fn2fm.m

演示matlab曲线拟和与插直的基本方法

function g = fn2fm(f,form,sconds)
%FN2FM Convert to specified form.
%
%   FN2FM(F,FORM)  converts the function in F to the specified FORM.
%   Present choices for FORM are: 'pp', 'B-', 'BB', 'rp', 'rB'.
%   See the relevant ..mak commands for specifics of such a form.
%
%   FN2FM(F,FORM,SCONDS)  uses additional information in SCONDS to help in
%   the conversion from ppform to B-form. In this conversion, the actual 
%   smoothness of the function in F across each of its interior breaks has to 
%   be guessed. 
%   If 0 < SCONDS(1) < 1, then this is done by looking for the derivative
%   of smallest order whose jump across that break is, in absolute value,
%   greater than SCONDS(1) times the absolute value of that derivative 
%   nearby.  The default value for this tolerance SCONDS(1) is 1.e-12. 
%   Otherwise, if SCONDS is a vector with exactly as many entries as F has 
%   interior breaks, then SCONDS(i) smoothness conditions are imposed at the 
%   i-th interior break. 
%   If the function in F is a tensor product, then SCONDS, if given, must 
%   be a cell array. 
%
%   FN2FM(F)  converts a possibly old version of a form into its present 
%   version.
%
%   Examples:
%   
%      sp = fn2fm(spline(x,y),'B-');
%
%   gives the interpolating cubic spline provided by MATLAB's SPLINE,
%   but in B-form (i.e., described as a linear combination of B-splines).
%
%      p0 = ppmak([0 1],[3 0 0]); 
%      p1 = fn2fm(fn2fm(fnrfn(p0,[.4 .6]),'B-'),'pp');
%
%   gives  p1  identical to  p0  (up to round-off) since the spline has no
%   discontinuity in any derivative across the additional breaks introduced
%   by FNRFN, hence conversion to B-form ignores these additional breaks, and 
%   conversion to ppform does not retain any knot multiplicities (like 
%   the knot multiplicities introduced, by conversion to B-form, at the 
%   endpoints of the spline's basic interval).
%
%      nurb = fn2fm(f,'rB');
%
%   converts the (d+1)-valued spline in  f  into B-form if need be, then
%   converts to rBform, i.e., into a d-valued rational spline or nurbs, 
%   by designating its (d+1)st component as the denominator, provided d>0. 
%
%   See also PP2SP, SP2PP, SP2BB.

%   Copyright 1987-2003 C. de Boor and The MathWorks, Inc.
%   $Revision: 1.18 $

%%%%%%%%%%%%%%%%%%%%  array versions of univariate forms (from FNBRK)
%   case 10, out1 = 'ppform, univariate';
%   case 11, out1 = 'B-form, univariate';
%   case 12, out1 = 'BBform, univariate';
%   case 15, out1 = 'polynomial in Newton form';
%   case 25, out1 = 'thinplate spline';

if nargin==1 % set FORM to convert the form in F into its current version
   if isstruct(f)
      form = f.form;
   else
      switch f(1)
      case 10,  form = 'pp';
      case 11,  form = 'B-';
      case 12,  form = 'BB';
      case 15,  form = 'NP';
      case 25,  form = 'st-tp00';
      otherwise
         error('SPLINES:FN2FM:unknownform','Input is of unknown (function) form.')
      end
   end
else
   if length(form)<2
      error('SPLINES:FN2FM:unknownform',...
           ['The specified form, ''',form,''', is not (yet) recognized.']),
   end 
end

switch form(1:2)
case 'pp'
   if isstruct(f)
      switch f.form(1:2)
      case 'pp',          g = f;
      case 'rp',          g = f; g.form = 'pp'; g.dim = g.dim+1;
      case 'rB',          f.form = 'B-'; f.dim = f.dim+1; g = sp2pp(f);
      case {'B-','BB'},   g = sp2pp(f);
      end
   else
      switch f(1)
      case 10,      [b,c,l,k,d] = ppbrk(f); g = ppmak(b,c,d);
      case {11,12}, g = sp2pp(f);
      case 15
       error('SPLINES:FN2FM:nonnewtonhere',...
             'Use NP2PP directly to convert from Newton form to ppform.')
      end
   end
case {'sp','B-'}
   form = 'B-';
   if isstruct(f)
      switch f.form(1:2)
      case 'pp'
         if nargin<3, g = pp2sp(f);
         else         g = pp2sp(f,sconds);
         end
      case 'rp',      f.form = 'pp'; f.dim = f.dim+1; g = pp2sp(f);
      case 'rB',      g = f; g.form = 'B-'; g.dim = g.dim+1;
      case 'B-',      g = f;
      case 'BB',      g = pp2sp(sp2pp(f));
      end
   else
      switch f(1)
      case 10, g = pp2sp(f);
      case 11, [knots,coefs] = spbrk(f); g = spmak(knots,coefs);
      case 12, g = pp2sp(sp2pp(f));
      end
   end
case {'bb','BB'}
   form = 'BB';
   if isstruct(f)
      switch f.form(1:2)
      case 'pp', g = sp2bb(pp2sp(f));          
      case 'rp', f.form = 'pp'; f.dim = f.dim+1; g = sp2bb(pp2sp(f));
      case 'rB', f.form = 'B-'; f.dim = f.dim+1; g = sp2bb(f);
      case 'B-', g = sp2bb(f);
      case 'BB',  g = f;
      end
   else
      switch f(1)
      case 10, g = sp2bb(pp2sp(f));
      case 11, g = sp2bb(f);
      case 12, [knots,coefs] = spbrk(f); g = spmak(knots,coefs); g.form = 'BB';
      end
   end
case 'st' 
   if isstruct(f)
      switch f.form(1:2)
      case 'st', g = f;
      end
   else
      switch f(1)
      case 25, [ce,co] = stbrk(f); g = stmak(ce,co,'tp00');
      end
   end
case 'rp'  % switch first into ppform if need be
   if isstruct(f)
      switch f.form(1:2)
      case 'pp',      g = f;
      case 'rp',      g = f; g.dim = g.dim+1;
      case 'rB',      g = sp2pp(fn2fm(f,'B-'));
      case {'B-','BB','sp','bb'}, g = sp2pp(f);
      end
   else
      switch f(1)
      case 10,      [b,c,l,k,d] = ppbrk(f); g = ppmak(b,c,d);
      case {11,12}, g = sp2pp(f);
      end
   end
   if exist('g','var')
      g.form = 'rp'; 
      if length(g.dim)>1
         warning('SPLINES:FN2FM:noNDrats', ...
	 ['While the given function has values of size [', num2str(g.dim), ...
	  '], the function returned is ', num2str(prod(g.dim)-1), ...
	  '-vector valued.'])
	 g.dim = prod(g.dim);
      end
      g.dim = g.dim-1;
      if g.dim<1
         error('SPLINES:FN2FM:ratneedsmoredim',...
               'A rational spline must have more than one component.')
      end
   end
case 'rB'  % switch first into B-form if need be
   if isstruct(f)
      switch f.form(1:2)
      case 'pp'
         if nargin<3, g = pp2sp(f);
         else         g = pp2sp(f,sconds);
         end
      case 'rp',      g = fn2fm(fn2fm(f,'pp'),'B-');
      case 'rB',      g = f; g.dim = g.dim+1;
      case {'B-','BB','sp','bb'}, g = f;
      end   
   else 
      switch f(1)
      case 10, g = pp2sp(f);
      case 11, g = spmak(fnbrk(f,'knots'),fnbrk(f,'coefs'));
      case 12, g = pp2sp(sp2pp(f));
      end
   end
   if exist('g','var')
      g.form = 'rB';
      if length(g.dim)>1
         warning('SPLINES:FN2FM:noNDrats', ...
	 ['While the given function has values of size [', num2str(g.dim), ...
	  '], the function returned is ', num2str(prod(g.dim)-1), ...
	  '-vector valued.'])
	 g.dim = prod(g.dim);
      end
      g.dim = g.dim-1;
      if g.dim<1
         error('SPLINES:FN2FM:ratneedsmoredim',...
               'A rational spline must have more than one component.')
      end
   end
case 'NP' % at present, this form only exists in the old version
   if isstruct(f)
      switch f.form(1:2)
      case 'NP', g = f;
      end
   else
      switch f(1)
      case 15, g = f;
      end
   end
case 'MA'     % convert univariate spline to old, nonstructure ppform.
   if isstruct(f)&&~iscell(f.order)
      switch f.form(1:2)
      case 'pp' 
         g = [10 f.dim f.pieces f.breaks(:).' f.order f.coefs(:).'];
      case {'B-','BB'}
         f = sp2pp(f);
         g = [10 f.dim f.pieces f.breaks(:).' f.order f.coefs(:).'];
      otherwise
         error('SPLINES:FN2FM:notintoMA',...
               'Cannot convert the given function into MATLAB''s ppform.')
      end
   else
      g = f;
   end
case 'ol'    % convert univariate structured form to corresponding 
             % formerly used array form.
   if ~isstruct(f)
      error('SPLINES:FN2FM:olneedsstruct',...
            'When FORM=''ol'', F must be a structure.')
   else
      if length(f.order)>1
         error('SPLINES:FN2FM:nonstructneedsuni',...
              ['Reversion to array is only possible for',...
                ' univariate functions.'])
      else
         switch f.form(1:2)
         case 'pp' 
            g = [10 f.dim f.pieces f.breaks(:).' f.order f.coefs(:).'];
         case {'B-','BB'}
            g = [11 f.dim f.number f.coefs(:).' f.order f.knots(:).'];
            if f.form(1:2)=='BB', g(1) = 12; end
         otherwise
            error('SPLINES:FN2FM:unknownfn','Unknown function type encountered.')
         end
      end
   end
otherwise
   error('SPLINES:FN2FM:unknownfn',...
    ['The specified form, ''',form,''', is not (yet) recognized.']),
end

if ~exist('g','var')
   error('SPLINES:FN2FM:notintothatform',...
        ['Cannot convert the given function into ',form,'form.'])
end