📄 findq.m.htm

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<html>function Q = FindQ(d, j, normalization)% Q = FindQ(d, j, normalization) returns the Q matrix for B-spline scaling% functions of degree d, level j.  If normalization is 'min' (or is not% specified) then the smallest entry in each column is made 1.  If% normalization is 'max' then the largest entry in each column is made 1.% If normalization is 'L2' then the L_2 norm of each wavelet is made 1.if nargin < 3  normalization = 'min';elseif ~strcmp(normalization, 'min') & ~strcmp(normalization, 'max') ...      & ~strcmp(normalization, 'L2') & ~strcmp(normalization, 'lcd')  error('FindQ: normalization must be ''min'', ''max'', or ''L2''.');end;P = FindP(d, j);I = Inner(d, j);if ~strcmp(normalization, 'L2')%  P = round(P*LCD(P));%  I = round(I*LCD(I));end;M = P'*I;[m1, m2] = size(M);n = m2 - rank(M);Q = zeros(m2, n);found = 0;start_col = 0;while (found < n/2) & (start_col < m2)  start_col = start_col + 1 + (found > d);  width = 0;  rank_def = 0;  while ~rank_def & (width < m2 - start_col + 1)    width = width + 1;    submatrix = M(:,start_col:start_col+width-1);    rank_def = width - rank(submatrix);  end;  if rank_def    % find null space of submatrix (should be just one column)    q_col = null(submatrix);        if strcmp(normalization, 'min')      % normalize column so smallest nonzero entry has magnitude 1      q_col = q_col/min(abs(q_col + 1e38*(abs(q_col) < 1e-10)));    elseif strcmp(normalization, 'max')      % normalize column so largest entry has magnitude 1      q_col = q_col/max(abs(q_col));    elseif strcmp(normalization, 'lcd')      % normalize column to get all integers      q_col = q_col/min(abs(q_col + 1e38*(abs(q_col) < 1e-10)));      q_col = q_col*LCD(q_col);    end;    % change sign to give consistent orientation    q_col = q_col*(-1)^(start_col + floor((d+1)/2) + (q_col(1,1) > 0));          % correct any slight error for answers that should be integers    if all(abs(submatrix*round(q_col)) < 1e-10) & any(round(q_col) ~= 0)      q_col = round(q_col);    end;        % put column into left half of Q    found = found + 1;    Q(start_col:start_col+width-1,found) = q_col;        % use symmetry to put column into right half of Q in reverse order    % and negated if degree is even    Q(:,n-found+1) = flipud(Q(:,found))*(-1)^(d+1);  end;end;if strcmp(normalization, 'L2')  % normalize matrix so each column has L_2 norm of 1  ip = Q'*I*Q;  Q = Q*diag(1./sqrt(diag(ip)));end;return;
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