dqed, a Fortran90 code which solves (square) systems of nonlinear equations, or minimizes the residual in a set of nonlinear equations, using least squares, by Richard Hanson and Fred Krogh.
The user may include simple bounds or linear constraints on variables.
DQED can solve, in the least squares sense, systems of linear or nonlinear equations, with inequality bounds or equality constraints.
DQED was written by Richard Hanson and Fred Krogh of Sandia National Laboratory.
The original Fortran77 source code is available through NETLIB at
https://www.netlib.org/opt/dqed.f.
DQED solves the constrained nonlinear least squares problem:
Minimize the sum of squares of MEQUA generally nonlinear equations,
f(1:MEQUA)(x) = 0, Equation (1)where x is a set of NVARS unknowns. The vector function with these MEQUA components is called f(x) in the discussion that follows.
The components of x may have upper and lower bounds given by the user. In fact, all of the possible cases can be specified:
Linear constraints on the unknowns, more general than simple bounds, can also be given. These linear constraints can be of the equality or inequality type:
a(L,1) x(1)+ ... + a(L,NVARS) x(NVARS) = y(L), L = 1,...,MCON, Equation (2)with bounds specified on the right hand side values y(L), again given by the user. The constraints can actually be slightly nonlinear. In this case the constraints can be described as:
g(L)(x) = y(L), L = 1,...,MCON, Equation (2')where bounds are specified on each y(L). The functions g(L)(x) must be defined for all x in the set described by the simple bounds.
Experienced users may wish to turn directly to the examples before reading the subprogram documentation.
There is no size relation required for the problem dimensions MEQUA, NVARS, and MCON except that MEQUA and NVARS are both positive, and MCON is nonnegative.
DQED will do a decent job of solving most nonlinear least squares problems that can be expressed as Equations (1) and (2) above, provided that continuous derivatives of the functions with respect to the parameters can be computed. This can also include problems where the derivatives must be computed using some form of numerical differentiation. Numerical differentiation is not provided with this software for solving nonlinear least squares problems.
The authors also plan to develop methods that will do a much better job of coping with constraints more general than the essentially linear ones indicated above in Equations (2)-(2'). There are nonlinear least squares problems with innocent-looking but highly nonlinear constraints where this package will fail to work. The authors also hope to reduce the overhead required by the software. This high overhead is due primarily to the method used to solve the inner-loop quadratic model problem. The authors recommend that users consider using the option number 14, to suppress use of the quadratic model. The user may find that the software works quite well without the quadratic model. This may be important when the function and derivatives evaluations are not expensive but many individual problems are being solved.
There are two fundamental ways to use the subprogram DQED.
The most straightforward way is to make one call to DQED and obtain values for the unknowns, x. The user provides a subprogram DQEDEV that gives DQED the values of the functions f(x) and g(x), and the derivative or Jacobian matrices for f(x) and g(x) at each desired point x. This usage is called 'forward communication.'
An alternate way to use DQED is to provide an option that allows the user to communicate these values by 'reverse communication.' DQED returns to the user calling program and requests values for f(x) and g(x), and the Jacobian matrices for f(x) and g(x) for a given value of x. This framework is often required in applications that have complicated algorithmic requirements for evaluation of the functions.
Examples using both 'forward' and 'reverse' communication are provided.
The information on this web page is distributed under the MIT license.
dqed is available in a Fortran90 version,
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Richard Hanson and Fred Krogh.