NAG Library Routine Document

E04HEF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     M – INTEGERInput

2:     N – INTEGERInput

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

Constraint: $1\le \mathbf{N}\le \mathbf{M}$.

3:     LSQFUN – SUBROUTINE, supplied by the user.External Procedure

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 a parameter to cause E04HEF to terminate immediately.)

The specification of LSQFUN is:

SUBROUTINE LSQFUN ( IFLAG, M, N, XC, FVEC, FJAC, LDFJAC, IW, LIW, W, LW) INTEGER  IFLAG, M, N, LDFJAC, IW(LIW), LIW, LW REAL (KIND=nag_wp)  XC(N), FVEC(M), FJAC(LDFJAC,N), W(LW)

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

1:     IFLAG – INTEGERInput/Output

On entry: to LSQFUN, IFLAG will be set to $2$.

On exit: 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 E04HEF. E04HEF will then terminate immediately, with IFAIL set to your setting of IFLAG.

2:     M – INTEGERInput

On entry: $m$, the numbers of residuals.

3:     N – INTEGERInput

On entry: $n$, the numbers of variables.

4:     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.

5:     FVEC(M) – REAL (KIND=nag_wp) arrayOutput

On exit: unless IFLAG is reset to a negative number, $\mathbf{FVEC}\left(\mathit{i}\right)$ must contain the value of $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$.

6:     FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayOutput

On exit: unless IFLAG is reset to a negative number, $\mathbf{FJAC}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

7:     LDFJAC – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which E04HEF is called.

8:     IW(LIW) – INTEGER arrayWorkspace

9:     LIW – INTEGERInput

10:   W(LW) – REAL (KIND=nag_wp) arrayWorkspace

11:   LW – INTEGERInput

LSQFUN is called with E04HEF's parameters IW, LIW, W, LW as these parameters. 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. Parameters denoted as Input must not be changed by this procedure.

Note:  LSQFUN should be tested separately before being used in conjunction with E04HEF.

4:     LSQHES – SUBROUTINE, supplied by the user.External Procedure

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 E04HEF to terminate immediately.)

The specification of LSQHES is:

SUBROUTINE LSQHES ( IFLAG, M, N, FVEC, XC, B, LB, IW, LIW, W, LW) INTEGER  IFLAG, M, N, LB, IW(LIW), LIW, LW REAL (KIND=nag_wp)  FVEC(M), XC(N), B(LB), W(LW)

1:     IFLAG – INTEGERInput/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 – INTEGERInput

On entry: $m$, the numbers of residuals.

3:     N – INTEGERInput

On entry: $n$, the numbers of variables.

4:     FVEC(M) – REAL (KIND=nag_wp) arrayInput

On entry: the value of the residual $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$, so that the values of the $f_{\mathit{i}}$ can be used in the calculation of the elements of B.

5:     XC(N) – REAL (KIND=nag_wp) arrayInput

On entry: the point $x$ at which the elements of B are to be evaluated.

6:     B(LB) – REAL (KIND=nag_wp) arrayOutput

On exit: unless IFLAG is reset to a negative number, B must contain the lower triangle of the matrix $\mathbf{B}\left(x\right)$, evaluated at the point $x$, stored by rows. (The upper triangle is not required because the matrix is symmetric.) More precisely, $\mathbf{B}\left(\mathit{j}\left(\mathit{j}-1\right)/2+\mathit{k}\right)$ must contain ${\displaystyle \sum _{\mathit{i}=1}^m}{f}_{\mathit{i}}\frac{{\partial }^2{f}_{\mathit{i}}}{\partial x_{\mathit{j}}\partial x_{\mathit{k}}}$ evaluated at the point $x$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{k}=1,2,\dots ,\mathit{j}$.

7:     LB – INTEGERInput

On entry: the length of the array B.

8:     IW(LIW) – INTEGER arrayWorkspace

9:     LIW – INTEGERInput

10:   W(LW) – REAL (KIND=nag_wp) arrayWorkspace

11:   LW – INTEGERInput

As in LSQFUN, these parameters correspond to the parameters 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. Parameters denoted as Input must not be changed by this procedure.

Note:  LSQHES should be tested separately before being used in conjunction with E04HEF.

5:     LSQMON – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

If $\mathbf{IPRINT}\ge 0$, you must supply LSQMON which is suitable for monitoring the minimization process. LSQMON must not change the values of any of its parameters.

If $\mathbf{IPRINT}<0$, the dummy routine E04FDZ can be used as LSQMON.

The specification of LSQMON is:

SUBROUTINE LSQMON ( M, N, XC, FVEC, FJAC, LDFJAC, S, IGRADE, NITER, NF, IW, LIW, W, LW) INTEGER  M, N, LDFJAC, IGRADE, NITER, NF, IW(LIW), LIW, LW REAL (KIND=nag_wp)  XC(N), FVEC(M), FJAC(LDFJAC,N), S(N), W(LW)

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 coordinates of the current point $x$.

4:     FVEC(M) – REAL (KIND=nag_wp) arrayInput

On entry: the values of the residuals $f_i$ at the current point $x$.

5:     FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayInput

On entry: $\mathbf{FJAC}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the current point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

6:     LDFJAC – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which E04HEF is called.

7:     S(N) – REAL (KIND=nag_wp) arrayInput

On entry: the singular values of the current Jacobian matrix. Thus S may be useful as information about the structure of your problem. (If $\mathbf{IPRINT}>0$, 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 – INTEGERInput

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 – INTEGERInput

On entry: the number of iterations which have been performed in E04HEF.

10:   NF – INTEGERInput

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 arrayWorkspace

12:   LIW – INTEGERInput

13:   W(LW) – REAL (KIND=nag_wp) arrayWorkspace

14:   LW – INTEGERInput

As in LSQFUN and LSQHES, these parameters correspond to the parameters IW, LIW, W, LW of E04HEF. They are included in LSQMON's parameter 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. Parameters 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 $F\left(x\right)$ mentioned in Section 7. It is usually helpful to also print XC, the gradient of the sum of squares, NITER and NF.

6:     IPRINT – INTEGERInput

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

$\mathbf{IPRINT}>0$

LSQMON is called once every IPRINT iterations and just before exit from E04HEF.

$\mathbf{IPRINT}=0$

LSQMON is just called at the final point.

$\mathbf{IPRINT}<0$

LSQMON is not called at all.

IPRINT should normally be set to a small positive number.

Suggested value: $\mathbf{IPRINT}=1$.

7:     MAXCAL – INTEGERInput

On entry: this parameter 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: $\mathbf{MAXCAL}=50\times n$.

Constraint: $\mathbf{MAXCAL}\ge 1$.

8:     ETA – REAL (KIND=nag_wp)Input

On entry: every iteration of E04HEF involves a linear minimization (i.e., minimization of $F\left(x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)}\right)$ with respect to ${\alpha }^{\left(k\right)}$). ETA must lie in the range $0.0\le \mathbf{ETA}<1.0$, 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 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: $\mathbf{ETA}=0.5$ ($\mathbf{ETA}=0.0$ if $\mathbf{N}=1$).

Constraint: $0.0\le \mathbf{ETA}<1.0$.

9:     XTOL – REAL (KIND=nag_wp)Input

On entry: the accuracy in $x$ to which the solution is required.

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

$$‖x_{\mathrm{sol}}-x_{\mathrm{true}}‖<\mathbf{XTOL}\times \left(1.0+‖x_{\mathrm{true}}‖\right)\text{,}$$

where $‖y‖=\sqrt{{\displaystyle \sum _{j=1}^n}{y}_j^2}$. For example, if the elements of $x_{\mathrm{sol}}$ are not much larger than $1.0$ in modulus and if $\mathbf{XTOL}=\text{1.0E−5}$, then $x_{\mathrm{sol}}$ is usually accurate to about five decimal places. (For further details see Section 7.)

If $F\left(x\right)$ and the variables are scaled roughly as described in Section 8 and $\epsilon$ is the machine precision, then a setting of order $\mathbf{XTOL}=\sqrt{\epsilon }$ will usually be appropriate. If XTOL is set to $0.0$ or some positive value less than $10\epsilon$, E04HEF will use $10\epsilon$ instead of XTOL, since $10\epsilon$ is probably the smallest reasonable setting.

Constraint: $\mathbf{XTOL}\ge 0.0$.

10:   STEPMX – REAL (KIND=nag_wp)Input

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

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

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 $x^{\left(k\right)}$ 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.

Suggested value: $\mathbf{STEPMX}=100000.0$.

Constraint: $\mathbf{STEPMX}\ge \mathbf{XTOL}$.

11:   X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: $\mathbf{X}\left(\mathit{j}\right)$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$.

On exit: the final point $x^{\left(k\right)}$. Thus, if $\mathbf{IFAIL}=\mathbf{0}$ on exit, $\mathbf{X}\left(j\right)$ is the $j$th component of the estimated position of the minimum.

12:   FSUMSQ – REAL (KIND=nag_wp)Output

On exit: 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.

13:   FVEC(M) – REAL (KIND=nag_wp) arrayOutput

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

14:   FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the value of the first derivative $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ evaluated at the final point given in X, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

15:   LDFJAC – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which E04HEF is called.

Constraint: $\mathbf{LDFJAC}\ge \mathbf{M}$.

16:   S(N) – REAL (KIND=nag_wp) arrayOutput

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.

17:   V(LDV,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the matrix $V$ associated with the singular value decomposition

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

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 $J^{\mathrm{T}}J$.

18:   LDV – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which E04HEF is called.

Constraint: $\mathbf{LDV}\ge \mathbf{N}$.

19:   NITER – INTEGEROutput

On exit: the number of iterations which have been performed in E04HEF.

20:   NF – INTEGEROutput

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

21:   IW(LIW) – INTEGER arrayCommunication Array

22:   LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which E04HEF is called.

Constraint: $\mathbf{LIW}\ge 1$.

23:   W(LW) – REAL (KIND=nag_wp) arrayCommunication Array

24:   LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which E04HEF is called.

Constraints:

• if $\mathbf{N}>1$, $\mathbf{LW}\ge 7\times \mathbf{N}+2\times \mathbf{M}\times \mathbf{N}+\mathbf{M}+\mathbf{N}\times \mathbf{N}$;
• if $\mathbf{N}=1$, $\mathbf{LW}\ge 9+3\times \mathbf{M}$.

25:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ 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 $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}<0$

A negative value of IFAIL indicates an exit from E04HEF because you have set IFLAG negative in LSQFUN or LSQHES. The value of IFAIL will be the same as your setting of IFLAG.

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{N}<1$,
or	$\mathbf{M}<\mathbf{N}$,
or	$\mathbf{MAXCAL}<1$,
or	$\mathbf{ETA}<0.0$,
or	$\mathbf{ETA}\ge 1.0$,
or	$\mathbf{XTOL}<0.0$,
or	$\mathbf{STEPMX}<\mathbf{XTOL}$,
or	$\mathbf{LDFJAC}<\mathbf{M}$,
or	$\mathbf{LDV}<\mathbf{N}$,
or	$\mathbf{LIW}<1$,
or	$\mathbf{LW}<7\times \mathbf{N}+\mathbf{M}\times \mathbf{N}+2\times \mathbf{M}+\mathbf{N}\times \mathbf{N}$ when $\mathbf{N}>1$,
or	$\mathbf{LW}<9+3\times \mathbf{M}$ when $\mathbf{N}=1$.

When this exit occurs, no values will have been assigned to FSUMSQ, or to the elements of FVEC, FJAC, S or V.

$\mathbf{IFAIL}=2$

There have been MAXCAL calls of LSQFUN. If steady reductions in the sum of squares, $F\left(x\right)$, were monitored up to the point where this exit occurred, then the exit probably occurred simply because MAXCAL was set too small, so the calculations should be restarted from the final point held in X. This exit may also indicate that $F\left(x\right)$ has no minimum.

$\mathbf{IFAIL}=3$

The conditions for a minimum have not all been satisfied, but a lower point could not be found. This could be because XTOL has been set so small that rounding errors in the evaluation of the residuals and derivatives make attainment of the convergence conditions impossible.

$\mathbf{IFAIL}=4$

The method for computing the singular value decomposition of the Jacobian matrix has failed to converge in a reasonable number of sub-iterations. It may be worth applying E04HEF again starting with an initial approximation which is not too close to the point at which the failure occurred.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (e04hefe.f90)

9.2  Program Data

Program Data (e04hefe.d)

9.3  Program Results

Program Results (e04hefe.r)