积分与微分方程

李秋锐

function [Q,cnt] = quad8(funfcn,a,b,tol,trace,varargin)
%数值积分
%z=quad8('Fun',A,B,Tol,trace,p1,p2,L)
%   其中:"Fun"-表示被积函数的M函数名.
%        A,B-上﹑下限.
%        Tol-精度,缺省值为1e-3.
%        trace-非零时显示计算过程,缺省值为0.
%        p1,p2,L-参变量,一般缺省.
%例如求 exp(-x^2)在[-1，1]积分
%   先写M函数quadeg3fun.m
%                   function  y=quadeg3fun(x)
%                   y=exp(-x.^2);
%   用
%   z=quad8('quadeg3fun',-1,1)   
%
%QUAD8  Numerically evaluate integral, higher order method.
%   Q = QUAD8('F',A,B) approximates the integral of F(X) from A to B to
%   within a relative error of 1e-3 using an adaptive recursive Newton
%   Cotes 8 panel rule.  'F' is a string containing the name of the
%   function.  The function must return a vector of output values if
%   given a vector of input values.  Q = Inf is returned if an excessive
%   recursion level is reached, indicating a possibly singular integral.
%
%   Q = QUAD8('F',A,B,TOL) integrates to a relative error of TOL.  Use
%   a two element tolerance, TOL = [rel_tol abs_tol], to specify a
%   combination of relative and absolute error.
%
%   Q = QUAD8('F',A,B,TOL,TRACE) integrates to a relative error of TOL and
%   for non-zero TRACE traces the function evaluations with a point plot
%   of the integrand.
%
%   Q = QUAD8('F',A,B,TOL,TRACE,P1,P2,...) allows coefficients P1, P2, ...
%   to be passed directly to function F:   G = F(X,P1,P2,...).
%   To use default values for TOL or TRACE, you may pass in the empty
%   matrix ([]).
%
%   See also QUAD, DBLQUAD.

%   Cleve Moler, 5-08-88.
%   Copyright (c) 1984-98 by The MathWorks, Inc.
%   $Revision: 5.16 $  $Date: 1997/11/21 23:31:14 $

% [Q,cnt] = quad8(F,a,b,tol) also returns a function evaluation count.

if nargin < 4, tol = [1.e-3 0]; trace = 0; end
if nargin < 5, trace = 0; end
if isempty(tol), tol = [1.e-3 0]; end
if length(tol)==1, tol = [tol 0]; end
if isempty(trace), trace = 0; end
% QUAD8 usually does better than the default 1e-3.
h = b - a;

% Top level initialization, Newton-Cotes weights
w = [3956 23552 -3712 41984 -18160 41984 -3712 23552 3956]/14175;
x = a + (0:8)*(b-a)/8;
y = feval(funfcn,x,varargin{:});

yllow = min([min(real(y)) min(imag(y))]);
ylhi = max([max(real(y)) max(imag(y))]);
lims = [min(x) max(x) yllow ylhi];
ind = find(~isfinite(lims));
if ~isempty(ind)
    [mind,nind] = size(ind);
    lims(ind) = 1.e30*(-ones(mind,nind) .^ rem(ind,2));
end
if trace
    axis(lims);
% doesn't take care of complex case
    plot([a b],[real(y(1)) real(y(9))],'.'), hold on
    if ~isreal(y)
        plot([a b],[imag(y(1)) imag(y(9))],'+')
    end
    drawnow
end
lev = 1;

% Adaptive, recursive Newton-Cotes 8 panel quadrature
if ~isreal(y)
    Q0 = 1e30;
else
    Q0 = inf;
end
recur_lev_excess = 0;
[Q,cnt,recur_lev_excess] = quad8stp(funfcn,a,b,tol,lev,w,x,y,Q0,trace,recur_lev_excess,varargin{:});
cnt = cnt + 9;
if trace
    hold off
    axis('auto');
end
if (recur_lev_excess > 1)
    warning(sprintf('Recursion level limit reached %d times.', ...
         recur_lev_excess ))
end

%------------------------------------------------------------------
function [Q,cnt,recur_lev_excess] = quad8stp(FunFcn,a,b,tol,lev,w,x0,f0,Q0,trace,recur_lev_excess,varargin)
%QUAD8STP Recursive function used by QUAD8.
%   [Q,cnt] = quad8stp(F,a,b,tol,lev,w,f,Q0) tries to approximate
%   the integral of f(x) from a to b to within a relative error of tol.  
%   F is a string containing the name of f.  The remaining arguments are
%   generated by quad8 or by the recursion.  lev is the recursion level.
%   w is the weights in the 8 panel Newton Cotes formula.
%   x0 is a vector of 9 equally spaced abscissa is the interval.
%   f0 is a vector of the 9 function values at x.
%   Q0 is an approximate value of the integral.
%   See also QUAD8 and QUAD.

%   Cleve Moler, 5-08-88.

LEVMAX = 10;

% Evaluate function at midpoints of left and right half intervals.
x = zeros(1,17);
f = zeros(1,17);
x(1:2:17) = x0;
f(1:2:17) = f0;
x(2:2:16) = (x0(1:8) + x0(2:9))/2;
f(2:2:16) = feval(FunFcn,x(2:2:16),varargin{:});
if trace
    plot(x(2:2:16),f(2:2:16),'.');
    if ~isreal(f)
        plot(x(2:2:16),imag(f(2:2:16)),'+');
    end
    drawnow
end
cnt = 8;

% Integrate over half intervals.
h = (b-a)/16;
Q1 = h*w*f(1:9).';
Q2 = h*w*f(9:17).';
Q = Q1 + Q2;

% Recursively refine approximations.
if (abs(Q - Q0) > tol(1)*abs(Q)+tol(2)) & (lev <= LEVMAX)
   c = (a+b)/2;
   [Q1,cnt1,recur_lev_excess] = quad8stp(FunFcn,a,c,tol/2,lev+1,w,x(1:9),f(1:9),Q1,trace,recur_lev_excess,varargin{:});
   [Q2,cnt2,recur_lev_excess] = quad8stp(FunFcn,c,b,tol/2,lev+1,w,x(9:17),f(9:17),Q2,trace,recur_lev_excess,varargin{:});
   Q = Q1 + Q2;
   cnt = cnt + cnt1 + cnt2;
elseif (lev > LEVMAX)
   if ~recur_lev_excess
      warning('Recursion level limit reached in quad8. Singularity likely.')
      recur_lev_excess = 1;
   else
      recur_lev_excess = recur_lev_excess + 1;
   end
end

李秋锐

不错不错。。。。。。。。。。。。。。。。。。。。。。。。。。。。。