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function [X, info, perf] = nonlinsys(fun, x0, opts, B0, varargin)
%NONLINSYS Powell's dog-leg method for non-linear system of equations.
% Find an n-vector xm, such that f_i(xm) = 0 , i=1,...,n.
% The vector function f(x) (with components f_i, i=1,...,n) must be
% given by a MATLAB function with declaration
% function f = fun(x, p1,p2,...)
% % % p1,p2,... are parameters of the function.
%
% Call
% [X, info] = nonlinsys(fun, x0)
% [X, info] = nonlinsys(fun, x0, opts)
% [X, info] = nonlinsys(fun, x0, opts, B0, p1,p2,...)
% [X, info, perf] = nonlinsys(.....)
%
% Input parameters
% fun : Handle to the function.
% x0 : Starting guess for xm .
% opts : Vector with five elements:
% opts(1) initial trust region radius Delta
% opts(2:4) used in stopping criteria:
% ||f||inf <= opts(2) or
% ||dx||2 <= opts(3)*(opts(3) + ||x||2) or
% no. of iteration steps exceeds opts(4) .
% opts(5) "relative" step length for difference approximations.
% Default opts = [0.1(1+||x0||) 1e-6 1e-8 100 1e-6]
% If the input opts has less than 5 elements, it is
% augmented by the default values.
% B0 : Initial approximation to the Jacobian of f.
% If B0 is not given, a forward difference approximation
% to it is used.
% p1,p2,.. are passed dirctly to the function FUN .
%
% Output parameters
% X : If perf is present, then array, holding the iterates
% columnwise. Otherwise, computed solution vector.
% info : Performance information, vector with 6 elements:
% info(1:3) = final values of
% [||f(x)||inf ||dx||2 Delta]
% info(4:5) = no. of iteration steps and function avaluations
% info(6) = 1 : Stopped by small f-vector
% 2 : Stopped by small x-step
% 3 : No. of iteration steps exceeded
% -1 : x is not a real valued vector
% -2 : f is not a real valued column vector
% -3 : Dimension mismatch in x, f, B0
% -4 : Maybe started at a saddle point
% -5 : Overflow during computation
% perf : Array, holding
% perf(1,:) = values of || f(x) ||inf
% perf(2,:) = Delta-values.
% Version 04.04.10. hbn(a)imm.dtu.dk
% Check parameters and function call
if nargin < 2
[stop,x,f,opts,B,D,info,n] = checkinput(fun, [], [], []);
elseif nargin < 3
[stop,x,f,opts,B,D,info,n] = checkinput(fun, x0, [], []);
elseif nargin < 4
[stop,x,f,opts,B,D,info,n] = checkinput(fun, x0, opts, []);
else
[stop,x,f,opts,B,D,info,n] = checkinput(fun, x0, opts, B0, varargin{:});
end
if stop
perf = []; X = x; return
end
% Finish initialization
Delta = opts(1); kmax = opts(4);
Trace = nargout > 2;
if Trace
X = repmat(x,1,kmax+1);
perf = repmat([norm(f,inf); Delta],1,kmax+1);
end
k = 1; nx = norm(x); nf = norm(f,inf); F = .5*norm(f)^2;
ku = 0; refac = 0; % For "extra" updates
newD = 0; % Signifies whether D should be recomputed
nstep = 0;
% Iterate
while ~stop
if isinf(nf), stop = -5;
elseif nf <= opts(2), stop = 1;
elseif Delta <= opts(3)*(opts(3) + nx), stop = 2;
elseif k >= kmax, stop = 3;
else
% Newton step
if newD, [stop D hN] = getDandh(B,f);
else, hN = D*(-f); end
nhN = norm(hN);
if nhN <= opts(3)*(opts(3) + nx), stop = 2;
elseif ~stop
if nhN > Delta % Include gradient
g = B'*(-f); ng = norm(g); alpha = (ng / norm(B*g))^2;
gn = alpha*g; ngn = alpha*ng;
if ngn >= Delta
h = (Delta/ngn) * gn;
else % Dog leg
b = hN - gn; bb = b'*b; gb = gn'*b;
c = (Delta + ngn)*(Delta - ngn);
if gb > 0
beta = c / (gb + sqrt(gb^2 + c * bb));
else
beta = (sqrt(gb^2 + c * bb) - gb)/bb;
end
h = gn + beta*b;
end
else
h = hN;
end
end
end
if ~stop
ku = mod(ku,n) + 1;
if abs(h(ku)) < .8*norm(h) % extra step for updating B and D
xu = x;
if x(ku) == 0, xu(ku) = opts(5)^2;
else, xu(ku) = x(ku) + opts(5)*abs(x(ku)); end
[stop,B,D,Fu,fu,hu,nhu,newD] = updBD(fun,x,xu,f,B,D,0,varargin{:});
info(5) = info(5)+1;
end
if ~stop
xnew = x + h;
[stop,B,D,Fn,fn,h,nstep,newD] = updBD(fun,x,xnew,f,B,D,newD,varargin{:});
info(5) = info(5)+1;
if ~stop
% Check trust region
dF = F - Fn;
if dF > 0 % Update x
dL = F - .5*norm(f + B*h)^2;
x = xnew; F = Fn; f = fn;
nf = norm(f,inf); nx = norm(x);
if dL > 0, rho = dF / dL; else, rho = 0; end
else, rho = 0; end
if rho > .75, Delta = max(Delta, 3*nstep);
elseif rho < .25, Delta = Delta/2; end
k = k + 1;
if Trace, X(:,k) = x; perf(:,k) = [nf Delta]'; end
end
end
end
end
% Set return values
if Trace
X = X(:,1:k); perf = perf(:,1:k);
else, X = x; end
if stop < 0, nf = NaN; end
info = [nf nstep Delta k-1 info(5) stop];
%%%%%%%%%%%%%%%%%%%% Auxiliary functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [stop,x,f,opts,B,D,info,n] = checkinput(fun, x0, opts, B0, varargin)
% Check indata
info = [NaN zeros(1,5)]; f = NaN; B = NaN; D = NaN; n = NaN;
stop = 0;
if isempty(x0), stop = -1; x = [];
else
[stop x n] = checkx(x0);
if ~stop
[stop F f] = checkfJ(fun,x0,varargin{:});
info([1 5]) = [norm(f,inf) 1];
nf = norm(f, inf);
if ~stop & length(f) ~= n, stop = -3; end
if ~stop
% Finish initialization
opts = checkopts(opts, [.1*(1+norm(x,inf)) 1e-6 1e-8 100 1e-6]);
% Jacobian
sB = size(B0);
if sum(sB) == 0 % placeholder
[stop B] = Dapprox(fun,x,opts(5),f,varargin{:});
info(5) = info(5) + n;
elseif any(sB ~= n), stop = -3;
else, B = B0; end
% Check gradient
if ~stop
g = B'*(-f); ng = norm(g,inf);
if isinf(ng), stop = -5; end
end
% Get initial inverse Jacobian and check D*f
if ~stop
[stop D hN] = getDandh(B,f);
end
end
end
end
if stop, info(6) = stop; end
function [stop, D, hN] = getDandh(B,f)
% Get inverse of approximate Jacobian, and check for overflow
[U S V] = svd(B);
s = diag(S); i = find(s > 100*eps*s(1));
if isempty(i), stop = -4; D = NaN; hN = NaN;
else
D = V(:,i) * diag(1./s(i)) * U(:,i)';
hN = D*(-f);
if isinf(norm(hN)), stop = -5; else, stop = 0; end
end
function [stop,B,D,Fn,fn,h,nh,newD] = updBD(fun,x,xn,f,B,D,newD,varargin)
% Evaluate at new point and update B and D
[stop Fn fn] = checkfJ(fun,xn,varargin{:});
if ~stop
h = xn - x; nh = norm(h); y = fn - f;
B = B + ((y - B*h)/nh^2) * h';
if ~newD
Dy = D*y; hDy = dot(h,Dy);
if abs(hDy) < sqrt(eps)*nh, newD = 1;
else
D = D + ((h - Dy)/hDy)*(h'*D);
end
end
end