pdf_MF_M2S.m

function [s FVAL NITER]=pdf_MF_M2S(d,s0)
%pdf_MF_M2S: transforms the first moments into the proper singular values
%   s=pdf_MF_M2S(d,s0) numerically solves the following equations for s
%
%       \frac{1}{c(S)}\frac{\partial c(S)}{s_i} - d_i = 0,
%
%   to find the proper singular values of the matrix Fisher distribution
%   whose first moments match with d. It is solved via Newton-Armijo
%   iteration with a polynomial line search, and the optional input variable 
%   s0 specifis the initial guess of the iteration.
%
%   [s, FVAL, niter]=pdf_MF_M2S(d,s0) returns the value of the equation at
%   s, and the number of iterations
%
%   See T. Lee, "Bayesian Attitude Estimation with the Matrix Fisher
%   Distribution on SO(3)", 2017, http://arxiv.org/abs/1710.03746
%   C. T. Kelly, "Iterative Methods For Linear And Nonlinear Equations,"
%   SIAM, 1995, Section 8.3.1.
%
%   See also PDF_MF_M2S_APPROX

bool_scaled=1;
if nargin < 2
    if max(abs(d)) < 0.5
        s0 = pdf_MF_M2S_approx(d,0);
    elseif min(abs(d)) > 0.5
        s0 = pdf_MF_M2S_approx(d,1);
        if min(abs(d)) > 0.99
            s = pdf_MF_M2S_approx(d,1);
            return;
        end
    else
        s0=rand(3,1).*sign(d);
    end
end

s=s0;
eps=1e-12;
alpha=0.05;
nf=2*eps;

NITER=0;
MAX_ITER=100;

sigma0=0.1;
sigma1=0.5;

while nf > eps && NITER < MAX_ITER
    
    NITER=NITER+1;

    f_stack=[];
    lambda_stack=[];

    [f, df]=func(s,d,1,bool_scaled);        
    s_dir=-df\f;    
    f_stack=stack3(f_stack,f);
    lambda_stack=stack3(lambda_stack,0);
    
    lambda=1;        
    s_trial=s+lambda*s_dir;
    f_trial=func(s_trial,d,0,bool_scaled);
    f_stack=stack3(f_stack,f_trial);
    lambda_stack=stack3(lambda_stack,lambda);
    
    % polynomial line search: three-point parabolic method
    N_SUBITER=0;
    
    while norm(f_stack(end)) > (1-alpha*lambda)*norm(f) && N_SUBITER < MAX_ITER 
        
        N_SUBITER=N_SUBITER+1;
        if length(lambda_stack) < 3
            lambda=sigma1;
        else
            lam2=lambda_stack(2);
            lam3=lambda_stack(3);
            f2=f_stack(2);
            f3=f_stack(3);
            poly_coff_2=1/lam2/lam3/(lam2-lam3)*(lam3*(norm(f2)-norm(f))-lam2*(norm(f3)-norm(f)));
            if poly_coff_2 > 0
                poly_coff_1=1/(lam2-lam3)*(-lam3*(norm(f2)-norm(f))/lam2+lam2*(norm(f3)-norm(f))/lam3);
                lambda_t=-poly_coff_1/2/poly_coff_2;
                lambda=saturation(lambda_t,sigma0*lam3,sigma1*lam3)              
            else
                lambda=sigma1*lam3;
            end

        end
        
        s_trial=s+lambda*s_dir;
        f_trial=func(s_trial,d,0,bool_scaled);
        f_stack=stack3(f_stack,f_trial);
        lambda_stack=stack3(lambda_stack,lambda);
   end
    
    s=s_trial;
    nf=max(abs(f_trial));    
end

FVAL=f_trial;

if NITER == MAX_ITER || N_SUBITER == MAX_ITER
    disp('Warning: MAX iteration reached');
    disp('diag(D) = ');
    disp(d);
    disp('diag(S) = ');
    disp(s);
end

end

function Y=stack3(X,x)

Y=[X x];
n=size(Y,2);
if n > 3
    Y=Y(:,n-2:n);
end

end

function y=saturation(x,min_x,max_x)

if x <= min_x
    y=min_x;
elseif x >= max_x
    y=max_x;
else
    y=x;
end    
end


function varargout=func(s,d,bool_df,bool_scaled)
if ~bool_scaled
    c=pdf_MF_normal(s,0);
    if ~bool_df
        dc=pdf_MF_normal_deriv(s,0,0);
        f=dc-c*d;
        varargout{1}=f;
    else
        [dc, ddc]=pdf_MF_normal_deriv(s,1,0);
        f=dc-c*d;
        df=ddc-d*dc';
        varargout{1}=f;
        varargout{2}=df;
    end    
else
    c_bar=pdf_MF_normal(s,1);
    if ~bool_df
        dc_bar=pdf_MF_normal_deriv(s,0,1);
        f=dc_bar/c_bar-(d-ones(3,1));
        varargout{1}=f;
    else
        [dc_bar, ddc_bar]=pdf_MF_normal_deriv(s,1,1);
        %f=dc_bar-c_bar*(d-ones(3,1));
        %df=ddc_bar-(d-ones(3,1))*dc_bar';        
        f=dc_bar/c_bar-(d-ones(3,1));
        df=ddc_bar/c_bar-dc_bar*dc_bar'/c_bar^2;
        varargout{1}=f;
        varargout{2}=df;
    end
end
end