e04hef (lsq_uncon_mod_deriv2_comp) : NAG Library, Mark 27

On entry: the number

m

of residuals,

f_{i} (x)

, and the number

n

of variables,

x_{j}

.

Constraint:

1 \leq n \leq m

.

lsqfun must calculate the vector of values

f_{i} (x)

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 e04hef to terminate immediately.)

The specification of lsqfun is:

Fortran Interface

Subroutine lsqfun (

iflag, m, n, xc, fvec, fjac, ldfjac, iw, liw, w, lw)

Integer, Intent (In)	::	m, n, ldfjac, liw, lw
Integer, Intent (Inout)	::	iflag, iw(liw)
Real (Kind=nag_wp), Intent (In)	::	xc(n)
Real (Kind=nag_wp), Intent (Inout)	::	fjac(ldfjac,n), w(lw)
Real (Kind=nag_wp), Intent (Out)	::	fvec(m)

C Header Interface

void	lsqfun (Integer iflag, const Integer m, const Integer n, const double xc[], double fvec[], double fjac[], const Integer ldfjac, Integer iw[], const Integer liw, double w[], const Integer lw)

Important: the dimension declaration fjac must contain the variable ldfjac, not an integer constant.

1: $iflag$ – Integer Input/Output: On entry: to lsqfun, iflag will be set to $2$ .

On exit: if it is not possible to evaluate the $f_{i} (x)$ 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 e04hef. e04hef will then terminate immediately, with ifail set to your setting of iflag.
2: $m$ – Integer Input: On entry: $m$ , the numbers of residuals.
3: $n$ – Integer Input: On entry: $n$ , the numbers of variables.
4: $xc (n)$ – Real (Kind=nag_wp) array Input: On entry: the point $x$ at which the values of the $f_{i}$ and the $\frac{\partial f_{i}}{\partial x_{j}}$ are required.
5: $fvec (m)$ – Real (Kind=nag_wp) array Output: On exit: unless iflag is reset to a negative number, $fvec (i)$ must contain the value of $f_{i}$ at the point $x$ , for $i = 1, 2, \dots, m$ .
6: $fjac (ldfjac, n)$ – Real (Kind=nag_wp) array Output: On exit: unless iflag is reset to a negative number, $fjac (i, j)$ must contain the value of $\frac{\partial f_{i}}{\partial x_{j}}$ at the point $x$ , for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ .
7: $ldfjac$ – Integer Input: On entry: the first dimension of the array fjac as declared in the (sub)program from which e04hef is called.
8: $iw (liw)$ – Integer array Workspace
9: $liw$ – Integer Input
10: $w (lw)$ – Real (Kind=nag_wp) array Workspace
11: $lw$ – Integer Input: lsqfun is called with e04hef's arguments iw, liw, w, lw as these arguments. They are present so that, when other library routines require the solution of a minimization subproblem, constants needed for the evaluation of residuals can be passed through iw and w. Similarly, you could pass quantities of lsqfun from the segment which calls e04hef by using partitions of iw and w beyond those used as workspace by e04hef. However, because of the danger of mistakes in partitioning, it is recommended that you should pass information to lsqfun via COMMON global variables and not use iw or w at all. In any case you must not change the elements of iw and w used as workspace by e04hef.

lsqfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hef is called. Arguments denoted as Input must not be changed by this procedure.

Note: lsqfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04hef. If your code inadvertently does return any NaNs or infinities, e04hef is likely to produce unexpected results.

lsqfun should be tested separately before being used in conjunction with e04hef.

lsqhes must calculate the elements of the symmetric matrix

B (x) = \sum_{i = 1}^{m} f_{i} (x) G_{i} (x),

at any point

x

, where

G_{i} (x)

is the Hessian matrix of

f_{i} (x)

. (As with lsqfun, there is the option of causing e04hef to terminate immediately.)

The specification of lsqhes is:

Fortran Interface

Subroutine lsqhes (

iflag, m, n, fvec, xc, b, lb, iw, liw, w, lw)

Integer, Intent (In)	::	m, n, lb, liw, lw
Integer, Intent (Inout)	::	iflag, iw(liw)
Real (Kind=nag_wp), Intent (In)	::	fvec(m), xc(n)
Real (Kind=nag_wp), Intent (Inout)	::	w(lw)
Real (Kind=nag_wp), Intent (Out)	::	b(lb)

C Header Interface

void	lsqhes (Integer iflag, const Integer m, const Integer n, const double fvec[], const double xc[], double b[], const Integer lb, Integer iw[], const Integer liw, double w[], const Integer lw)

1: $iflag$ – Integer Input/Output: On entry: is set to a non-negative number.

On exit: if lsqhes resets iflag to some negative number, e04hef will terminate immediately, with ifail set to your setting of iflag.
2: $m$ – Integer Input: On entry: $m$ , the numbers of residuals.
3: $n$ – Integer Input: On entry: $n$ , the numbers of variables.
4: $fvec (m)$ – Real (Kind=nag_wp) array Input: On entry: the value of the residual $f_{i}$ at the point $x$ , for $i = 1, 2, \dots, m$ , so that the values of the $f_{i}$ can be used in the calculation of the elements of b.
5: $xc (n)$ – Real (Kind=nag_wp) array Input: On entry: the point $x$ at which the elements of b are to be evaluated.
6: $b (lb)$ – Real (Kind=nag_wp) array Output: On exit: unless iflag is reset to a negative number, b must contain the lower triangle of the matrix $b (x)$ , evaluated at the point $x$ , stored by rows. (The upper triangle is not required because the matrix is symmetric.) More precisely, $b (j (j - 1) / 2 + k)$ must contain $\sum_{i = 1}^{m} f_{i} \frac{\partial^{2} f_{i}}{\partial x_{j} \partial x_{k}}$ evaluated at the point $x$ , for $j = 1, 2, \dots, n$ and $k = 1, 2, \dots, j$ .
7: $lb$ – Integer Input: On entry: the length of the array b.
8: $iw (liw)$ – Integer array Workspace
9: $liw$ – Integer Input
10: $w (lw)$ – Real (Kind=nag_wp) array Workspace
11: $lw$ – Integer Input: As in lsqfun, these arguments correspond to the arguments iw, liw, w, lw of e04hef. lsqhes must not change the sections of iw and w required as workspace by e04hef. Again, it is recommended that you should pass quantities to lsqhes via COMMON global variables and not use iw or w at all.

lsqhes must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hef is called. Arguments denoted as Input must not be changed by this procedure.

Note: lsqhes should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04hef. If your code inadvertently does return any NaNs or infinities, e04hef is likely to produce unexpected results.

lsqhes should be tested separately before being used in conjunction with e04hef.

If

iprint \geq 0

, you must supply lsqmon which is suitable for monitoring the minimization process. lsqmon must not change the values of any of its arguments.

If

iprint < 0

, the dummy routine e04fdz can be used as lsqmon.

The specification of lsqmon is:

Fortran Interface

Subroutine lsqmon (

m, n, xc, fvec, fjac, ldfjac, s, igrade, niter, nf, iw, liw, w, lw)

Integer, Intent (In)	::	m, n, ldfjac, igrade, niter, nf, liw, lw
Integer, Intent (Inout)	::	iw(liw)
Real (Kind=nag_wp), Intent (In)	::	xc(n), fvec(m), fjac(ldfjac,n), s(n)
Real (Kind=nag_wp), Intent (Inout)	::	w(lw)

C Header Interface

void	lsqmon (const Integer m, const Integer n, const double xc[], const double fvec[], const double fjac[], const Integer ldfjac, const double s[], const Integer igrade, const Integer niter, const Integer nf, Integer iw[], const Integer liw, double w[], const Integer lw)

Important: the dimension declaration for fjac must contain the variable ldfjac, not an integer constant.

1: $m$ – Integer Input: On entry: $m$ , the numbers of residuals.
2: $n$ – Integer Input: On entry: $n$ , the numbers of variables.
3: $xc (n)$ – Real (Kind=nag_wp) array Input: On entry: the coordinates of the current point $x$ .
4: $fvec (m)$ – Real (Kind=nag_wp) array Input: On entry: the values of the residuals $f_{i}$ at the current point $x$ .
5: $fjac (ldfjac, n)$ – Real (Kind=nag_wp) array Input: On entry: $fjac (i, j)$ contains the value of $\frac{\partial f_{i}}{\partial x_{j}}$ at the current point $x$ , for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ .
6: $ldfjac$ – Integer Input: On entry: the first dimension of the array fjac as declared in the (sub)program from which e04hef is called.
7: $s (n)$ – Real (Kind=nag_wp) array Input: On entry: 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.)
8: $igrade$ – Integer Input: On entry: e04hef 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.
9: $niter$ – Integer Input: On entry: the number of iterations which have been performed in e04hef.
10: $nf$ – Integer Input: On entry: 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.
11: $iw (liw)$ – Integer array Workspace
12: $liw$ – Integer Input
13: $w (lw)$ – Real (Kind=nag_wp) array Workspace
14: $lw$ – Integer Input: As in lsqfun and lsqhes, these arguments correspond to the arguments iw, liw, w, lw of e04hef. They are included in lsqmon's argument list primarily for when e04hef is called by other library routines.

lsqmon must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hef is called. Arguments denoted as Input must not be changed by this procedure.

Note: you should normally print the sum of squares of residuals, so as to be able to examine the sequence of values of

F (x)

mentioned in Section 7. It is usually helpful to also print xc, the gradient of the sum of squares, niter and nf.

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

$iprint > 0$: lsqmon is called once every iprint iterations and just before exit from e04hef.
$iprint = 0$: 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.

Suggested value:

iprint = 1

.

On entry: this argument is present so as to enable you to limit the number of times that lsqfun is called by e04hef. There will be an error exit (see Section 6) after maxcal calls of lsqfun.

Suggested value:

maxcal = 50 \times n

.

Constraint:

maxcal \geq 1

.

On entry: every iteration of e04hef involves a linear minimization (i.e., minimization of

F (x^{(k)} + α^{(k)} p^{(k)})

with respect to

α^{(k)}

). eta must lie in the range

0.0 \leq eta < 1.0

, and specifies how accurately these linear minimizations are to be performed. The minimum with respect to

α^{(k)}

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 e04hef, 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:

eta = 0.5

(

eta = 0.0

if

n = 1

).

Constraint:

0.0 \leq eta < 1.0

.

On entry: 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

‖ x_{sol} - x_{true} ‖ < xtol \times (1.0 + ‖ x_{true} ‖),

where

‖ y ‖ = \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 Section 7.)

If

F (x)

and the variables are scaled roughly as described in Section 9 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

10 ε

, e04hef will use

10 ε

instead of xtol, since

10 ε

is probably the smallest reasonable setting.

Constraint:

xtol \geq 0.0

.

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.)

e04hef will ensure that, for each iteration

\sum_{j = 1}^{n} {(x_{j}^{(k)} - x_{j}^{(k - 1)})}^{2} \leq {(stepmx)}^{2},

where

k

is the iteration number. Thus, if the problem has more than one solution, e04hef is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence of

x^{(k)}

entering a region where the problem is ill-behaved and can help avoid overflow in the evaluation of

F (x)

. However, an underestimate of stepmx can lead to inefficiency.

Suggested value:

stepmx = 100000.0

.

Constraint:

stepmx \geq xtol

.

On entry:

x (j)

must be set to a guess at the

j

th component of the position of the minimum, for

j = 1, 2, \dots, n

.

On exit: the final point

x^{(k)}

. Thus, if

ifail = 0

on exit,

x (j)

is the

j

th component of the estimated position of the minimum.

On exit: the value of

F (x)

, the sum of squares of the residuals

f_{i} (x)

, at the final point given in x.

On exit: the value of the residual

f_{i} (x)

at the final point given in x, for

i = 1, 2, \dots, m

.

On exit: the value of the first derivative

\frac{\partial f_{i}}{\partial x_{j}}

evaluated at the final point given in x, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

.

On entry: the first dimension of the array fjac as declared in the (sub)program from which e04hef is called.

Constraint:

ldfjac \geq m

.

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.

On exit: the matrix

V

associated with the singular value decomposition

J = U S 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

.

On entry: the first dimension of the array v as declared in the (sub)program from which e04hef is called.

Constraint:

ldv \geq n

.

On exit: the number of iterations which have been performed in e04hef.

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

On entry: the dimension of the array iw as declared in the (sub)program from which e04hef is called.

Constraint:

liw \geq 1

.

On entry: the dimension of the array w as declared in the (sub)program from which e04hef is called.

Constraints:

if $n > 1$ , $lw \geq 7 \times n + 2 \times m \times n + m + n \times n$ ;
if $n = 1$ , $lw \geq 9 + 3 \times m$ .

On entry: ifail must be set to

0

,

−1

or

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

or

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

NAG FL Interface
e04hef (lsq_uncon_mod_deriv2_comp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interfacee04hef (lsq_​uncon_​mod_​deriv2_​comp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
e04hef (lsq_uncon_mod_deriv2_comp)