初学数值分析——关于三次样条逼近

vib

%三次样条逼近
%由x计算h;
x=0:10;
y=[2.51 3.30 4.04 4.70 5.22 5.54 5.78 5.40 5.57 5.70 5.80];
h=zeros(1,10);
for i=1:10,
    h(i)=(x(i+1)-x(i));
end
%计算a,b;
a=zeros(1,11)
b=zeros(1,11)
for i=2:10,
   a(i)=(h(i-1)/(h(i-1)+h(i))),
   b(i)=3*((1-a(i))*(y(i)-y(i-1))/h(i-1)+a(i)*(y(i+1)-y(i))/h(i))
end

%A
for i=1:9,
    A(i,i)=1-a(i)
    A(i,i+1)=2
    A(i,i+2)=a(i)
end
%m
M=b(2:10)'\A
%s
m=M
t=sym('t')
    figure(1);
    AXIS([0 10 0 6])
for i=1:10,
    s=(1+2*(t-x(i))/(x(i+1)-x(i)))*((t-x(i+1))/(x(i)-x(i+1)))^2*y(i)+(1-2*(t-x(i+1))/(x(i)-x(i+1)))*((t-x(i))/(x(i+1)-x(i)))^2*y(i+1)+(t-x(i))*((t-x(i+1))/(x(i)-x(i+1)))^2+m(i)+(t-x(i+1))*((t-x(i))/(x(i+1)-x(i)))^2*m(i+1);
    ezplot(s,[x(i),x(i+1)])
    hold on;   
    AXIS([0 10 0 6])
end
grid on;

这是我按照徐翠薇的《计算方法引论》编得一个matlab程序，可是我在matlab中查spindle函数的帮助，却什莫也看不懂，能帮我解释一下吗？

function output = spline(x,y,xx)
%SPLINE Cubic spline data interpolation.
%   YY = SPLINE(X,Y,XX) uses cubic spline interpolation to find YY, the values
%   of the underlying function Y at the points in the vector XX.  The vector X
%   specifies the points at which the data Y is given.  If Y is a matrix, then
%   the data is taken to be vector-valued and interpolation is performed for
%   each column of Y and YY will be length(XX)-by-size(Y,2).
%
%   PP = SPLINE(X,Y) returns the piecewise polynomial form of the cubic spline
%   interpolant for later use with PPVAL and the spline utility UNMKPP.
%
%   Ordinarily, the not-a-knot end conditions are used. However, if Y contains
%   two more values than X has entries, then the first and last value in Y are
%   used as the endslopes for the cubic spline.  Namely:
%      f(X) = Y(:,2:end-1),   df(min(X)) = Y(:,1),   df(max(X)) = Y(:,end)
%
%   Example:
%   This generates a sine curve, then samples the spline over a finer mesh:
%       x = 0:10;  y = sin(x);
%       xx = 0:.25:10;
%       yy = spline(x,y,xx);
%       plot(x,y,'o',xx,yy)
%
%   Example:
%   This illustrates the use of clamped or complete spline interpolation where
%   end slopes are prescribed. Zero slopes at the ends of an interpolant to the
%   values of a certain distribution are enforced:
%      x = -4:4; y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];
%      cs = spline(x,[0 y 0]);
%      xx = linspace(-4,4,101);
%      plot(x,y,'o',xx,ppval(cs,xx),'-');
%
%   See also INTERP1, PPVAL, SPLINES (The Spline Toolbox).

%   Carl de Boor 7-2-86
%   Copyright 1984-2002 The MathWorks, Inc.
%   $Revision: 5.18 $  $Date: 2002/06/06 13:39:51 $

% Generate the cubic spline interpolant in ppform, depending on the
% number of data points (and usually using the not-a-knot end condition).

output=[];
n=length(x);
if n<2, error('There should be at least two data points.'), end

if any(diff(x)<0), [x,ind]=sort(x); else, ind=1:n; end

x=x(:); dx = diff(x);
if all(dx)==0, error('The data abscissae should be distinct.'), end

[yd,yn] = size(y); % if Y happens to be a column matrix, change it to
                   % the expected row matrix.
if yn==1, yn=yd; y=reshape(y,1,yn); yd=1; end

if yn==n
   notaknot = 1;
elseif yn==n+2
   notaknot = 0; endslopes = y(:,[1 n+2]).'; y(:,[1 n+2])=[];
else
   error('Abscissa and ordinate vector should be of the same length.')
end

yi=y(:,ind).'; dd = ones(1,yd);
dx = diff(x); divdif = diff(yi)./dx(:,dd);
if n==2
   if notaknot, % the interpolant is a straight line
      pp=mkpp(x.',[divdif.' yi(1,:).'],yd);
   else         % the interpolant is the cubic Hermite polynomial
      divdif2 = diff([endslopes(1,:);divdif;endslopes(2,:)])./dx([1 1],dd);
      pp = mkpp(x,...
      [(diff(divdif2)./dx(1,dd)).' ([2 -1]*divdif2).' ...
                                           endslopes(1,:).' yi(1,:).'],yd);
   end
elseif n==3&notaknot, % the interpolant is a parabola
   yi(2:3,:)=divdif;
   yi(3,:)=diff(divdif)/(x(3)-x(1));
   yi(2,:)=yi(2,:)-yi(3,:)*dx(1);
   pp = mkpp([x(1),x(3)],yi([3 2 1],:).',yd);
else % set up the sparse, tridiagonal, linear system for the slopes at  X .
   b=zeros(n,yd);
   b(2:n-1,:)=3*(dx(2:n-1,dd).*divdif(1:n-2,:)+dx(1:n-2,dd).*divdif(2:n-1,:));
   if notaknot
      x31=x(3)-x(1);xn=x(n)-x(n-2);
      b(1,:)=((dx(1)+2*x31)*dx(2)*divdif(1,:)+dx(1)^2*divdif(2,:))/x31;
      b(n,:)=...
      (dx(n-1)^2*divdif(n-2,:)+(2*xn+dx(n-1))*dx(n-2)*divdif(n-1,:))/xn;
   else
      x31 = 0; xn = 0; b([1 n],:) = dx([2 n-2],dd).*endslopes;
   end
   c = spdiags([ [dx(2:n-1);xn;0] ...
        [dx(2);2*[dx(2:n-1)+dx(1:n-2)];dx(n-2)] ...
        [0;x31;dx(1:n-2)] ],[-1 0 1],n,n);

   % sparse linear equation solution for the slopes
   mmdflag = spparms('autommd');
   spparms('autommd',0);
   s=c\b;
   spparms('autommd',mmdflag);
   % convert to pp form
   c4=(s(1:n-1,:)+s(2:n,:)-2*divdif(1:n-1,:))./dx(:,dd);
   c3=(divdif(1:n-1,:)-s(1:n-1,:))./dx(:,dd) - c4;
   pp=mkpp(x.', ...
      reshape([(c4./dx(:,dd)).' c3.' s(1:n-1,:).' yi(1:n-1,:).'], ...
               (n-1)*yd,4),yd);
end
if nargin==2
   output=pp;
else
   output=ppval(pp,xx);
end

eight

原帖由 vib 于 2007-4-4 15:48 发表
%三次样条逼近
%由x计算h;
x=0:10;
y=;
h=zeros(1,10);
for i=1:10,
    h(i)=(x(i+1)-x(i));
end
%计算a,b;
a=zeros(1,11)
b=zeros(1,11)
for i=2:10,
   a(i)=(h(i-1)/(h(i-1)+h(i))),
   b(i) ...

要全面了解matlab的实现，建议阅读 DeBoor 的 A practical guide to Splines 一书

hao1982

MATLAB里面的函数都经过优化,在原来基础上可能会有不少改变,所以它并不一定按书上写的方法一模一样.