NAG FL Interface
e04gyf (lsq_​uncon_​quasi_​deriv_​easy)

        Subroutine lsfun2(m,n,xc,fvec,fjac,ldfjac,iuser,ruser)

!       Routine to evaluate the residuals and their 1st derivatives.
!       .. Scalar Arguments ..
         Integer, Intent (In)           :: ldfjac, m, n
!       .. Array Arguments ..
         Real (Kind=nag_wp), Intent (Inout) :: fjac(ldfjac,n), ruser(*)
         Real (Kind=nag_wp), Intent (Out) :: fvec(m)
         Real (Kind=nag_wp), Intent (In) :: xc(n)
         Integer, Intent (Inout)        :: iuser(*)
!       .. Local Scalars ..
         Real (Kind=nag_wp)             :: denom, dummy
         Integer                        :: i
!       .. Executable Statements ..

         fvec(1:m) = ...
         fjac(1:ldfjac,1:n) = ...

         Return
        End Subroutine lsfun2

        Subroutine lsqfun(nvar,x,nres,rx,inform,iuser,ruser,cpuser)

!        .. Use Statements ..
          Use, Intrinsic                 :: iso_c_binding, Only: c_ptr
!        .. Scalar Arguments ..
          Type (c_ptr), Intent (In)      :: cpuser
          Integer, Intent (Inout)        :: inform
          Integer, Intent (In)           :: nres, nvar
!        .. Array Arguments ..
          Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
          Real (Kind=nag_wp), Intent (Out) :: rx(nres)
          Real (Kind=nag_wp), Intent (In) :: x(nvar)
          Integer, Intent (Inout)        :: iuser(*)
!        .. Executable Statements ..

!        Compute residuals
          rx(:) = ...

!        Inform evaluation was successful
          inform = 0
          Return
        End Subroutine lsqfun

        Subroutine lsqgrd(nvar,x,nres,nnzrd,rdx,inform,iuser,ruser,cpuser)

!        .. Use Statements ..
          Use, Intrinsic                 :: iso_c_binding, Only: c_ptr
!        .. Scalar Arguments ..
          Type (c_ptr), Intent (In)      :: cpuser
          Integer, Intent (Inout)        :: inform
          Integer, Intent (In)           :: nnzrd, nres, nvar
!        .. Array Arguments ..
          Real (Kind=nag_wp), Intent (Inout) :: rdx(nnzrd), ruser(*)
          Real (Kind=nag_wp), Intent (In) :: x(nvar)
          Integer, Intent (Inout)        :: iuser(*)
!        .. Executable Statements ..

!        Compute gradient of residuals
          rdx(:) = ...

!        Inform evaluation was successful
          inform = 0
          Return
        End Subroutine lsqgrd

       ifail = -1
       Call e04gyf(m,n,lsfun2,x,fsumsq,w,lw,iuser,ruser,ifail)

!      .. Use Statements ..
        Use, Intrinsic                   :: iso_c_binding, Only: c_null_ptr, c_ptr

!      ...

!      Initialize problem with n variables
        ifail = 0
        Call e04raf(handle,n,ifail)

!      Define nonlinear least-square problem with m residuals and dense structure
        isparse = 0
        nnzrd = 0
        Call e04rmf(handle,m,isparse,nnzrd,irowrd,icolrd,ifail)

!      Solve the problem
        cpuser = c_null_ptr
        ifail = -1
        Call e04ggf(handle,lsqfun,lsqgrd,e04ggu,e04ggv,e04ffu,n,x,m,rx,rinfo,    &
          stats,iuser,ruser,cpuser,ifail)

        fsumsq = 2.0_nag_wp * rinfo(1)

!      Free the handle memory
        ifail = 0
        Call e04rzf(handle,ifail)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{m}$ – Integer Input

2: $\mathbf{n}$ – Integer Input

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: $\mathbf{lsfun2}$ – Subroutine, supplied by the user. External Procedure

You must supply this routine to calculate the vector of values $f_i\left(x\right)$ 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 e04gyf (see the E04 Chapter Introduction).

The specification of lsfun2 is:

Fortran Interface copy

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 copy

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

C++ Header Interface copy

#include <nag.h> extern "C" { 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: $\mathbf{m}$ – Integer Input

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

2: $\mathbf{n}$ – Integer Input

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

3: $\mathbf{xc}\left(\mathbf{n}\right)$ – 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: $\mathbf{fvec}\left(\mathbf{m}\right)$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{fvec}\left(\mathit{i}\right)$ must contain the value of $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$.

5: $\mathbf{fjac}(\mathbf{ldfjac},\mathbf{n})$ – Real (Kind=nag_wp) array Output

On exit: $\mathbf{fjac}(\mathit{i},\mathit{j})$ 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 $\mathit{j}=1,2,\dots ,n$.

6: $\mathbf{ldfjac}$ – Integer Input

On entry: the first dimension of the array fjac, set to $m$ by e04gyf.

7: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

8: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

lsfun2 is called with the arguments iuser and ruser as supplied to e04gyf. 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 e04gyf 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 e04gyf. If your code inadvertently does return any NaNs or infinities, e04gyf is likely to produce unexpected results.

4: $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input/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 $\mathit{j}=1,2,\dots ,n$. The routine checks the first derivatives calculated by lsfun2 at the starting point and so is more likely to detect an error in your routine if the initial $\mathbf{x}\left(j\right)$ are nonzero and mutually distinct.

On exit: the lowest point found during the calculations. Thus, if $\mathbf{ifail}=\mathbf{0}$ on exit, $\mathbf{x}\left(j\right)$ is the $j$th component of the position of the minimum.

5: $\mathbf{fsumsq}$ – Real (Kind=nag_wp) Output

On exit: the value of the sum of squares, $F\left(x\right)$, corresponding to the final point stored in x.

6: $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) array Communication Array

7: $\mathbf{lw}$ – Integer Input

On entry: the dimension of the array w as declared in the (sub)program from which e04gyf is called.

Constraints:

• if $\mathbf{n}>1$, $\mathbf{lw}\ge 8\times \mathbf{n}+2\times \mathbf{n}\times \mathbf{n}+2\times \mathbf{m}\times \mathbf{n}+3\times \mathbf{m}$;
• if $\mathbf{n}=1$, $\mathbf{lw}\ge 11+5\times \mathbf{m}$.

8: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

9: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by e04gyf, but are passed directly to lsfun2 and may be used to pass information to this routine.

10: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when $\mathbf{ifail}\ne \mathbf{0}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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

$\mathbf{ifail}=1$

On entry, $\mathbf{m}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{m}\ge \mathbf{n}$.

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{n}=1$ and $\mathbf{lw}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{n}=1$ then $\mathbf{lw}\ge 11+5\times \mathbf{m}$; that is, $⟨\mathit{\text{value}}⟩$.

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

$\mathbf{ifail}=2$

There have been $50\times \mathbf{n}$ 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 e04gyf from the final point held in x.

$\mathbf{ifail}=3$

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

$\mathbf{ifail}=4$

Failure in computing SVD of Jacobian matrix.

$\mathbf{ifail}=5$

It is probable that a local minimum has been found, but it cannot be guaranteed.

$\mathbf{ifail}=6$

It is possible that a local minimum has been found, but it cannot be guaranteed.

$\mathbf{ifail}=7$

It is unlikely that a local minimum has been found.

$\mathbf{ifail}=8$

It is very unlikely that a local minimum has been found.

$\mathbf{ifail}=9$

It is very likely that you have made an error in forming the derivatives in lsfun2.

$\mathbf{ifail}=-99$

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.

$\mathbf{ifail}=-399$

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.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results