This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with n equations in n
unknowns). The library provides low level components for a variety of
iterative solvers and convergence tests. These can be combined by the
user to achieve the desired solution, with full access to the
intermediate steps of the iteration. Each class of methods uses the
same framework, so that you can switch between solvers at runtime
without needing to recompile your program. Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs. The solvers are based on the original Fortran
library MINPACK.
The header file `gsl_multiroots.h' contains prototypes for the
multidimensional root finding functions and related declarations.
The problem of multidimensional root finding requires the simultaneous
solution of n equations, f_i, in n variables,
x_i,
f_i (x_1, ..., x_n) = 0 for i = 1 ... n.
In general there are no bracketing methods available for n
dimensional systems, and no way of knowing whether any solutions
exist. All algorithms proceed from an initial guess using a variant of
the Newton iteration,
x -> x' = x - J^{-1} f(x)
where x, f are vector quantities and J is the
Jacobian matrix
J_{ij} = d f_i / d x_j.
Additional strategies can be used to enlarge the region of
convergence. These include requiring a decrease in the norm |f| on
each step proposed by Newton's method, or taking steepest-descent steps in
the direction of the negative gradient of |f|.
Several root-finding algorithms are available within a single framework.
The user provides a high-level driver for the algorithms, and the
library provides the individual functions necessary for each of the
steps. There are three main phases of the iteration. The steps are,
initialize solver state, s, for algorithm T
update s using the iteration T
test s for convergence, and repeat iteration if necessary
The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
n^2 terms of the matrix becomes too expensive. For these reasons
the algorithms provided by the library are divided into two classes according
to whether the derivatives are available or not.
The state for solvers with an analytic Jacobian matrix is held in a
gsl_multiroot_fdfsolver struct. The updating procedure requires
both the function and its derivatives to be supplied by the user.
The state for solvers which do not use an analytic Jacobian matrix is
held in a gsl_multiroot_fsolver struct. The updating procedure
uses only function evaluations (not derivatives). The algorithms
estimate the matrix J or
J^{-1} by approximate methods.
The following functions initialize a multidimensional solver, either
with or without derivatives. The solver itself depends only on the
dimension of the problem and the algorithm and can be reused for
different problems.
This function returns a pointer to a a newly allocated instance of a
solver of type T for a system of n dimensions.
For example, the following code creates an instance of a hybrid solver,
to solve a 3-dimensional system of equations.
const gsl_multiroot_fsolver_type * T
= gsl_multiroot_fsolver_hybrid;
gsl_multiroot_fsolver * s
= gsl_multiroot_fsolver_alloc (T, 3);
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
This function returns a pointer to a a newly allocated instance of a
derivative solver of type T for a system of n dimensions.
For example, the following code creates an instance of a Newton-Raphson solver,
for a 2-dimensional system of equations.
const gsl_multiroot_fdfsolver_type * T
= gsl_multiroot_fdfsolver_newton;
gsl_multiroot_fdfsolver * s =
gsl_multiroot_fdfsolver_alloc (T, 2);
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
You must provide n functions of n variables for the root
finders to operate on. In order to allow for general parameters the
functions are defined by the following data types:
Data Type:gsl_multiroot_function
This data type defines a general system of functions with parameters.
this function should store the vector result
f(x,params) in f for argument x and parameters params,
returning an appropriate error code if the function cannot be computed.
size_t n
the dimension of the system, i.e. the number of components of the
vectors x and f.
this function should store the vector result
f(x,params) in f for argument x and parameters params,
returning an appropriate error code if the function cannot be computed.
this function should store the n-by-n matrix result
J_ij = d f_i(x,params) / d x_j in J for argument x
and parameters params, returning an appropriate error code if the
function cannot be computed.
This function should set the values of the f and J as above,
for arguments x and parameters params. This function provides
an optimization of the separate functions for f(x) and J(x) --
it is always faster to compute the function and its derivative at the
same time.
size_t n
the dimension of the system, i.e. the number of components of the
vectors x and f.
void * params
a pointer to the parameters of the function.
The example of Powell's test function defined above can be extended to
include analytic derivatives using the following code,
The following functions drive the iteration of each algorithm. Each
function performs one iteration to update the state of any solver of the
corresponding type. The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.
Function: int gsl_multiroot_fsolver_iterate(gsl_multiroot_fsolver * s)
Function: int gsl_multiroot_fdfsolver_iterate(gsl_multiroot_fdfsolver * s)
These functions perform a single iteration of the solver s. If the
iteration encounters an unexpected problem then an error code will be
returned,
GSL_EBADFUNC
the iteration encountered a singular point where the function or its
derivative evaluated to Inf or NaN.
GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from
continuing.
The solver maintains a current best estimate of the root at all times.
This information can be accessed with the following auxiliary functions,
A root finding procedure should stop when one of the following conditions is
true:
A multidimensional root has been found to within the user-specified precision.
A user-specified maximum number of iterations has been reached.
An error has occurred.
The handling of these conditions is under user control. The functions
below allow the user to test the precision of the current result in
several standard ways.
This function tests for the convergence of the sequence by comparing the
last step dx with the absolute error epsabs and relative
error epsrel to the current position x. The test returns
GSL_SUCCESS if the following condition is achieved,
|dx_i| < epsabs + epsrel |x_i|
for each component of x and returns GSL_CONTINUE otherwise.
Function: int gsl_multiroot_test_residual(const gsl_vector * f, double epsabs)
This function tests the residual value f against the absolute
error bound epsabs. The test returns GSL_SUCCESS if the
following condition is achieved,
\sum_i |f_i| < epsabs
and returns GSL_CONTINUE otherwise. This criterion is suitable
for situations where the the precise location of the root, x, is
unimportant provided a value can be found where the residual is small
enough.
The root finding algorithms described in this section make use of both
the function and its derivative. They require an initial guess for the
location of the root, but there is no absolute guarantee of convergence
-- the function must be suitable for this technique and the initial
guess must be sufficiently close to the root for it to work. When the
conditions are satisfied then convergence is quadratic.
This is a modified version of Powell's Hybrid method as implemented in
the HYBRJ algorithm in MINPACK. Minpack was written by Jorge
J. Mor'e, Burton S. Garbow and Kenneth E. Hillstrom. The Hybrid
algorithm retains the fast convergence of Newton's method but will also
reduce the residual when Newton's method is unreliable.
The algorithm uses a generalized trust region to keep each step under
control. In order to be accepted a proposed new position x' must
satisfy the condition |D (x' - x)| < \delta, where D is a
diagonal scaling matrix and \delta is the size of the trust
region. The components of D are computed internally, using the
column norms of the Jacobian to estimate the sensitivity of the residual
to each component of x. This improves the behavior of the
algorithm for badly scaled functions.
On each iteration the algorithm first determines the standard Newton
step by solving the system J dx = - f. If this step falls inside
the trust region it is used as a trial step in the next stage. If not,
the algorithm uses the linear combination of the Newton and gradient
directions which is predicted to minimize the norm of the function while
staying inside the trust region.
dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2
This combination of Newton and gradient directions is referred to as a
dogleg step.
The proposed step is now tested by evaluating the function at the
resulting point, x'. If the step reduces the norm of the function
sufficiently then it is accepted and size of the trust region is
increased. If the proposed step fails to improve the solution then the
size of the trust region is decreased and another trial step is
computed.
The speed of the algorithm is increased by computing the changes to the
Jacobian approximately, using a rank-1 update. If two successive
attempts fail to reduce the residual then the full Jacobian is
recomputed. The algorithm also monitors the progress of the solution
and returns an error if several steps fail to make any improvement,
GSL_ENOPROG
the iteration is not making any progress, preventing the algorithm from
continuing.
GSL_ENOPROGJ
re-evaluations of the Jacobian indicate that the iteration is not
making any progress, preventing the algorithm from continuing.
Derivative Solver:gsl_multiroot_fdfsolver_hybridj
This algorithm is an unscaled version of hybridsj. The steps are
controlled by a spherical trust region |x' - x| < \delta, instead
of a generalized region. This can be useful if the generalized region
estimated by hybridsj is inappropriate.
Derivative Solver:gsl_multiroot_fdfsolver_newton
Newton's Method is the standard root-polishing algorithm. The algorithm
begins with an initial guess for the location of the solution. On each
iteration a linear approximation to the function F is used to
estimate the step which will zero all the components of the residual.
The iteration is defined by the following sequence,
x -> x' = x - J^{-1} f(x)
where the Jacobian matrix J is computed from the derivative
functions provided by f. The step dx is obtained by solving
the linear system,
J dx = - f(x)
using LU decomposition.
Derivative Solver:gsl_multiroot_fdfsolver_gnewton
This is a modified version of Newton's method which attempts to improve
global convergence by requiring every step to reduce the Euclidean norm
of the residual, |f(x)|. If the Newton step leads to an increase
in the norm then a reduced step of relative size,
t = (\sqrt(1 + 6 r) - 1) / (3 r)
is proposed, with r being the ratio of norms
|f(x')|^2/|f(x)|^2. This procedure is repeated until a suitable step
size is found.
The algorithms described in this section do not require any derivative
information to be supplied by the user. Any derivatives needed are
approximated from by finite difference.
Solver:gsl_multiroot_fsolver_hybrids
This is a version of the Hybrid algorithm which replaces calls to the
Jacobian function by its finite difference approximation. The finite
difference approximation is computed using gsl_multiroots_fdjac
with a relative step size of GSL_SQRT_DBL_EPSILON.
Solver:gsl_multiroot_fsolver_hybrid
This is a finite difference version of the Hybrid algorithm without
internal scaling.
Solver:gsl_multiroot_fsolver_dnewton
The discrete Newton algorithm is the simplest method of solving a
multidimensional system. It uses the Newton iteration
x -> x - J^{-1} f(x)
where the Jacobian matrix J is approximated by taking finite
differences of the function f. The approximation scheme used by
this implementation is,
J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j
where \delta_j is a step of size \sqrt\epsilon |x_j| with
\epsilon being the machine precision
(
\epsilon \approx 2.22 \times 10^-16).
The order of convergence of Newton's algorithm is quadratic, but the
finite differences require n^2 function evaluations on each
iteration. The algorithm may become unstable if the finite differences
are not a good approximation to the true derivatives.
Solver:gsl_multiroot_fsolver_broyden
The Broyden algorithm is a version of the discrete Newton
algorithm which attempts to avoids the expensive update of the Jacobian
matrix on each iteration. The changes to the Jacobian are also
approximated, using a rank-1 update,
where the vectors dx and df are the changes in x
and f. On the first iteration the inverse Jacobian is estimated
using finite differences, as in the discrete Newton algorithm.
This approximation gives a fast update but is unreliable if the changes
are not small, and the estimate of the inverse Jacobian becomes worse as
time passes. The algorithm has a tendency to become unstable unless it
starts close to the root. The Jacobian is refreshed if this instability
is detected (consult the source for details).
This algorithm is not recommended and is included only for demonstration
purposes.
The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms. This first example
demonstrates the hybrids scaled-hybrid algorithm, which does not
require derivatives. The program solves the Rosenbrock system of equations,
f_1 (x, y) = a (1 - x)
f_2 (x, y) = b (y - x^2)
with a = 1, b = 10. The solution of this system lies at
(x,y) = (1,1) in a narrow valley.
The first stage of the program is to define the system of equations,
The main program begins by creating the function object f, with
the arguments (x,y) and parameters (a,b). The solver
s is initialized to use this function, with the hybrids
method.
int
main (void)
{
const gsl_multiroot_fsolver_type *T;
gsl_multiroot_fsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function f = {&rosenbrock_f, n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fsolver_hybrids;
s = gsl_multiroot_fsolver_alloc (T, 2);
gsl_multiroot_fsolver_set (s, &f, x);
print_state (iter, s);
do
{
iter++;
status = gsl_multiroot_fsolver_iterate (s);
print_state (iter, s);
if (status) /* check if solver is stuck */
break;
status =
gsl_multiroot_test_residual (s->f, 1e-7);
}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fsolver_free (s);
gsl_vector_free (x);
return 0;
}
Note that it is important to check the return status of each solver
step, in case the algorithm becomes stuck. If an error condition is
detected, indicating that the algorithm cannot proceed, then the error
can be reported to the user, a new starting point chosen or a different
algorithm used.
The intermediate state of the solution is displayed by the following
function. The solver state contains the vector s->x which is the
current position, and the vector s->f with corresponding function
values.
Here are the results of running the program. The algorithm is started at
(-10,-5) far from the solution. Since the solution is hidden in
a narrow valley the earliest steps follow the gradient of the function
downhill, in an attempt to reduce the large value of the residual. Once
the root has been approximately located, on iteration 8, the Newton
behavior takes over and convergence is very rapid.
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 1 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 2 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01
iter = 3 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01
iter = 4 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01
iter = 5 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01
iter = 6 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01
iter = 7 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00
iter = 8 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00
iter = 9 x = 1.000 0.878 f(x) = 1.268e-10 -1.218e+00
iter = 10 x = 1.000 0.989 f(x) = 1.124e-11 -1.080e-01
iter = 11 x = 1.000 1.000 f(x) = 0.000e+00 0.000e+00
status = success
Note that the algorithm does not update the location on every
iteration. Some iterations are used to adjust the trust-region
parameter, after trying a step which was found to be divergent, or to
recompute the Jacobian, when poor convergence behavior is detected.
The next example program adds derivative information, in order to
accelerate the solution. There are two derivative functions
rosenbrock_df and rosenbrock_fdf. The latter computes both
the function and its derivative simultaneously. This allows the
optimization of any common terms. For simplicity we substitute calls to
the separate f and df functions at this point in the code
below.
The main program now makes calls to the corresponding fdfsolver
versions of the functions,
int
main (void)
{
const gsl_multiroot_fdfsolver_type *T;
gsl_multiroot_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function_fdf f = {&rosenbrock_f,
&rosenbrock_df,
&rosenbrock_fdf,
n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fdfsolver_gnewton;
s = gsl_multiroot_fdfsolver_alloc (T, n);
gsl_multiroot_fdfsolver_set (s, &f, x);
print_state (iter, s);
do
{
iter++;
status = gsl_multiroot_fdfsolver_iterate (s);
print_state (iter, s);
if (status)
break;
status = gsl_multiroot_test_residual (s->f, 1e-7);
}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fdfsolver_free (s);
gsl_vector_free (x);
return 0;
}
The addition of derivative information to the hybrids solver does
not make any significant difference to its behavior, since it able to
approximate the Jacobian numerically with sufficient accuracy. To
illustrate the behavior of a different derivative solver we switch to
gnewton. This is a traditional newton solver with the constraint
that it scales back its step if the full step would lead "uphill". Here
is the output for the gnewton algorithm,
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 1 x = -4.231 -65.317 f(x) = 5.231e+00 -8.321e+02
iter = 2 x = 1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
iter = 3 x = 1.000 1.000 f(x) = -2.220e-16 -4.441e-15
status = success
The convergence is much more rapid, but takes a wide excursion out to
the point (-4.23,-65.3). This could cause the algorithm to go
astray in a realistic application. The hybrid algorithm follows the
downhill path to the solution more reliably.
The original version of the Hybrid method is described in the following
articles by Powell,
M.J.D. Powell, "A Hybrid Method for Nonlinear Equations" (Chap 6, p
87-114) and "A Fortran Subroutine for Solving systems of Nonlinear
Algebraic Equations" (Chap 7, p 115-161), in Numerical Methods for
Nonlinear Algebraic Equations, P. Rabinowitz, editor. Gordon and
Breach, 1970.
The following papers are also relevant to the algorithms described in
this section,
J.J. Mor'e, M.Y. Cosnard, "Numerical Solution of Nonlinear Equations",
ACM Transactions on Mathematical Software, Vol 5, No 1, (1979), p 64-85
C.G. Broyden, "A Class of Methods for Solving Nonlinear
Simultaneous Equations", Mathematics of Computation, Vol 19 (1965),
p 577-593
J.J. Mor'e, B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
Optimization Software", ACM Transactions on Mathematical Software, Vol
7, No 1 (1981), p 17-41