NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04hy: FL CL CPP AD PY MB

NAG FL Interface
e04hyf (lsq_uncon_mod_deriv2_easy)

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NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04hy: FL CL CPP AD PY MB

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2022

1 Purpose

e04hyf is an easy-to-use modified Gauss–Newton algorithm for finding an unconstrained minimum of a sum of squares of

m

nonlinear functions in

n

variables

(m \geq n)

. First and second derivatives are required.

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

2 Specification

Fortran Interface

Subroutine e04hyf (

m, n, lsfun2, lshes2, x, fsumsq, w, lw, iuser, ruser, ifail)

Integer, Intent (In)	::	m, n, lw
Integer, Intent (Inout)	::	iuser(*), ifail
Real (Kind=nag_wp), Intent (Inout)	::	x(n), ruser(*)
Real (Kind=nag_wp), Intent (Out)	::	fsumsq, w(lw)
External	::	lsfun2, lshes2

C Header Interface

#include <nag.h>

void

e04hyf_ (const Integer *m, const Integer *n,
void (NAG_CALL *lsfun2)(const Integer *m, const Integer *n, const double xc[], double fvec[], double fjac[], const Integer *ldfjac, Integer iuser[], double ruser[]),
void (NAG_CALL *lshes2)(const Integer *m, const Integer *n, const double fvec[], const double xc[], double b[], const Integer *lb, Integer iuser[], double ruser[]),
double x[], double *fsumsq, double w[], const Integer *lw, Integer iuser[], double ruser[], Integer *ifail)

The routine may be called by the names e04hyf or nagf_opt_lsq_uncon_mod_deriv2_easy.

3 Description

e04hyf is similar to the subroutine LSSDN2 in the NPL Algorithms Library. It is applicable to problems of the form:

Minimize F (x) = \sum_{i = 1}^{m} {[f_{i} (x)]}^{2}

where

x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}

and

m \geq n

. (The functions

f_{i} (x)

are often referred to as ‘residuals’.)

You must supply a subroutine to evaluate the residuals and their first derivatives at any point

x

, and a subroutine to evaluate the elements of the second derivative term of the Hessian matrix of

F (x)

.

Before attempting to minimize the sum of squares, the algorithm checks the user-supplied subroutines for consistency. Then, from a starting point supplied by you, a sequence of points is generated which is intended to converge to a local minimum of the sum of squares. These points are generated using estimates of the curvature of

F (x)

.

4 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

5 Arguments

1: $m$ – Integer Input

2: $n$ – Integer Input

On entry: the number

m

of residuals,

f_{i} (x)

, and the number

n

of variables,

x_{j}

.

Constraint:

1 \leq n \leq m

.

3: $lsfun2$ – Subroutine, supplied by the user. External Procedure

You must supply this routine to calculate the vector of values

f_{i} (x)

and the Jacobian matrix of first derivatives

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

at any point

x

. It should be tested separately before being used in conjunction with e04hyf (see the E04 Chapter Introduction).

The specification of lsfun2 is:

Fortran Interface

Subroutine lsfun2 (

m, n, xc, fvec, fjac, ldfjac, iuser, ruser)

Integer, Intent (In)	::	m, n, ldfjac
Integer, Intent (Inout)	::	iuser(*)
Real (Kind=nag_wp), Intent (In)	::	xc(n)
Real (Kind=nag_wp), Intent (Inout)	::	fjac(ldfjac,n), ruser(*)
Real (Kind=nag_wp), Intent (Out)	::	fvec(m)

C Header Interface

void	lsfun2 (const Integer m, const Integer n, const double xc[], double fvec[], double fjac[], const Integer *ldfjac, Integer iuser[], double ruser[])

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 point $x$ at which the values of the $f_{i}$ and the $\frac{\partial f_{i}}{\partial x_{j}}$ are required.
4: $fvec (m)$ – Real (Kind=nag_wp) array Output: On exit: $fvec (i)$ must be set to the value of $f_{i}$ at the point $x$ , for $i = 1, 2, \dots, m$ .
5: $fjac (ldfjac, n)$ – Real (Kind=nag_wp) array Output: On exit: $fjac (i, j)$ must be set to the value of $\frac{\partial f_{i}}{\partial x_{j}}$ at the point $x$ , for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ .
6: $ldfjac$ – Integer Input: On entry: the first dimension of the array fjac as declared in the (sub)program from which e04hyf is called.
7: $iuser (*)$ – Integer array User Workspace
8: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace: lsfun2 is called with the arguments iuser and ruser as supplied to e04hyf. You should use the arrays iuser and ruser to supply information to lsfun2.

lsfun2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: lsfun2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04hyf. If your code inadvertently does return any NaNs or infinities, e04hyf is likely to produce unexpected results.

4: $lshes2$ – Subroutine, supplied by the user. External Procedure

You must supply this routine to 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)

. It should be tested separately before being used in conjunction with e04hyf (see the E04 Chapter Introduction).

The specification of lshes2 is:

Fortran Interface

Subroutine lshes2 (

m, n, fvec, xc, b, lb, iuser, ruser)

Integer, Intent (In)	::	m, n, lb
Integer, Intent (Inout)	::	iuser(*)
Real (Kind=nag_wp), Intent (In)	::	fvec(m), xc(n)
Real (Kind=nag_wp), Intent (Inout)	::	ruser(*)
Real (Kind=nag_wp), Intent (Out)	::	b(lb)

C Header Interface

void	lshes2 (const Integer m, const Integer n, const double fvec[], const double xc[], double b[], const Integer *lb, Integer iuser[], double ruser[])

1: $m$ – Integer Input: On entry: $m$ , the number of residuals.
2: $n$ – Integer Input: On entry: $n$ , the number of residuals.
3: $fvec (m)$ – 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.
4: $xc (n)$ – Real (Kind=nag_wp) array Input: On entry: the point $x$ at which the elements of b are to be evaluated.
5: $b (lb)$ – Real (Kind=nag_wp) array Output: On exit: 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 $k = 1, 2, \dots, j$ .
6: $lb$ – Integer Input: On entry: the length of the array b.
7: $iuser (*)$ – Integer array User Workspace
8: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace: lshes2 is called with the arguments iuser and ruser as supplied to e04hyf. You should use the arrays iuser and ruser to supply information to lshes2.

lshes2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: lshes2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04hyf. If your code inadvertently does return any NaNs or infinities, e04hyf is likely to produce unexpected results.

5: $x (n)$ – Real (Kind=nag_wp) array Input/Output

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

. The routine checks lsfun2 and lshes2 at the starting point and so is more likely to detect any error in your routines if the initial

x (j)

are nonzero and mutually distinct.

On exit: the lowest point found during the calculations. Thus, if

ifail = 0

on exit,

x (j)

is the

j

th component of the position of the minimum.

6: $fsumsq$ – Real (Kind=nag_wp) Output

On exit: the value of the sum of squares,

F (x)

, corresponding to the final point stored in x.

7: $w (lw)$ – Real (Kind=nag_wp) array Communication Array

8: $lw$ – Integer Input

On entry: the dimension of the array w as declared in the (sub)program from which e04hyf is called.

Constraints:

if $n > 1$ , $lw \geq 8 \times n + 2 \times n \times n + 2 \times m \times n + 3 \times m$ ;
if $n = 1$ , $lw \geq 11 + 5 \times m$ .

9: $iuser (*)$ – Integer array User Workspace

10: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by e04hyf, but are passed directly to lsfun2 and lshes2 and may be used to pass information to these routines.

11: $ifail$ – Integer Input/Output

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).

6 Error Indicators and Warnings

If on entry

ifail = 0

or

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

Note: in some cases e04hyf may return useful information.

If you are not satisfied with the result (e.g., because

ifail = 3

,

4

,

5

,

6

,

7

or

8

), it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure. Repeated failure may indicate some defect in the formulation of the problem.

$ifail = 1$: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $m \geq n$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $n = 1$ and $lw = ⟨ value ⟩$ .
Constraint: if $n = 1$ then $lw \geq 11 + 5 \times m$ ; that is, $⟨ value ⟩$ .

On entry, $n > 1$ and $lw = ⟨ value ⟩$ .
Constraint: if $n > 1$ then $lw \geq 8 \times n + 2 \times n^{2} + 2 \times m \times n + 3 \times m$ ; that is, $⟨ value ⟩$ .

$ifail = 2$: There have been $50 \times n$ calls to lsfun2.
The algorithm does not seem to have converged. This may be due to an awkward function or to a poor starting point, so it is worth restarting e04hyf from the final point held in x.

$ifail = 3$: The conditions for a minimum have not all been satisfied, but a lower point could not be found. See Section 7 for further information.

$ifail = 4$: Failure in computing SVD of Jacobian matrix.

$ifail = 5$: It is probable that a local minimum has been found, but it cannot be guaranteed.

$ifail = 6$: It is possible that a local minimum has been found, but it cannot be guaranteed.

$ifail = 7$: It is unlikely that a local minimum has been found.

$ifail = 8$: It is very unlikely that a local minimum has been found.

$ifail = 9$: It is very likely that you have made an error in forming the derivatives in lsfun2.

$ifail = 10$: It is very likely that you have made an error in setting up the array b in lshes2.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If the problem is reasonably well scaled and a successful exit is made, then, for a computer with a mantissa of

t

decimals, one would expect to get about

t / 2 - 1

decimals accuracy in the components of

x

and between

t - 1

(if

F (x)

is of order

1

at the minimum) and

2 t - 2

(if

F (x)

is close to zero at the minimum) decimals accuracy in

F (x)

.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e04hyf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e04hyf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The number of iterations required depends on the number of variables, the number of residuals and their behaviour, and the distance of the starting point from the solution. The number of multiplications performed per iteration of e04hyf varies, but for

m ≫ n

is approximately

n \times m^{2} + O (n^{3})

. In addition, each iteration makes at least one call of lsfun2 and some iterations may call lshes2. So, unless the residuals and their derivatives can be evaluated very quickly, the run time will be dominated by the time spent in lsfun2 (and, to a lesser extent, in lshes2).

Ideally, the problem should be scaled so that the minimum value of the sum of squares is in the range

(0, + 1)

and so that at points a unit distance away from the solution the sum of squares is approximately a unit value greater than at the minimum. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that e04hyf will take less computer time.

When the sum of squares represents the goodness-of-fit of a nonlinear model to observed data, elements of the variance-covariance matrix of the estimated regression coefficients can be computed by a subsequent call to e04ycf, using information returned in segments of the workspace array w. See e04ycf for further details.

10 Example

This example finds least squares estimates of

x_{1}

,

x_{2}

and

x_{3}

in the model

y = x_{1} + \frac{t_{1}}{x_{2} t_{2} + x_{3} t_{3}}

using the

15

sets of data given in the following table.

\begin{array}{r} y & t_{1} & t_{2} & t_{3} \\ 0.14 & 1.0 & 15.0 & 1.0 \\ 0.18 & 2.0 & 14.0 & 2.0 \\ 0.22 & 3.0 & 13.0 & 3.0 \\ 0.25 & 4.0 & 12.0 & 4.0 \\ 0.29 & 5.0 & 11.0 & 5.0 \\ 0.32 & 6.0 & 10.0 & 6.0 \\ 0.35 & 7.0 & 9.0 & 7.0 \\ 0.39 & 8.0 & 8.0 & 8.0 \\ 0.37 & 9.0 & 7.0 & 7.0 \\ 0.58 & 10.0 & 6.0 & 6.0 \\ 0.73 & 11.0 & 5.0 & 5.0 \\ 0.96 & 12.0 & 4.0 & 4.0 \\ 1.34 & 13.0 & 3.0 & 3.0 \\ 2.10 & 14.0 & 2.0 & 2.0 \\ 4.39 & 15.0 & 1.0 & 1.0 \end{array}

The program uses

(0.5, 1.0, 1.5)

as the initial guess at the position of the minimum.

10.1 Program Text

Program Text (e04hyfe.f90)

10.2 Program Data

Program Data (e04hyfe.d)

10.3 Program Results

Program Results (e04hyfe.r)

NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04hy: FL CL CPP AD PY MB

© The Numerical Algorithms Group Ltd. 2022