spaps.m

演示matlab曲线拟和与插直的基本方法

% SPAPS   暯妸壔僗僾儔僀儞
%
% [SP,VALUES] = SPAPS(X,Y,TOL)偼丄僨乕僞 X, Y 傊偺僉儏乕價僢僋暯妸壔
% 僗僾儔僀儞 f 偺B-宆偲丄昁梫偱偁傟偽丄X 偱偺抣傪弌椡偟傑偡丅
% 暯妸壔僗僾儔僀儞偼丄j=1:length(X) 偲偟偰丄梌偊傜傟偨僨乕僞抣 Y(:,j) 
% 傪僨乕僞僒僀僩 X(j) 偱嬤帡偟傑偡丅僨乕僞抣偼丄僗僇儔丄儀僋僩儖丄峴楍丄
% 傑偨偼N師尦攝楍偱偡丅
% 摨偠僒僀僩偺僨乕僞揰偼丄偦傟傜偺(壛廳)暯嬒偵傛偭偰抲偒姺偊傜傟丄
% 嫋梕岆嵎 TOL 偼丄偦傟偵墳偠偰尭彮偟傑偡丅
%
% 暯妸壔僗僾儔僀儞 f 偼丄岆嵎婎弨
%
%       E(f) :=  sum_j { W(j)*( Y(:,j) - f(X(j)) )^2 : j=1,...,n }
%
% 偑 TOL 偲斾傋偰彮偟傕戝偒偔側偄偡傋偰偺娭悢 f 忋偱丄慹偝(roughness)
% 婎弨
%
%       F(D^M f) := integral |D^M f(t)|^2 dt  on  min(X) < t < max(X)
%
% 傪嵟彫偵偟傑偡丅
%
% 偙偙偱丄X 偲 Y 偼丄摨偠僒僀僩傪傕偮擟堄偺僨乕僞揰傪丄懳墳偡傞廳傒偺
% 榓傪廳傒偲偡傞偦傟傜偺暯嬒偵傛偭偰丄抲偒姺偊偨寢壥偱偡丅傑偨丄梌偊
% 傜傟偨 TOL 偼丄偦傟偵懳偟偰尭彮偝傟傑偡丅
% D^M f 偼丄f 偺M師旝暘傪帵偟傑偡丅
% 廳傒 W 偼丄E(f) 偑 F(y-f) 偵懳偡傞暋崌戜宍懃偺嬤帡偵側傞傛偆偵慖戰
% 偝傟傑偡丅
% 惍悢 M 偼丄2偑慖戰偝傟傑偡丅廬偭偰丄D^M f 偼丄f 偺2師旝暘偱偡丅 M 偺
% 偙偺慖戰偵傛傝丄f 偼僉儏乕價僢僋暯妸壔僗僾儔僀儞偱偡丅
% 偙傟偼丄f 偑丄E(f) 偲 TOL 偑摍偟偔側傞傛偆慖戰偝傟偨暯妸壔僷儔儊乕僞 
% RHO 傪傕偮
%
%                     rho*E(f) + F(D^2 f) ,
%
% 偺儐僯乕僋側儈僯儅僀僓偲偟偰峔惉偝傟傞偨傔偱偡丅
% 備偊偵丄FN2FM(SP,'pp') 偼丄CPAPS(X,Y,RHO/(1+RHO)) 偐傜偺弌椡偲(娵傔傑偱)
% 摨偠偵側傞昁梫偑偁傝傑偡丅
%
% [SP,VALUES,RHO] = SPAPS(X,Y,...) 偼丄RHO 傕弌椡偟傑偡丅
%
% TOL 偑0偺偲偒丄曄摦偡傞曗娫僉儏乕價僢僋僗僾儔僀儞偑摼傜傟傑偡丅
% TOL 偑晧偱偁傞応崌丄-TOL 偼丄巊梡偝傟傞 RHO 偺抣偲偟偰夝庍偝傟傑偡丅
%
% 僨乕僞抣 Y(:,j) 偑丄僗僇儔偱偼側偔丄戝偒偝 d 偱偁傞偲偒丄奺 t 偵懳偟偰丄
% f(t) 偲 D^M f(t) 傕傑偨戝偒偝 d 偱偡丅廬偭偰丄慡懱偼prod(d)屄偺梫慺傪
% 傕偪丄幃 |Y(:,j) - f(X(j))|^2 偲 |D^M f(t)|^2 偑丄偙傟傜偺prod(d)屄偺
% 梫慺偺擇忔榓傪帵偡偙偲偵側傝傑偡丅
%
% 僌儕僢僪壔偝傟偨僨乕僞偺応崌偼丄埲壓偱榑偠傑偡丅
%
% 椺戣:
%
%      x = linspace(0,2*pi,21); y = sin(x) + (rand(1,21)-.5)*.2;
%      sp = spaps(x,y, (.05)^2*(x(end)-x(1)) );
%
% 偼丄撪嵼偡傞僲僀僘偺側偄僨乕僞偵偐側傝嬤偄偙偲偑梊憐偝傟傞丄僲僀僘晅偒
% 僨乕僞傊偺嬤帡傪弌椡偟傑偡丅偙傟偼丄撪嵼僨乕僞偑備偭偔傝曄壔偡傞娭悢偐傜
% 側傝丄巊梡偝傟傞 TOL 偑僲僀僘偺戝偒偝偵懳偟揔摉側戝偒偝偱偁傞偨傔偱偡丅
%
%   SPAPS(X,Y,TOL,W)
%   SPAPS(X,Y,TOL,M)
%   SPAPS(X,Y,TOL,W,M)
%   SPAPS(X,Y,TOL,M,W)
% 偙傟傜偡傋偰偼丄M = 1(慄宍)偍傛傃 M = 3(5師)(僨僼僅儖僩偺 M = 2偲偼暿偵)
% 偲偟偰尰嵼惂尷偝傟偰偄傞 M 傪丄W 偍傛傃/傑偨偼 M 偵柧帵揑偵梌偊傑偡丅
%
% TOL 偑僗僇儔偱偼側偔傓偟傠楍偲偡傞偙偲偵傛傝丄妸傜偐偝偺梫媮傪曄偊傞
% 偙偲傕壜擻偱偡丅偦偺応崌丄慹偝(roughness)婎弨偼丄偮偓偺幃偱摼傜傟傑偡丅
%
%                    integral lambda(t) (D^M f(t))^2 dt ,
%
% 偙偙偱丄lambda 偼丄僽儗乕僋億僀儞僩 X 傪傕偮嬫暘揑側掕悢娭悢偱丄偦偺
% 嬫娫 (X(i-1) .. X(i)) 偱偺抣偼丄偡傋偰偺 i 偵偍偗傞 TOL(i) 偱偡丅堦曽丄
% TOL(1) 偼丄堷偒懕偒梫媮偝傟傞嫋梕岆嵎 TOL 偲偟偰夝庍偝傟傑偡丅
%
% 椺戣:
%
%    sp1 = spaps(x,y, [(.025)^2*(x(end)-x(1)),ones(1,10),repmat(.1,1,10)] );
%
% 偼丄慜偺椺偲摨偠僨乕僞偲嫋梕岆嵎傪巊梡偟傑偡偑丄慹偝(roughness)偺廳傒傪
% 嬫娫偺塃敿暘偱偨偭偨.1偵偡傞偙偲偱丄偦偙偱偼峳偄偗傟偳傕傛傝揔愗側
% 僼傿僢僥傿儞僌傪梌偊傑偡丅
% 斾妑偺偨傔偵椉曽偺椺傪帵偡僾儘僢僩偼丄偮偓偵傛傝摼傜傟傑偡丅
%
%    fnplt(sp); hold on, fnplt(sp1,'r'), plot(x,y,'ok'), hold off
%    title('Two cubic smoothing splines')
%    xlabel('the red one has reduced smoothness requirement in right half.')
%
% SPAPS({X1,...,Xr},Y,...) 偼丄r屄偺儀僋僩儖 X1, ..., Xr 偵傛偭偰巜掕
% 偝傟傞r師尦挿曽宍偺奿巕忋偺僨乕僞抣 Y 偵懳偡傞暯妸壔僗僾儔僀儞傪梌偊
% 傑偡丅偙傟傜偺儀僋僩儖 X1, ..., Xr 偼丄堎側傞挿偝傪傕偮偐傕偟傟傑偣傫丅
% Y 偼丄[d,length(X1), ..., length(Xr)] 偺戝偒偝傪傕偪傑偡丅
% Y 偑丄r師尦偺攝楍偱偁傞偑丄[length(X1), ..., length(Xr)] 偺戝偒偝偺
% 応崌丄d 偼丄[1] 偱偁傞偲夝庍偝傟傑偡丅偡側傢偪丄娭悢偼丄僗僇儔抣偱偡丅
% 奿巕僨乕僞偺応崌丄僆僾僔儑儞偺堷悢 M 偼丄1偮偺惍悢偱偁傞偐丄偁傞偄偼丄 
% (廤崌{1,2,3}偐傜偺)惍悢偺r梫慺偺儀僋僩儖偱偡丅傑偨丄僆僾僔儑儞偺堷悢 
% W 偼丄挿偝 r 偺僙儖攝楍偱偁傞偙偲偑梫媮偝傟傑偡丅偙偙偱 W{i} 偼(僨僼僅
% 儖僩偺慖戰傪摼傞偨傔)偵嬻偐丄偁傞偄偼丄i=1:r 偺 X{i} 偲摨偠挿偝偺
% 儀僋僩儖偺偄偢傟偐偱偡丅
% 傑偨丄偙偺応崌丄傕偟 TOL 偑僙儖攝楍偱側偗傟偽丄TOL 偼丄r屄偺昁梫側
% 嫋梕岆嵎傪巜掕偡傞r梫慺偺儀僋僩儖偱偁傞偐丄偝傕側偔偽丄嵟弶偺梫慺偑
% r屄偺曄悢偦傟偧傟偵偍偄偰梫媮偝傟傞嫋梕岆嵎偲偟偰巊梡偝傟側偗傟偽側傝
% 傑偣傫丅
%
% 椺戣:
%      x = -2:.2:2; y=-1:.25:1; [xx,yy] = ndgrid(x,y); 
%      z = exp(-(xx.^2+yy.^2)) + (rand(size(xx))-.5)/30; 
%      sp = spaps({x,y},z,8/(60^2));  fnplt(sp)
%
% 偼丄2曄悢娭悢偐傜偺僲僀僘偺偁傞僨乕僞傪妸傜偐偵嬤帡偟偨恾傪惗惉偟傑偡丅
% 偙偙偱 NDGRID 偱偼側偔丄MESHGRID 傪巊梡偡傞偲丄僄儔乕偵側傝傑偡丅
%
% 嶲峫 SPAPI, SPAP2, CSAPS, TPAPS.


%   Copyright 1987-2004 C. de Boor and The MathWorks, Inc.