Home > vbmeg > functions > leadfield > eeg > vb_legendre_grad.m

vb_legendre_grad

PURPOSE ^

vb_legendre_grad - Evaluate the first N Legendre polynomials

SYNOPSIS ^

function [P,dP] = vb_legendre_grad(N,x)

DESCRIPTION ^

 vb_legendre_grad - Evaluate the first N Legendre polynomials
  [P,dP] = vb_legendre_grad(N,x)

  Evaluates the first N Legendre polynomials and 
  the first derivatives at the vector x.

  Input:
    N  (scalar)   : Polynomial order to evaluate
    x  (M x 1)    : Input vector of values for evaluation

  Outputs:
   P  (N x M)   : Legendre Polynomial
                = [ P1(x1) P1(x2)    ...    P1(xM);
                    P2(x1) P2(x2)    ...    P2(xM);
                      .
                      .
                    PN(x1) PN(x2)    ...    PN(xM)]
 
  dP = dP/dx  :  Derivative of P  (N x M) 
 ---------------------- --------- 

 Copyright (C) 2011, ATR All Rights Reserved.
 License : New BSD License(see VBMEG_LICENSE.txt)

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function [P,dP] = vb_legendre_grad(N,x)
0002 % vb_legendre_grad - Evaluate the first N Legendre polynomials
0003 %  [P,dP] = vb_legendre_grad(N,x)
0004 %
0005 %  Evaluates the first N Legendre polynomials and
0006 %  the first derivatives at the vector x.
0007 %
0008 %  Input:
0009 %    N  (scalar)   : Polynomial order to evaluate
0010 %    x  (M x 1)    : Input vector of values for evaluation
0011 %
0012 %  Outputs:
0013 %   P  (N x M)   : Legendre Polynomial
0014 %                = [ P1(x1) P1(x2)    ...    P1(xM);
0015 %                    P2(x1) P2(x2)    ...    P2(xM);
0016 %                      .
0017 %                      .
0018 %                    PN(x1) PN(x2)    ...    PN(xM)]
0019 %
0020 %  dP = dP/dx  :  Derivative of P  (N x M)
0021 % ---------------------- ---------
0022 %
0023 % Copyright (C) 2011, ATR All Rights Reserved.
0024 % License : New BSD License(see VBMEG_LICENSE.txt)
0025 
0026 if size(x,1)~=1, x=x'; end;
0027 
0028 M  = size(x,2);      % # of evaluation points
0029 
0030 P  = zeros(N,M); 
0031 dP = zeros(N,M);
0032 
0033 P(1,:)    = x;              % initial of P
0034 P(2,:)    = 1.5*x.^2-0.5;  
0035 dP(1,:) = ones(1,M);      % initial derivative
0036 
0037 % recursively compute P,dP
0038 for n=3:N,
0039   P(n,:) = ( (2*n-1).*x.*P(n-1,:) - (n-1)*P(n-2,:) )/n;
0040 end
0041 
0042 for n=2:N
0043   dP(n,:) = x.*dP(n-1,:) + n.*P(n-1,:);     
0044 end

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