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# dlegpoly

## (Toolbox/dlegpoly.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    [P,dP1] = dlegpoly(N,x)

### Help Text

    DLEGPOLY - Evaluate the first N Legendre polynomials
     function [P,dP1] = dlegpoly(N,x)
    *****************************************************************************
      function [P,dP1,dP2] = dlegpoly(N,x,flag)
      This function evaluates the first N Legendre polynomials,the first and
      second derivatives at the vector x.

      Input:
        N       (scalar)       last polynomial order to evaluate
        x       (m x 1)        input vector of values to eval. the polys at
        flag    (scalar)       if flag=1, compute dP2

      Outputs:
        P       (N x m)      = [P1(x1) P1(x2)    ...    P1(xm);
                                P2(x1) P2(x2)    ...    P2(xm);
                                 .
                                 .
                                PN(x1) PN(x2)    ...    PN(xm)]

        dP1      (N x m)      =  derivative w.r.t. x of P
    *****************************************************************************

### Cross-Reference Information

This function is called by

    eeg_sph                            C:\BrainStorm_2001\Toolbox\eeg_sph.m

### Listing of function C:\BrainStorm_2001\Toolbox\dlegpoly.m

    function [P,dP1] = dlegpoly(N,x)
    %DLEGPOLY - Evaluate the first N Legendre polynomials
    % function [P,dP1] = dlegpoly(N,x)
    %*****************************************************************************
    %  function [P,dP1,dP2] = dlegpoly(N,x,flag)
    %  This function evaluates the first N Legendre polynomials,the first and
    %  second derivatives at the vector x.
    %
    %  Input:
    %    N       (scalar)       last polynomial order to evaluate
    %    x       (m x 1)        input vector of values to eval. the polys at
    %    flag    (scalar)       if flag=1, compute dP2
    %
    %  Outputs:
    %    P       (N x m)      = [P1(x1) P1(x2)    ...    P1(xm);
    %                            P2(x1) P2(x2)    ...    P2(xm);
    %                             .
    %                             .
    %                            PN(x1) PN(x2)    ...    PN(xm)]
    %
    %    dP1      (N x m)      =  derivative w.r.t. x of P
    %*****************************************************************************

    %<autobegin> ---------------------- 26-May-2004 11:30:11 -----------------------
    % --------- Automatically Generated Comments Block Using AUTO_COMMENTS ---------
    %
    % CATEGORY: Utility - Numeric
    %
    % At Check-in: $Author: Mosher $  $Revision: 12 $  $Date: 5/26/04 9:59a $
    %
    % This software is part of BrainStorm Toolbox Version 2.0 (Alpha) 24-May-2004
    %
    % Principal Investigators and Developers:
    % ** Richard M. Leahy, PhD, Signal & Image Processing Institute,
    %    University of Southern California, Los Angeles, CA
    % ** John C. Mosher, PhD, Biophysics Group,
    %    Los Alamos National Laboratory, Los Alamos, NM
    % ** Sylvain Baillet, PhD, Cognitive Neuroscience & Brain Imaging Laboratory,
    %    CNRS, Hopital de la Salpetriere, Paris, France
    %
    % See BrainStorm website at http://neuroimage.usc.edu for further information.
    %
    % Copyright (c) 2004 BrainStorm by the University of Southern California
    % This software distributed  under the terms of the GNU General Public License
    % as published by the Free Software Foundation. Further details on the GPL
    % license can be found at http://www.gnu.org/copyleft/gpl.html .
    %
    % FOR RESEARCH PURPOSES ONLY. THE SOFTWARE IS PROVIDED "AS IS," AND THE
    % UNIVERSITY OF SOUTHERN CALIFORNIA AND ITS COLLABORATORS DO NOT MAKE ANY
    % WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF
    % MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, NOR DO THEY ASSUME ANY
    % LIABILITY OR RESPONSIBILITY FOR THE USE OF THIS SOFTWARE.
    %<autoend> ------------------------ 26-May-2004 11:30:11 -----------------------

    if size(x,1)~=1, x=x'; end      % make it a row vector

    m=size(x,2);  % # evaluation points

    P=zeros(N,m);     % preallocate
    dP1=zeros(N,m);

    P(1,:)=x;              % initials of P
    P(2,:)=1.5*x.^2-0.5;
    dP1(1,:)=ones(1,m);    % initial derivative

    % recursively compute P,dP1
    for n=3:N,
      P(n,:)=( (2*n-1).*x.*P(n-1,:)-(n-1)*P(n-2,:) )/n;
    end

    for n=2:N
      dP1(n,:)=x.*dP1(n-1,:)+n.*P(n-1,:);
    end

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