quad_m.m

分别给出利用自适应Simpson积分的数值积分方法（QUAD.m）和自适应Lobatto的数值积分算法（QUADL.m）。

function [Q,fcnt] = quad_m(funfcn,a,b,tol,trace,varargin)
%QUAD_M   Numerically evaluate integral, adaptive Simpson quadrature.
%   difference with QUAD: QUAD_M accepts vector arguments
%
%   Q = QUAD(FUN,A,B) tries to approximate the integral of function
%   FUN from A to B to within an error of 1.e-6 using recursive
%   adaptive Simpson quadrature.  The function Y = FUN(X) should
%   accept a column-vector argument X and return a column-vector
%   result Y, the integrand evaluated at each element of X.
%   The function should return a matrix when one of the additional
%   arguments (see below) is a row-vecor. The resulting matrix should
%   have integrand evaluated at each element of X in the column-direction
%   and of the argument in the row-direction.
%
%
%   Q = QUAD(FUN,A,B,TOL) uses an absolute error tolerance of TOL 
%   instead of the default, which is 1.e-6.  Larger values of TOL
%   result in fewer function evaluations and faster computation,
%   but less accurate results.  The QUAD function in MATLAB 5.3 used
%   a less reliable algorithm and a default tolerance of 1.e-3.
%
%   [Q,FCNT] = QUAD(...) returns the number of function evaluations.
%
%   QUAD(FUN,A,B,TOL,TRACE) with non-zero TRACE shows the values
%   of [fcnt a b-a Q] during the recursion.
%
%   QUAD(FUN,A,B,TOL,TRACE,P1,P2,...) provides for additional 
%   arguments P1, P2, ... to be passed directly to function FUN,
%   FUN(X,P1,P2,...).  Pass empty matrices for TOL or TRACE to
%   use the default values.
%   The additional arguments can be vectors.
%
%   Use array operators .*, ./ and .^ in the definition of FUN
%   so that it can be evaluated with a vector argument.
%
%   Function QUADL may be more efficient with high accuracies
%   and smooth integrands.
%
%   Example:
%       FUN can be specified three different ways.
%
%       A string expression involving a single variable:
%          Q = quad('1./(x.^3-2*x-5)',0,2);
%
%       An inline object:
%          F = inline('1./(x.^3-2*x-5)');
%          Q = quad(F,0,2);
%
%       A function handle:
%          Q = quad(@myfun,0,2);
%          where myfun.m is an M-file:
%             function y = myfun(x)
%             y = 1./(x.^3-2*x-5);
%
%   See also QUADL, DBLQUAD, INLINE, @.

%   Based on "adaptsim" by Walter Gander.  
%   Ref: W. Gander and W. Gautschi, "Adaptive Quadrature Revisited", 1998.
%   http://www.inf.ethz.ch/personal/gander
%   Copyright 1984-2001 The MathWorks, Inc. 

%   Based on QUAD. (see above)
%   M.F.P. Tolsma, Signals, Systems and Control Group, Applied Physics, TU Delft
%   http://www.tn.tudelft.nl/mmr
%   copyright remains by author

%   $Revision: 1.4 $  $Date: 2002/01/21 11:45:00 $

mxsize=1;
for cnt=1:length(varargin)
    tval=length(varargin{cnt});
    if (tval>1)
        if mxsize==1
            mxsize=tval;
        else
            if ~(mxsize==tval)
                error('The optional arguments must either be a scalar or an fixed sized row-vector')
            end;
        end;
    end;
end;

f = fcnchk(funfcn);
if nargin < 4 | isempty(tol), tol = 1.e-6; end;
if nargin < 5 | isempty(trace), trace = 0; end;

% Initialize with three unequal subintervals.
h = 0.13579*(b-a);
x = [a a+h a+2*h (a+b)/2 b-2*h b-h b];
y = feval(f, x, varargin{:});
fcnt = 7;

% Fudge endpoints to avoid infinities.
% for each element of vector y
if ~all(isfinite(y(:,1))) | ~all(isfinite(y(:,7))) %check only when there are at least some problems
    for cnt=1:mxsize
        if ~isfinite(y(cnt,1))
            for cnt2=1:length(varargin)
                varsubset{cnt2}=varargin{cnt2}(1);
            end;
            y(cnt,1) = feval(f,a+eps*(b-a),varsubset{:});
            fcnt = fcnt+1;
        end;
        if ~isfinite(y(cnt,7))
            for cnt2=1:length(varargin)
                if length(varargin{cnt2})>1
                    varsubset{cnt2}=varargin{cnt2}(13)
                else
                    varsubset{cnt2}=varargin{cnt2}(1);
                end;
            end;
            y(cnt,13) = feval(f,a+eps*(b-a),varsubset{:});
            fcnt = fcnt+1;
        end;
    end;
end;

% Call the recursive core integrator.
hmin = eps/1024*abs(b-a);
[Q(:,1),fcnt,warn(1)] = ...
   quadstep(f,x(1),x(3),y(:,1),y(:,2),y(:,3),tol,trace,fcnt,hmin,varargin{:});
[Q(:,2),fcnt,warn(2)] = ...
   quadstep(f,x(3),x(5),y(:,3),y(:,4),y(:,5),tol,trace,fcnt,hmin,varargin{:});
[Q(:,3),fcnt,warn(3)] = ...
   quadstep(f,x(5),x(7),y(:,5),y(:,6),y(:,7),tol,trace,fcnt,hmin,varargin{:});
Q = sum(Q,2);
warn = max(warn);

switch warn
   case 1
      warning('Minimum step size reached; singularity possible.')
   case 2
      warning('Maximum function count exceeded; singularity likely.')
   case 3
      warning('Infinite or Not-a-Number function value encountered.')
   otherwise
      if ~all(isfinite(Q))
          warning('Some Infinite or Not-a-Number function values encountered.')
      end
end

% ------------------------------------------------------------------------

function [Q,fcnt,warn] = quadstep (f,a,b,fa,fc,fb,tol,trace,fcnt,hmin,varargin)
%QUADSTEP  Recursive core routine for function QUAD.

maxfcnt = 10000;

% Evaluate integrand twice in interior of subinterval [a,b].
h = b - a;
c = (a + b)/2;
if abs(h) < hmin | c == a | c == b
   % Minimum step size reached; singularity possible.
   Q = h*fc;
   warn = 1;
   return
end
x = [(a + c)/2, (c + b)/2];
y = feval(f, x, varargin{:});
fcnt = fcnt + 2;
if fcnt > maxfcnt
   % Maximum function count exceeded; singularity likely.
   Q = h*fc;
   warn = 2;
   return
end
fd = y(:,1);
fe = y(:,2);

% Three point Simpson's rule.
Q1 = (h/6)*(fa + 4*fc + fb);

% Five point double Simpson's rule.
Q2 = (h/12)*(fa + 4*fd + 2*fc + 4*fe + fb);

% One step of Romberg extrapolation.
Q = Q2 + (Q2 - Q1)/15;

if ~any(isfinite(Q))    %all are infinite or NAN (no one is finite)
   % Infinite or Not-a-Number function value encountered.
   warn = 3;
   return
end

% Check accuracy of integral over this subinterval.
if all((abs(Q2 - Q) <= tol))   %all are below than tolerance
   warn = 0;
   return
end;

varlist=find(isfinite(Q) & (abs(Q2 - Q) > tol));    %these must still be done
if ~(length(varlist)==length(fa))             %more arguments as the ones that must be done
    targ=varargin;
    for cnt=1:length(targ)
        if length(targ{cnt})>1
            varargin{cnt}=targ{cnt}(varlist);
        else
            varargin{cnt}=targ{cnt};
        end;
    end;
end;


if trace
   disp(sprintf('%8.0f %16.10f %18.8e %16.10f',fcnt,a,h,Q))
end

% Subdivide into two subintervals.
[Qac,fcnt,warnac] = quadstep(f,a,c,fa(varlist),fd(varlist),fc(varlist),tol,trace,fcnt,hmin,varargin{:});
[Qcb,fcnt,warncb] = quadstep(f,c,b,fc(varlist),fe(varlist),fb(varlist),tol,trace,fcnt,hmin,varargin{:});
Q(varlist) = Qac + Qcb;
warn = max(warnac,warncb);