| Optimization Toolbox™ | |
Solves a problem specified by
F(x) = 0
for x, where x is a vector and F(x) is a function that returns a vector value.
x = fsolve(fun,x0)
x = fsolve(fun,x0,options)
x = fsolve(problem)
[x,fval] = fsolve(fun,x0)
[x,fval,exitflag] = fsolve(...)
[x,fval,exitflag,output] = fsolve(...)
[x,fval,exitflag,output,jacobian] = fsolve(...)
fsolve finds a root (zero) of a system of nonlinear equations.
x = fsolve(fun,x0) starts at x0 and tries to solve the equations described in fun.
x = fsolve(fun,x0,options) solves the equations with the optimization options specified in the structure options. Use optimset to set these options.
x = fsolve(problem) solves problem, where problem is a structure described in Input Arguments.
Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB® Workspace.
[x,fval] = fsolve(fun,x0) returns the value of the objective function fun at the solution x.
[x,fval,exitflag] = fsolve(...) returns a value exitflag that describes the exit condition.
[x,fval,exitflag,output] = fsolve(...) returns a structure output that contains information about the optimization.
[x,fval,exitflag,output,jacobian] = fsolve(...) returns the Jacobian of fun at the solution x.
Passing Extra Parameters explains how to parameterize the objective function fun, if necessary.
Function Arguments contains general descriptions of arguments passed into fsolve. This section provides function-specific details for fun and problem:
The nonlinear system of equations to solve. fun is a function that accepts a vector x and returns a vector F, the nonlinear equations evaluated at x. The function fun can be specified as a function handle for an M-file function x = fsolve(@myfun,x0) where myfun is a MATLAB® function such as function F = myfun(x) F = ... % Compute function values at x fun can also be a function handle for an anonymous function. x = fsolve(@(x)sin(x.*x),x0); If the user-defined values for x and F are matrices, they are converted to a vector using linear indexing. If the Jacobian can also be computed and the Jacobian option is 'on', set by options = optimset('Jacobian','on')
then the function fun must return, in a second output argument, the Jacobian value J, a matrix, at x. If fun returns a vector (matrix) of m components and x has length n, where n is the length of x0, then the Jacobian J is an m-by-n matrix where J(i,j) is the partial derivative of F(i) with respect to x(j). (Note that the Jacobian J is the transpose of the gradient of F.) | ||
| problem | objective | Objective function |
x0 | Initial point for x | |
solver | 'fsolve' | |
options | Options structure created with optimset | |
Function Arguments contains general descriptions of arguments returned by fsolve. For more information on the output headings for fsolve, see Function-Specific Output Headings.
This section provides function-specific details for exitflag and output:
exitflag | Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated. | |
1 | Function converged to a solution x. | |
2 | Change in x was smaller than the specified tolerance. | |
3 | Change in the residual was smaller than the specified tolerance. | |
4 | Magnitude of search direction was smaller than the specified tolerance. | |
0 | Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.FunEvals. | |
-1 | Algorithm was terminated by the output function. | |
-2 | Algorithm appears to be converging to a point that is not a root. | |
-3 | Trust radius became too small. | |
-4 | Line search cannot sufficiently decrease the residual along the current search direction. | |
output | Structure containing information about the optimization. The fields of the structure are | |
| iterations | Number of iterations taken | |
| funcCount | Number of function evaluations | |
| algorithm | Optimization algorithm used. | |
| cgiterations | Total number of PCG iterations (large-scale algorithm only) | |
| stepsize | Final displacement in x (Gauss-Newton and Levenberg-Marquardt algorithms) | |
| firstorderopt | Measure of first-order optimality (dogleg or large-scale algorithm, [ ] for others) | |
| message | Exit message | |
Optimization options used by fsolve. Some options apply to all algorithms, some are only relevant when using the large-scale algorithm, and others are only relevant when using the medium-scale algorithm. You can use optimset to set or change the values of these fields in the options structure, options. See Optimization Options for detailed information.
The LargeScale option specifies a preference for which algorithm to use. It is only a preference because certain conditions must be met to use the large-scale algorithm. For fsolve, the nonlinear system of equations cannot be underdetermined; that is, the number of equations (the number of elements of F returned by fun) must be at least as many as the length of x or else the medium-scale algorithm is used:
LargeScale | Use large-scale algorithm if possible when set to 'on'. Use medium-scale algorithm when set to 'off'. The default for fsolve is 'off'. |
These options are used by both the medium-scale and large-scale algorithms:
DerivativeCheck | Compare user-supplied derivatives (Jacobian) to finite-differencing derivatives. |
Diagnostics | Display diagnostic information about the function to be solved. |
DiffMaxChange | Maximum change in variables for finite differencing. |
DiffMinChange | Minimum change in variables for finite differencing. |
Display | Level of display. 'off' displays no output; 'iter' displays output at each iteration; 'final' (default) displays just the final output. |
| FunValCheck | Check whether objective function values are valid. 'on' displays an error when the objective function returns a value that is complex, Inf, or NaN. 'off' (the default) displays no error. |
Jacobian | If 'on', fsolve uses a user-defined Jacobian (defined in fun), or Jacobian information (when using JacobMult), for the objective function. If 'off', fsolve approximates the Jacobian using finite differences. |
MaxFunEvals | Maximum number of function evaluations allowed. |
MaxIter | Maximum number of iterations allowed. |
| OutputFcn | Specify one or more user-defined functions that an optimization function calls at each iteration. See Output Function. |
PlotFcns | Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Specifying @optimplotx plots the current point; @optimplotfunccount plots the function count; @optimplotfval plots the function value; @optimplotresnorm plots the norm of the residuals; @optimplotstepsize plots the step size; @optimplotfirstorderopt plots the first-order of optimality. |
TolFun | Termination tolerance on the function value. |
TolX | Termination tolerance on x. |
TypicalX | Typical x values. |
These options are used only by the large-scale algorithm:
JacobMult | Function handle for Jacobian multiply function. For large-scale structured problems, this function computes the Jacobian matrix product J*Y, J'*Y, or J'*(J*Y) without actually forming J. The function is of the form W = jmfun(Jinfo,Y,flag,p1,p2,...) where Jinfo and the additional parameters p1,p2,... contain the matrices used to compute J*Y (or J'*Y, or J'*(J*Y)). The first argument Jinfo must be the same as the second argument returned by the objective function fun, for example by |
[F,Jinfo] = fun(x) Y is a matrix that has the same number of rows as there are dimensions in the problem. flag determines which product to compute:
| |
See Nonlinear Minimization with a Dense but Structured Hessian and Equality Constraints for a similar example. | |
JacobPattern | Sparsity pattern of the Jacobian for finite differencing. If it is not convenient to compute the Jacobian matrix J in fun, lsqnonlin can approximate J via sparse finite differences provided the structure of J—i.e., locations of the nonzeros—is supplied as the value for JacobPattern. In the worst case, if the structure is unknown, you can set JacobPattern to be a dense matrix and a full finite-difference approximation is computed in each iteration (this is the default if JacobPattern is not set). This can be very expensive for large problems, so it is usually worth the effort to determine the sparsity structure. |
MaxPCGIter | Maximum number of PCG (preconditioned conjugate gradient) iterations (see Algorithm). |
PrecondBandWidth | The default PrecondBandWidth is 'Inf', which means a direct factorization (Cholesky) is used rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set PrecondBandWidth to 0 for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations. |
TolPCG | Termination tolerance on the PCG iteration. |
These options are used only by the medium-scale algorithm:
NonlEqnAlgorithm | Specify one of the following algorithms for solving nonlinear equations:
|
LineSearchType | Line search algorithm choice. This option applies to the 'lm' (Levenberg-Marquardt) and 'gn' (Gauss-Netwton) algorithms. |
This example finds a zero of the system of two equations and two unknowns:
You want to solve the following system for x
starting at x0 = [-5 -5].
First, write an M-file that computes F, the values of the equations at x.
function F = myfun(x)
F = [2*x(1) - x(2) - exp(-x(1));
-x(1) + 2*x(2) - exp(-x(2))];
Next, call an optimization routine.
x0 = [-5; -5]; % Make a starting guess at the solution
options=optimset('Display','iter'); % Option to display output
[x,fval] = fsolve(@myfun,x0,options) % Call optimizer
After 33 function evaluations, a zero is found.
Norm of First-order Trust-region
Iteration Func-count f(x) step optimality radius
0 3 23535.6 2.29e+004 1
1 6 6001.72 1 5.75e+003 1
2 9 1573.51 1 1.47e+003 1
3 12 427.226 1 388 1
4 15 119.763 1 107 1
5 18 33.5206 1 30.8 1
6 21 8.35208 1 9.05 1
7 24 1.21394 1 2.26 1
8 27 0.016329 0.759511 0.206 2.5
9 30 3.51575e-006 0.111927 0.00294 2.5
10 33 1.64763e-013 0.00169132 6.36e-007 2.5
Optimization terminated successfully:
First-order optimality is less than options.TolFun
x =
0.5671
0.5671
fval =
1.0e-006 *
-0.4059
-0.4059
Find a matrix x that satisfies the equation
starting at the point x= [1,1; 1,1].
First, write an M-file that computes the equations to be solved.
function F = myfun(x) F = x*x*x-[1,2;3,4];
Next, invoke an optimization routine.
x0 = ones(2,2); % Make a starting guess at the solution
options = optimset('Display','off'); % Turn off Display
[x,Fval,exitflag] = fsolve(@myfun,x0,options)
The solution is
x =
-0.1291 0.8602
1.2903 1.1612
Fval =
1.0e-009 *
-0.1619 0.0776
0.1161 -0.0469
exitflag =
1
and the residual is close to zero.
sum(sum(Fval.*Fval))
ans =
4.7915e-020If the system of equations is linear, use\ (matrix left division) for better speed and accuracy. For example, to find the solution to the following linear system of equations:
3x1 + 11x2 – 2x3 = 7
x1 + x2 –
2x3 = 4
x1 – x2 + x3 = 19.
You can formulate and solve the problem as
A = [ 3 11 -2; 1 1 -2; 1 -1 1];
b = [ 7; 4; 19];
x = A\b
x =
13.2188
-2.3438
3.4375
The Gauss-Newton, Levenberg-Marquardt, and large-scale methods are based on the nonlinear least-squares algorithms also used in lsqnonlin. Use one of these methods if the system may not have a zero. The algorithm still returns a point where the residual is small. However, if the Jacobian of the system is singular, the algorithm might converge to a point that is not a solution of the system of equations (see Limitations and Diagnostics following).
fsolve, with the LargeScale option set to 'on' with optimset, uses the large-scale algorithm if possible. This algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See Trust-Region Methods for Nonlinear Minimization and Preconditioned Conjugate Gradients.
By default fsolve chooses the medium-scale algorithm and uses the trust-region dogleg method. The algorithm is a variant of the Powell dogleg method described in [8]. It is similar in nature to the algorithm implemented in [7].
Alternatively, you can select a Gauss-Newton method [3] with line-search, or a Levenberg-Marquardt method [4], [5], and [6] with line-search. Use optimset to set NonlEqnAlgorithm option to 'dogleg' (default), 'lm', or 'gn'.
The default line search algorithm for the Levenberg-Marquardt and Gauss-Newton methods, i.e., the LineSearchType option, is 'quadcubic'. This is a safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation method. A safeguarded cubic polynomial method can be selected by setting LineSearchType to 'cubicpoly'. This method generally requires fewer function evaluations but more gradient evaluations. Thus, if gradients are being supplied and can be calculated inexpensively, the cubic polynomial line search method is preferable. The algorithms used are described fully in Standard Algorithms.
fsolve may converge to a nonzero point and give this message:
Optimizer is stuck at a minimum that is not a root Try again with a new starting guess
In this case, run fsolve again with other starting values.
For the trust-region dogleg method, fsolve stops if the step size becomes too small and it can make no more progress. fsolve gives this message:
The optimization algorithm can make no further progress: Trust region radius less than 10*eps
In this case, run fsolve again with other starting values.
The function to be solved must be continuous. When successful, fsolve only gives one root. fsolve may converge to a nonzero point, in which case, try other starting values.
fsolve only handles real variables. When x has complex variables, the variables must be split into real and imaginary parts.
The preconditioner computation used in the preconditioned conjugate gradient part of the large-scale method forms JTJ (where J is the Jacobian matrix) before computing the preconditioner; therefore, a row of J with many nonzeros, which results in a nearly dense product JTJ, might lead to a costly solution process for large problems.
The default trust-region dogleg method can only be used when the system of equations is square, i.e., the number of equations equals the number of unknowns. For the Levenberg-Marquardt and Gauss-Newton methods, the system of equations need not be square.
[1] Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds," SIAM Journal on Optimization, Vol. 6, pp. 418-445, 1996.
[2] Coleman, T.F. and Y. Li, "On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds," Mathematical Programming, Vol. 67, Number 2, pp. 189-224, 1994.
[3] Dennis, J. E. Jr., "Nonlinear Least-Squares," State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269-312.
[4] Levenberg, K., "A Method for the Solution of Certain Problems in Least-Squares," Quarterly Applied Mathematics 2, pp. 164-168, 1944.
[5] Marquardt, D., "An Algorithm for Least-squares Estimation of Nonlinear Parameters," SIAM Journal Applied Mathematics, Vol. 11, pp. 431-441, 1963.
[6] Moré, J. J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977.
[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom, User Guide for MINPACK 1, Argonne National Laboratory, Rept. ANL-80-74, 1980.
[8] Powell, M. J. D., "A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations," Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7, 1970.
@ (function_handle), \ (matrix left division), lsqcurvefit, lsqnonlin, optimset, optimtool, anonymous functions
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