. (The upper triangle is not required because the matrix is symmetric.) More precisely, $b[\text{j}(\text{j}-1)/2+\text{k}-1]$ must contain ${\sum }_{\text{i}=1}^m{f}_{\text{i}}\frac{{\partial }^2{f}_{\text{i}}}{\partial x_{\text{j}}\partial x_{\text{k}}}$ evaluated at the point $x$, for $\text{k}=1,2,\dots ,\text{j}$, for $\text{j}=1,2,\dots ,n$.

xtolfloat

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.)

If $F\left(x\right)$ and the variables are scaled roughly as described in Further Comments and $ϵ$ is the machine precision, then a setting of order $xtol=\sqrt{ϵ}$ will usually be appropriate.

If $xtol$ is set to $0.0$ or some positive value less than $10ϵ$, lsq_uncon_mod_deriv2_comp will use $10ϵ$ instead of $xtol$, since $10ϵ$ is probably the smallest reasonable setting.

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 Jacobian matrix. Thus $s$ may be useful as information about the structure of your problem. (If $iprint>0$, $lsqmon$ is called at the initial point before the singular values have been calculated, so the elements of $s$ are set to zero for the first call of $lsqmon$.)

igradeint

lsq_uncon_mod_deriv2_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_deriv2_comp.

nfint

The number of times that $lsqfun$ has been called so far. Thus $nf$ gives the number of evaluations of the residuals and the Jacobian matrix.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

iprintint, optional

Specifies the frequency with which $lsqmon$ is to be called.

$iprint>0$

$lsqmon$ is called once every $iprint$ iterations and just before exit from lsq_uncon_mod_deriv2_comp.

$iprint=0$

$lsqmon$ is just called at the final point.

$iprint<0$

$lsqmon$ 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: $50\times n$.

This argument is present so as to enable you to limit the number of times that $lsqfun$ is called by lsq_uncon_mod_deriv2_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_deriv2_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 ${\alpha }^{\left(k\right)}$). $eta$ must lie in the range $0.0\le eta<1.0$, 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 lsq_uncon_mod_deriv2_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.

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_deriv2_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_deriv2_comp 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.

dataarbitrary, optional

User-communication data for callback functions.

spiked_sorderstr, optional

If $fjac$ in $lsqfun$ is spiked (i.e., has unit extent in all but one dimension, or has size $1$), $spiked\_sorder$ selects the storage order to associate with it in the NAG Engine:

spiked_sorder = $\text{'C'}$

row-major storage will be used;

spiked_sorder = $\text{'F'}$

column-major storage will be used.

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 value of the first derivative $\frac{\partial f_{\text{i}}}{\partial x_{\text{j}}}$ evaluated 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 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 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_deriv2_comp.

nfint

The number of times that the residuals and Jacobian matrix 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: $n\ge 1$.

(errno $2$)

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

(errno $4$)

Failure in computing SVD of Jacobian matrix.

Warns

NagAlgorithmicWarning

(errno $i<0$)

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

(errno $3$)

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

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

lsq_uncon_mod_deriv2_comp is essentially identical to the function LSQSDN 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 functions to calculate the values of the $f_i\left(x\right)$ and their first derivatives and second derivatives 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 the Gauss–Newton direction; otherwise the second derivatives of the $f_i\left(x\right)$ are taken into account.

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