E04HYF (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

E04HYF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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

SUBROUTINE E04HYF (

M, N, LSFUN2, LSHES2, X, FSUMSQ, W, LW, IUSER, RUSER, IFAIL)

INTEGER	M, N, LW, IUSER(*), IFAIL
REAL (KIND=nag_wp)	X(N), FSUMSQ, W(LW), RUSER(*)
EXTERNAL	LSFUN2, LSHES2

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 Parameters

1: M – INTEGERInput

2: N – INTEGERInput

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:

SUBROUTINE LSFUN2 (

M, N, XC, FVEC, FJAC, LDFJAC, IUSER, RUSER)

INTEGER	M, N, LDFJAC, IUSER(*)
REAL (KIND=nag_wp)	XC(N), FVEC(M), FJAC(LDFJAC,N), RUSER(*)

Important: the dimension declaration for FJAC must contain the variable LDFJAC, not an integer constant.

1: M – INTEGERInput: On entry: $m$ , the numbers of residuals.
2: N – INTEGERInput: On entry: $n$ , the numbers of variables.
3: XC(N) – REAL (KIND=nag_wp) arrayInput: 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) arrayOutput: 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) arrayOutput: 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 – INTEGERInput: On entry: the first dimension of the array FJAC as declared in the (sub)program from which E04HYF is called.
7: IUSER( $*$ ) – INTEGER arrayUser Workspace
8: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: LSFUN2 is called with the parameters IUSER and RUSER as supplied to E04HYF. You are free to use the arrays IUSER and RUSER to supply information to LSFUN2 as an alternative to using COMMON global variables.

LSFUN2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04HYF is called. Parameters denoted as Input must not be changed by this procedure.

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:

SUBROUTINE LSHES2 (

M, N, FVEC, XC, B, LB, IUSER, RUSER)

INTEGER	M, N, LB, IUSER(*)
REAL (KIND=nag_wp)	FVEC(M), XC(N), B(LB), RUSER(*)

1: M – INTEGERInput: On entry: $m$ , the number of residuals.
2: N – INTEGERInput: On entry: $n$ , the number of residuals.
3: FVEC(M) – REAL (KIND=nag_wp) arrayInput: 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) arrayInput: On entry: the point $x$ at which the elements of B are to be evaluated.
5: B(LB) – REAL (KIND=nag_wp) arrayOutput: 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 – INTEGERInput: On entry: the length of the array B.
7: IUSER( $*$ ) – INTEGER arrayUser Workspace
8: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: LSHES2 is called with the parameters IUSER and RUSER as supplied to E04HYF. You are free to use the arrays IUSER and RUSER to supply information to LSHES2 as an alternative to using COMMON global variables.

LSHES2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04HYF is called. Parameters denoted as Input must not be changed by this procedure.

5: X(N) – REAL (KIND=nag_wp) arrayInput/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) arrayWorkspace

8: LW – INTEGERInput

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 arrayUser Workspace

10: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser 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 as an alternative to using COMMON global variables.

11: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

,

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. 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).

Note: E04HYF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$N < 1$ ,
or	$M < N$ ,
or	$LW < 8 \times N + 2 \times N \times N + 2 \times M \times N + 3 \times M$ , when $N > 1$ ,
or	$LW < 11 + 5 \times M$ , when $N = 1$ .

$IFAIL = 2$: There have been $50 \times n$ calls of LSFUN2, yet 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 final point does not satisfy the conditions for acceptance as a minimum, but no lower point could be found.

$IFAIL = 4$: An auxiliary routine has been unable to complete a singular value decomposition in a reasonable number of sub-iterations.

$IFAIL = 5$
$IFAIL = 6$
$IFAIL = 7$
$IFAIL = 8$: There is some doubt about whether the point $x$ found by E04HYF is a minimum of $F (x)$ . The degree of confidence in the result decreases as IFAIL increases. Thus, when $IFAIL = 5$ it is probable that the final $x$ gives a good estimate of the position of a minimum, but when $IFAIL = 8$ it is very unlikely that the routine has found a minimum.

$IFAIL = 9$: It is very likely that you have made an error in forming the derivatives $\frac{\partial f_{i}}{\partial x_{j}}$ in LSFUN2.

$IFAIL = 10$: It is very likely that you have made an error in forming the quantities $B_{j k}$ in LSHES2.

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

3

and

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.

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 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.

9 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.

9.1 Program Text

Program Text (e04hyfe.f90)

9.2 Program Data

Program Data (e04hyfe.d)

9.3 Program Results

Program Results (e04hyfe.r)

E04HYF (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012