[Master Index] [Index for PublicToolbox/regutools]

# parallax

## (PublicToolbox/regutools/parallax.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    [A,b] = parallax(n)

### Help Text

    PARALLAX Stellar parallax problem with 28 fixed, real observations.

     [A,b] = parallax(n)

     Stellar parallax problem with 28 fixed, real observations.

     The underlying problem is a Fredholm integral equation of the
     first kind with kernel
        K(s,t) = (1/sigma*sqrt(2*pi))*esp(-0.5*((s-t)/sigma)^2) ,
     and it is discretized by means of a Galerkin method with n
     orthonormal basis functions.  The right-hand side consists of
     a measured distribution function of stellar parallaxes, and its
     length is fixed, m = 26.  The exact solution, which represents
     the true distribution of stellar parallaxes, in not known.

### Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\parallax.m

    function [A,b] = parallax(n)
    %PARALLAX Stellar parallax problem with 28 fixed, real observations.
    %
    % [A,b] = parallax(n)
    %
    % Stellar parallax problem with 28 fixed, real observations.
    %
    % The underlying problem is a Fredholm integral equation of the
    % first kind with kernel
    %    K(s,t) = (1/sigma*sqrt(2*pi))*esp(-0.5*((s-t)/sigma)^2) ,
    % and it is discretized by means of a Galerkin method with n
    % orthonormal basis functions.  The right-hand side consists of
    % a measured distribution function of stellar parallaxes, and its
    % length is fixed, m = 26.  The exact solution, which represents
    % the true distribution of stellar parallaxes, in not known.

    % Reference: W. M. Smart, "Stellar Dynamics", Cambridge
    % University Press, 1938; p. 30.

    % Discretized by Galerkin method with orthonormal box functions;
    % 2-D integration is done by means of the computational molecule:
    %       1   4   1
    %       4  16   1
    %       1   4   1

    % Per Christian Hansen, UNI-C, 09/16/92.

    % Initialization.
    a = 0; b = 0.1; m = 26; sigma = 0.014234;
    hs = 0.130/m; hx = (b-a)/n; hsh = hs/2; hxh = hx/2;
    ss = (-0.03 + [0:m-1]'*hs)*ones(1,n);
    xx = ones(m,1)*(a + [0:n-1]*hx);

    % Set up the matrix.
    A =     16*exp(-0.5*((ss+hsh - xx-hxh)/sigma).^2);
    A = A + 4*(exp(-0.5*((ss+hsh - xx    )/sigma).^2) + ...
               exp(-0.5*((ss+hsh - xx-hx )/sigma).^2) + ...
               exp(-0.5*((ss     - xx-hxh)/sigma).^2) + ...
               exp(-0.5*((ss+hs  - xx-hxh)/sigma).^2));
    A = A +   (exp(-0.5*((ss     - xx    )/sigma).^2) + ...
               exp(-0.5*((ss+hs  - xx    )/sigma).^2) + ...
               exp(-0.5*((ss     - xx-hx )/sigma).^2) + ...
               exp(-0.5*((ss+hs  - xx-hx )/sigma).^2));
    A = sqrt(hs*hx)/(36*sigma*sqrt(2*pi))*A;

    % Set up the normalized right-hand side.
    b = [3;7;7;17;27;39;46;51;56;50;43;45;43;32;33;29;...
         21;12;17;13;15;12;6;6;5;5]/(sqrt(hs)*640);

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