The routine may be called by the names e04hyf or nagf_opt_lsq_uncon_mod_deriv2_easy.
e04hyf is similar to the subroutine LSSDN2 in the NPL Algorithms Library. It is applicable to problems of the form:
where and . (The functions are often referred to as ‘residuals’.)
You must supply a subroutine to evaluate the residuals and their first derivatives at any point , and a subroutine to evaluate the elements of the second derivative term of the Hessian matrix of .
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 .
Gill P E and Murray W (1978) Algorithms for the solution of the nonlinear least squares problem SIAM J. Numer. Anal.15 977–992
1: – IntegerInput
2: – IntegerInput
On entry: the number of residuals, , and the number of variables, .
3: – Subroutine, supplied by the user.External Procedure
You must supply this routine to calculate the vector of values and the Jacobian matrix of first derivatives at any point . It should be tested separately before being used in conjunction with e04hyf (see the E04 Chapter Introduction).
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: – Subroutine, supplied by the user.External Procedure
You must supply this routine to calculate the elements of the symmetric matrix
at any point , where is the Hessian matrix of . It should be tested separately before being used in conjunction with e04hyf (see the E04 Chapter Introduction).
On entry: the value of the residual
at the point , for , so that the values of the can be used in the calculation of the elements of b.
4: – Real (Kind=nag_wp) arrayInput
On entry: the point at which the elements of b are to be evaluated.
5: – Real (Kind=nag_wp) arrayOutput
On exit: must contain the lower triangle of the matrix , evaluated at the point , stored by rows. (The upper triangle is not required because the matrix is symmetric.) More precisely,
must contain evaluated at the point , for and .
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: – Real (Kind=nag_wp) arrayInput/Output
On entry: must be set to a guess at the th component of the position of the minimum, for . 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 are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if on exit, is the th component of the position of the minimum.
6: – Real (Kind=nag_wp)Output
On exit: the value of the sum of squares, , corresponding to the final point stored in x.
7: – Real (Kind=nag_wp) arrayCommunication Array
8: – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which e04hyf is called.
if , ;
if , .
9: – Integer arrayUser Workspace
10: – 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.
11: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended since useful values can be provided in some output arguments even when on exit. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , 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 , , , , or ), 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.
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.
The conditions for a minimum have not all been satisfied, but a lower point could not be found. See Section 7 for further information.
Failure in computing SVD of Jacobian matrix.
It is probable that a local minimum has been found, but it cannot be guaranteed.
It is possible that a local minimum has been found, but it cannot be guaranteed.
It is unlikely that a local minimum has been found.
It is very unlikely that a local minimum has been found.
It is very likely that you have made an error in forming the derivatives in lsfun2.
It is very likely that you have made an error in setting up the array b in lshes2.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
If the problem is reasonably well scaled and a successful exit is made, then, for a computer with a mantissa of decimals, one would expect to get about decimals accuracy in the components of and between (if is of order at the minimum) and (if is close to zero at the minimum) decimals accuracy in .
8Parallelism and Performance
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.
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 is approximately . 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 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.
This example finds least squares estimates of , and in the model
using the sets of data given in the following table.
The program uses as the initial guess at the position of the minimum.