spterms.m

来自「演示matlab曲线拟和与插直的基本方法」· M 代码 · 共 381 行 · 第 1/2 页

  ' for spaps, one gives the input  tol  as the n-vector ',...
  ' [tolerance, lam].']};
case 'kn'
   term = 'knots';
   mess = ...
   {'';'KNOTS are parameters used in the B-form of a spline.';
    'Any nondecreasing sequence  t  can serve as a knot sequence.';
 ['Assuming  t  has length  n+k , there are exactly  n  B-splines of order  k ',...
' associated with  t , with the j-th Bspline having the knots  t(j:j+k), all j.',...
' These  n  B-splines serve as a convenient basis for the space',...
' S_{k,t}  of splines of order  k  with knot sequence  t .'];'';
['The elements of this space are piecewise polynomial of order  k  with',...
' breaks at the knots, but their smoothness across each break depends on the',...
' multiplicity with which that break occurs in the knot sequence. The rule:'];'';
'    number of smoothness conditions + knot multiplicity  =  order .'}; 

case 'en'
   term = 'end knots';
   mess = ...
  {'';['When constructing the B-form of a cubic spline, the first four',...
   ' knots are usually chosen to equal the left end of the', ...
   ' interval of interest, and the',...
   ' last four knots are chosen to equal the right end of that',...
   ' interval (thereby making it the ''basic interval'' for the',...
   ' resulting B-form).'];'';
   ['These END KNOTS have no influence on the behavior of the spline on',...
   ' that interval of interest and could even be chosen to be distinct',...
   ' and lie outside that interval (which would increase the',...
   ' ''basic interval'' of the resulting B-form).'];'';
   ['More generally, for a spline of order  k , the first and last  k ',...
   ' knots in its B-form are usually chosen to equal the corresponding',...
  ' end points of the interval of interest and thereby become the end knots.']};
case 'er'
   term = 'error';
   mess = ...
  {['The error is, by definition, the difference between the exact value',...
    ' and the approximation.'];'';
   ['In least-squares approximation and in smoothing, the error in the',...
    ' approximation  f  to the data  (x(j), y(:,j)), j=1:length(x), is',...
    ' measured by the weighted sum of squares'];'';
    '         sum_j w(j) | y(:,j) - f(x(j)) |^2';'';
    'for some nonnegative weights  w .'};
case 'in'
   term = 'interior knots';
   mess = ...
  {['The knots of a spline inside its basic interval are its INTERIOR',...
   ' KNOTS. Usually, some derivative of a spline has a jump across',...
   ' such a knot. The lowest order derivative in which a jump may',...
   ' occur depends on the multiplicity  m  of that knot and the order',...
   '  k   of the spline: it''s the derivative of order  k-m .'];'';
   ['For example, across a simple knot (m=1), a cubic spline (k=4)',...
   ' may have a jump in the 3rd derivative (3 = 4-1), but not in any',...
   ' lower-order derivative (like the 2nd, 1st, or 0th derivative).']};

case 'le'
   term = 'least squares';
   mess = ...
  {['In (discrete) LEAST_SQUARES spline approximation, the element  s ',...
   ' from the specified spline space closest to the data (x,y) is',...
   ' constructed, with the distance measured by the ERROR: ',...
   ' sum( w.*(y - s(x)).^2 )  .'];'';
   'A SPLINE SPACE is specified by giving its knot sequence and its order.';
   ['The knot sequence is required to satisfy the Schoenberg-Whitney', ...
   ' Conditions wrto some subsequence of the given data sites, x .']};

case {'NU','nu'}
   term = 'NURBS';
   mess = ...
  {'A rational spline in which each coefficient of the denominator spline';
  ['is an explicit factor of the corresponding coefficient of the',...
   ' numerator spline.']};
case 'ra'
   term = 'rational spline';
   mess = ...
 {'A rational spline is a ratio of splines, that is a function of the form';...
   '';'          r(x) = s(x)/w(x),';''; ...
  'with both s and w splines, of the same order and with the same knots or';...
   'breaks, and with w scalar-valued, while s may be vector-valued.';'';...
   'A rational spline may be in rBform or rpform which, in effect, is the';...
   'B-form or ppform of the corresponding vector-valued spline';'';...
      '          R(x) = [s(x);w(x)].'};
case 'rB'
   term = 'rBform';
   mess = ...
  {'The rBform of a rational spline  r(x) = s(x)/w(x)  provides its order,',...
   ' its knot sequence, and the coefficients for the B-form of the',...
   ' corresponding (vector-valued) spline R(x) = [s(x);w(x)].'};
case 'rp'
   term = 'rpform';
   mess = ...
  {'The rpform of a rational spline  r(x) = s(x)/w(x)  provides its order,',...
   ' its break sequence, and the local polynomial coefficients for the',...
   ' ppform of the corresponding (vector-valued) spline R(x) = [s(x);w(x)].'};
case 'st'
   term = 'stform';
   mess = ...
  {['The stform provides the centers, coefficients, and type of a function ',...
   'of the form  sum_j coefs(:,j)*psi_j(x) + p(x) , with  p  a lower-', ...
   'order polynomial, and psi_j(x) = psi(x-c_j), all j. Here, the c_j are ',...
   'the centers, and the function psi, called the basis function, and the ',...
   'degree of the polynomial p are specified by the particular type. '];'';
   'The thin-plate spline is a standard example.'};
case 'th'
   term = 'thin-plate spline';
   mess = ...
  {'A thin-plate spline is any bivariate function of the form  f(x) = ',...
   'sum_j coefs(:,j) phi(norm(x-centers(:,j))^2) + coefs(:,end-2)*x(1) ',...
   '+ coefs(:,end-1)*x(2) + coefs(:,end), with phi(t) := t log(t) .'};
case 'ce'
   term = 'centers';
   mess = ...
  {'The centers of a stform are the sites by which the basis function of ',...
   'that form is translated to provide the individual terms of that form.'};
case 'qu'
   term = 'quintic smoothing spline';
      mess = {...
   ['The QUINTIC smoothing spline differs from the cubic smoothing spline',...
   ' only in that its roughness measure involves the 3rd rather than the',...
   ' 2nd derivative. Correspondingly, it is a spline of order 6 rather than',...
   ' of order 4.'];'';
   ['In the Spline Toolbox, the quintic smoothing spline is provided by',...
   ' the command  spaps . The level of desired smoothness is specified',...
   ' indirectly, by specifying a certain tolerance, and the command',...
   ' returns the smoothest spline within that tolerance of the given data.']};
case {'sp','Sc'} % this one, we have to split further
   if isempty(findstr(' ',term))&&term(2)~='c'
      term = 'spline';
      mess = ...
{['That''s a hard one. Roughly speaking, a (univariate, polynomial) SPLINE',...
 ' is any more or less smooth piecewise polynomial function.']};
   else
      switch term(2)
      case 'p'
         term = 'spline interpolation';
      case 'c'
         term = 'Schoenberg-Whitney conditions';
      end

      mess =...
{['General spline interpolation to data (x,y) is carried out in this', ...
 ' toolbox with the command SPAPI, and requires, in addition to the data,',...
 ' specification of a knot sequence, t, that satisfies the SCHOENBERG-',...
 'WHITNEY CONDITIONS wrto the given data sites, in the following sense.'];
 '';['Assuming that  x  is nondecreasing and of length  n , and assuming',...
 ' that also  t  is nondecreasing, the difference  k := length(t) - n',...
 '  must be positive, and, further,'];'';
 '                 t(i)  <=  x(i)  <=  t(i+k),   for  i=1:n ,';'';
['with equality allowed only when the relevant knot has multiplicity k', ...
' (as is typically the case for the endknots).'];'';
['If the knot sequence  t  satisfies the Schoenberg-Whitney conditions',...
' wrto the data sites  x , then there is exactly one spline of order  k',...
'  with knot sequence  t  that matches the given data.'];'';
'It is this spline that is being supplied by the command spapi(t,x,y).';...
['The variant  spapi(k,x,y), with  k  a natural number, supplies a knot', ...
' sequence  t  of length  k+length(x)  that satisfies the Schoenberg-',...
'Whitney conditions wrto  x . See the help for spapi for further detail.']};
   end

case 'or'
   term = 'order';
   mess = ...
   {['The  ORDER  of a spline counts the number of coefficients in any',...
    ' one of its polynomial pieces.'];'';
    'For example, a cubic spline is of order 4.'};

case 'pp'
   term = 'ppform';
   mess = ...
  {['The PPFORM of a spline provides its breaks, its order, and the',...
   ' coefficients for the shifted power form of its polynomial pieces.']};

case 'si'
   term = 'sites_etc';
   mess = ...
  {['The terms  SITES  and  VALUES  are used here instead of their',...
   ' oldfashioned counterparts,  ABSCISSAE  and  ORDINATES.'];'';
   ['The data POINT  (x(i),y(i))  tells us that, at the SITE  x(i) ',...
   ', we would like the spline to take the VALUE  y(i) .']};

otherwise

   mess = {'nothing yet'};
end

if nargout>0 
   expl = mess;
else
   msgbox(mess,['Explanation: ',term])
end