Gram Schmidt Process Algorithm

Question

Ellenna K 2015-10-5

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/246844-gram-schmidt-process-algorithm

回答： Sebastián Soto 2022-11-9

So I have an assignment to write algorithm code for Gram Schmidt process...I have done it in Matlab ,but when I run the code with input argument rand(some number), it gives me a result with variable ans... and some numbers in matrix... I want to create an output variable which won't be ans ,and can anyone see the code to tell me if it's okay? Thanks in advance,I would appreciate ur help..

    % function Q = gramschmidt( A )
% FUNCTION computes orthonormal basis from independent vectors in A.
%
%   Q = gramschmidt( A );
%
% INPUT :
%   A   - a matrix with *INDEPENDENT* vectors in columns
%         (e.g. A = rand(3) will produce one)
%
% OUTPUT :
%   Q   - a matrix with orthonomal basis of A
%
%

% The vectors in A are independent BUT NOT YET orthonormal. Check A'*A. % If it is orthonormal, you should get strictly an identity matrix.

% number of vectors n = size( A, 2 );

% initialize output Q = zeros( n );

% turn every independent vector into a basis vector % (1) jth basis vector will be perpendicular to 1..j-1 previous found basis % (2) will be of length 1 (norm will be equal to 1) for j = 1 : n

    % pick the jth independent vector
    u = A( :, j );
    % special case for j = 1: we will not run the following for loop. Will
    % just normalize it and put as the first found basis. There are no
    % previous basis to make orthogonal to.
    % remove from raw "u" all components spanned on 1..j-1 bases, their
    % contributions will be removed
    % ==> this effectively makes jth independent vector orthogonal to all
    % previous bases found in the previous steps.
    % ==> enforcing orthogonality principle here. Not orthonormality yet.
    for i = 1 : j - 1
        u = u - proj( Q(:,i), A(:,j) );
    end
    % normalize it to length of 1 and store it
    Q(:,j) = u ./ norm( u );

end

% projects a vector "a" on a direction "e" function p = proj( e, a )

% project "a" onto "e": (e' * a) / (e' * e) is the length (!) of "a" on "e" % multiplication by "e" is necessary to output the resulting vector which is % (colinear with "e") p = (e' * a) / (e' * e) .* e;

end