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wurongjun at 2017-07-31 11:49:49

你好!我的Matlab是舊版!

麻煩你把你的quadgk函數代碼貼一個上來,謝謝啦!

用命令 type quadgk就可以看到源代碼,在復制粘貼即可!

function [q,errbnd] = quadgk(FUN,a,b,varargin)

%QUADGK??Numerically evaluate integral, adaptive Gauss-Kronrod quadrature.

%? ?Q = QUADGK(FUN,A,B) attempts to approximate the integral of

%? ?scalar-valued function FUN from A to B using high order global adaptive

%? ?quadrature and default error tolerances. The function Y=FUN(X) should

%? ?accept a vector argument X and return a vector result Y, the integrand

%? ?evaluated at each element of X. FUN must be a function handle. A and B

%? ?can be -Inf or Inf. If both are finite, they can be complex. If at

%? ?least one is complex, the integral is approximated over a straight line

%? ?path from A to B in the complex plane.

%

%? ?[Q,ERRBND] = QUADGK(...). ERRBND is an approximate upper bound on the

%? ?absolute error, |Q - I|, where I denotes the exact value of the

%? ?integral.

%

%? ?[Q,ERRBND] = QUADGK(FUN,A,B,PARAM1,VAL1,PARAM2,VAL2,...) performs

%? ?the integration with specified values of optional parameters. The

%? ?available parameters are

%

%? ?'AbsTol', absolute error tolerance

%? ?'RelTol', relative error tolerance

%

%? ?? ? QUADGK attempts to satisfy ERRBND <= max(AbsTol,RelTol*|Q|). This

%? ?? ? is absolute error control when |Q| is sufficiently small and

%? ?? ? relative error control when |Q| is larger. A default tolerance

%? ?? ? value is used when a tolerance is not specified. The default value

%? ?? ? of 'AbsTol' is 1.e-10 (double), 1.e-5 (single). The default value

%? ?? ? of 'RelTol' is 1.e-6 (double), 1.e-4 (single). For pure absolute

%? ?? ? error control use

%? ?? ?? ?Q = quadgk(FUN,A,B,'AbsTol',ATOL,'RelTol',0)

%? ?? ? where ATOL > 0. For pure relative error control use

%? ?? ?? ?Q = quadgk(FUN,A,B,'RelTol',RTOL,'AbsTol',0)

%? ?? ? Except when using pure absolute error control, the minimum relative

%? ?? ? tolerance is 100*eps(class(Q)).

%

%? ?'Waypoints', vector of integration waypoints

%

%? ?? ? If FUN(X) has discontinuities in the interval of integration, the

%? ?? ? locations should be supplied as a 'Waypoints' vector. When A, B,

%? ?? ? and the waypoints are all real, only the waypoints between A and B

%? ?? ? are used, and they are used in sorted order.??Note that waypoints

%? ?? ? are not intended for singularities in FUN(X). Singular points

%? ?? ? should be handled by making them endpoints of separate integrations

%? ?? ? and adding the results.

%

%? ?? ? If A, B, or any entry of the waypoints vector is complex, the

%? ?? ? integration is performed over a sequence of straight line paths in

%? ?? ? the complex plane, from A to the first waypoint, from the first

%? ?? ? waypoint to the second, and so forth, and finally from the last

%? ?? ? waypoint to B.

%

%? ?'MaxIntervalCount', maximum number of intervals allowed

%

%? ?? ? The 'MaxIntervalCount' parameter limits the number of intervals

%? ?? ? that QUADGK will use at any one time after the first iteration. A

%? ?? ? warning is issued if QUADGK returns early because of this limit.

%? ?? ? The default value is 650. Increasing this value is not recommended,

%? ?? ? but it may be appropriate when ERRBND is small enough that the

%? ?? ? desired accuracy has nearly been achieved.

%

%? ?Consider using INTEGRAL instead of QUADGK. INTEGRAL uses the same

%? ?method as QUADGK and also supports vector-valued integrands.

%

%? ?Example:

%? ?Integrate f(x) = exp(-x^2)*log(x)^2 from 0 to infinity:

%? ?? ?f = @(x) exp(-x.^2).*log(x).^2

%? ?? ?Q = quadgk(f,0,Inf)

%

%? ?Example:

%? ?To use a parameter in the integrand:

%? ?? ?f = @(x,c) 1./(x.^3-2*x-c);

%? ?? ?Q = quadgk(@(x)f(x,5),0,2);

%

%? ?Example:

%? ?Integrate f(z) = 1/(2z-1) in the complex plane over the triangular

%? ?path from 0 to 1+1i to 1-1i to 0:

%? ?? ?Q = quadgk(@(z)1./(2*z-1),0,0,'Waypoints',[1+1i,1-1i])

%

%? ?Class support for inputs A, B, and the output of FUN:

%? ?? ?float: double, single

%

%? ?See also INTEGRAL, INTEGRAL2, INTEGRAL3, QUAD, QUADL, QUADV, QUAD2D,

%? ?DBLQUAD, TRIPLEQUAD, FUNCTION_HANDLE

%? ?Based on "quadva" by Lawrence F. Shampine.

%? ?Ref: L.F. Shampine, "Vectorized Adaptive Quadrature in Matlab",

%? ?Journal of Computational and Applied Mathematics 211, 2008, pp.131-140.

%? ?Copyright 2007-2013 The MathWorks, Inc.

% Variable names in all caps are referenced in nested functions.

% Validate the first three inputs.

narginchk(3,inf);

if ~isa(FUN,'function_handle')

error(message('MATLAB:quadgk:funArgNotHandle'));

end

if ~(isscalar(a) && isfloat(a) && isscalar(b) && isfloat(b))

error(message('MATLAB:quadgk:invalidEndpoint'));

end

% Gauss-Kronrod (7,15) pair. Use symmetry in defining nodes and weights.

pnodes = [ ...

0.2077849550078985; 0.4058451513773972; 0.5860872354676911; ...

0.7415311855993944; 0.8648644233597691; 0.9491079123427585; ...

0.9914553711208126];

pwt = [ ...

0.2044329400752989, 0.1903505780647854, 0.1690047266392679, ...

0.1406532597155259, 0.1047900103222502, 0.06309209262997855, ...

0.02293532201052922];

pwt7 = [0,0.3818300505051189,0,0.2797053914892767,0,0.1294849661688697,0];

NODES = [-pnodes(end:-1:1); 0; pnodes];

WT = [pwt(end:-1:1), 0.2094821410847278, pwt];

EWT = WT - [pwt7(end:-1:1), 0.4179591836734694, pwt7];

% Fixed parameters.

DEFAULT_DOUBLE_ABSTOL = 1.e-10;

DEFAULT_SINGLE_ABSTOL = 1.e-5;

DEFAULT_DOUBLE_RELTOL = 1.e-6;

DEFAULT_SINGLE_RELTOL = 1.e-4;

MININTERVALCOUNT = 10; % Minimum number subintervals to start.

% Process optional input.

p = inputParser;

p.addParamValue('AbsTol',[],@validateAbsTol);

p.addParamValue('RelTol',[],@validateRelTol);

p.addParamValue('MaxIntervalCount',650,@validateMaxIntervalCount);

p.addParamValue('Waypoints',[],@validateWaypoints);

p.parse(varargin{:});

optionStruct = p.Results;

ATOL = optionStruct.AbsTol;

RTOL = optionStruct.RelTol;

MAXINTERVALCOUNT = optionStruct.MaxIntervalCount;

WAYPOINTS = optionStruct.Waypoints(

.';

% Initialize the FIRSTFUNEVAL variable.??Some checks will be done just

% after the first evaluation.

FIRSTFUNEVAL = true;

% Handle contour integration.

if ~(isreal(a) && isreal(b) && isreal(WAYPOINTS))

tinterval = [a,WAYPOINTS,b];

if any(~isfinite(tinterval))

error(message('MATLAB:quadgk:nonFiniteContourError'));

end

% A and B should not be needed, so we do not define them here.

% Perform the contour integration.

[q,errbnd] = vadapt(@evalFun,tinterval);

return

end

% Define A and B and note the direction of integration on real axis.

if b < a

% Integrate left to right and change sign at the end.

reversedir = true;

A = b;

B = a;

else

reversedir = false;

A = a;

B = b;

end

% Trim and sort the waypoints vector.

WAYPOINTS = sort(WAYPOINTS(WAYPOINTS>A & WAYPOINTS

% Construct interval vector with waypoints.

interval = [A, WAYPOINTS, B];

% Extract A and B from interval vector to regularize possible mixed

% single/double inputs.

A = interval(1);

B = interval(end);

% Identify the task and perform the integration.

if A == B

% Handles both finite and infinite cases.

% Return zero or nan of the appropriate class.

q = midpArea(@evalFun,A,B);

errbnd = q;

elseif isfinite(A) && isfinite(B)

if numel(interval) > 2

% Analytical transformation suggested by K.L. Metlov:

alpha = 2*sin( asin((A + B - 2*interval(2:end-1))/(A - B))/3 );

interval = [-1,alpha,1];

else

interval = [-1,1];

end

[q,errbnd] = vadapt(@f1,interval);

elseif isfinite(A) && isinf(B)

if numel(interval) > 2

alpha = sqrt(interval(2:end-1) - A);

interval = [0,alpha./(1+alpha),1];

else

interval = [0,1];

end

[q,errbnd] = vadapt(@f2,interval);

elseif isinf(A) && isfinite(B)

if numel(interval) > 2

alpha = sqrt(B - interval(2:end-1));

interval = [-1,-alpha./(1+alpha),0];

else

interval = [-1,0];

end

[q,errbnd] = vadapt(@f3,interval);

elseif isinf(A) && isinf(B)

if numel(interval) > 2

% Analytical transformation suggested by K.L. Metlov:

alpha = tanh( asinh(2*interval(2:end-1))/2 );

interval = [-1,alpha,1];

else

interval = [-1,1];

end

[q,errbnd] = vadapt(@f4,interval);

else % i.e., if isnan(a) || isnan(b)

q = midpArea(@evalFun,A,B);

errbnd = q;

end

% Account for integration direction.

if reversedir

q = -q;

end

%==Nested functions=========================================================

function [q,errbnd] = vadapt(f,tinterval)

% Iterative routine to perform the integration.

% Compute the path length and split tinterval if needed.

[tinterval,pathlen] = split(tinterval,MININTERVALCOUNT);

if pathlen == 0

% Test case: quadgk(@(x)x,1+1i,1+1i);

q = midpArea(f,tinterval(1),tinterval(end));

errbnd = q;

return

end

% Initialize array of subintervals of [a,b].

subs = [tinterval(1:end-1);tinterval(2:end)];

% Initialize partial sums.

q_ok = 0;

err_ok = 0;

% Initialize main loop

while true

% SUBS contains subintervals of [a,b] where the integral is not

% sufficiently accurate. The first row of SUBS holds the left end

% points and the second row, the corresponding right endpoints.

midpt = sum(subs)/2;? ?% midpoints of the subintervals

halfh = diff(subs)/2;??% half the lengths of the subintervals

x = bsxfun(@plus,NODES*halfh,midpt);

x = reshape(x,1,[]);? ?% function f expects a row vector

[fx,too_close] = f(x);

% Quit if mesh points are too close.

if too_close

break

end

fx = reshape(fx,numel(WT),[]);

% Quantities for subintervals.

qsubs = (WT*fx) .* halfh;

errsubs = (EWT*fx) .* halfh;

% Calculate current values of q and tol.

q = sum(qsubs) + q_ok;

tol = max(ATOL,RTOL*abs(q));

% Locate subintervals where the approximate integrals are

% sufficiently accurate and use them to update the partial

% error sum.

ndx = find(abs(errsubs) <= (2*tol/pathlen)*abs(halfh));

err_ok = err_ok + sum(errsubs(ndx));

% Remove errsubs entries for subintervals with accurate

% approximations.

errsubs(ndx) = [];

% The approximate error bound is constructed by adding the

% approximate error bounds for the subintervals with accurate

% approximations to the 1-norm of the approximate error bounds

% for the remaining subintervals.??This guards against

% excessive cancellation of the errors of the remaining

% subintervals.

errbnd = abs(err_ok) + norm(errsubs,1);

% Check for nonfinites.

if ~(isfinite(q) && isfinite(errbnd))

warning(message('MATLAB:quadgk:NonFiniteValue'));

break

end

% Test for convergence.

if errbnd <= tol

break

end

% Remove subintervals with accurate approximations.

subs(:,ndx) = [];

if isempty(subs)

break

end

% Update the partial sum for the integral.

q_ok = q_ok + sum(qsubs(ndx));

% Split the remaining subintervals in half. Quit if splitting

% results in too many subintervals.

nsubs = 2*size(subs,2);

if nsubs > MAXINTERVALCOUNT

warning(message('MATLAB:quadgk:MaxIntervalCountReached',sprintf('%9.1e',errbnd),nsubs));

break

end

midpt(ndx) = []; % Remove unneeded midpoints.

subs = reshape([subs(1,

; midpt; midpt; subs(2,

],2,[]);

end

end % vadapt

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

function q = midpArea(f,a,b)

% Return q = (b-a)*f((a+b)/2). Although formally correct as a low

% order quadrature formula, this function is only used to return

% nan or zero of the appropriate class when a == b, isnan(a), or

% isnan(b).

x = (a+b)/2;

if isfinite(a) && isfinite(b) && ~isfinite(x)

% Treat overflow, e.g. when finite a and b > realmax/2

x = a/2 + b/2;

end

fx = f(x);

if ~isfinite(fx)

warning(message('MATLAB:quadgk:NonFiniteValue'));

end

q = (b-a)*fx;

end % midpArea

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

function [fx,too_close] = evalFun(x)

% Evaluate the integrand.

if FIRSTFUNEVAL

% Don't check the closeness of the mesh on the first iteration.

too_close = false;

fx = FUN(x);

finalInputChecks(x,fx);

FIRSTFUNEVAL = false;

else

too_close = checkSpacing(x);

if too_close

fx = [];

else

fx = FUN(x);

end

end

end % evalFun

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

function [y,too_close] = f1(t)

% Transform to weaken singularities at both ends: [a,b] -> [-1,1]

tt = 0.25*(B-A)*t.*(3 - t.^2) + 0.5*(B+A);

[y,too_close] = evalFun(tt);

if ~too_close

y = 0.75*(B-A)*y.*(1 - t.^2);

end

end % f1

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

function [y,too_close] = f2(t)

% Transform to weaken singularity at left end: [a,Inf) -> [0,Inf).

% Then transform to finite interval: [0,Inf) -> [0,1].

tt = t ./ (1 - t);

t2t = A + tt.^2;

[y,too_close] = evalFun(t2t);

if ~too_close

y =??2*tt .* y ./ (1 - t).^2;

end

end % f2

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

function [y,too_close] = f3(t)

% Transform to weaken singularity at right end: (-Inf,b] -> (-Inf,b].

% Then transform to finite interval: (-Inf,b] -> (-1,0].

tt = t ./ (1 + t);

t2t = B - tt.^2;

[y,too_close] = evalFun(t2t);

if ~too_close

y = -2*tt .* y ./ (1 + t).^2;

end

end % f3

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

function [y,too_close] = f4(t)

% Transform to finite interval: (-Inf,Inf) -> (-1,1).

tt = t ./ (1 - t.^2);

[y,too_close] = evalFun(tt);

if ~too_close

y = y .* (1 + t.^2) ./ (1 - t.^2).^2;

end

end % f4

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

function too_close = checkSpacing(x)

ax = abs(x);

tcidx = find(abs(diff(x)) <= 100*eps(class(x))*max(ax(1:end-1),ax(2:end)),1);

too_close = ~isempty(tcidx);

if too_close

warning(message('MATLAB:quadgk:MinStepSize',num2str(x(tcidx),6)));

end

end % checkSpacing

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

function [x,pathlen] = split(x,minsubs)

% Split subintervals in the interval vector X so that, to working

% precision, no subinterval is longer than 1/MINSUBS times the

% total path length. Removes subintervals of zero length, except

% that the resulting X will always has at least two elements on

% return, i.e., if the total path length is zero, X will be

% collapsed into a single interval of zero length.??Also returns

% the integration path length.

absdx = abs(diff(x));

if isreal(x)

pathlen = x(end) - x(1);

else

pathlen = sum(absdx);

end

if pathlen > 0

udelta = minsubs/pathlen;

nnew = ceil(absdx*udelta) - 1;

idxnew = find(nnew > 0);

nnew = nnew(idxnew);

for j = numel(idxnew):-1:1

k = idxnew(j);

nnj = nnew(j);

% Calculate new points.

newpts = x(k) + (1:nnj)./(nnj+1)*(x(k+1)-x(k));

% Insert the new points.

x = [x(1:k),newpts,x(k+1:end)];

end

end

% Remove useless subintervals.

x(abs(diff(x))==0) = [];

if isscalar(x)

% Return at least two elements.

x = [x(1),x(1)];

end

end % split

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

function finalInputChecks(x,fx)

% Do final input validation with sample input and outputs to the

% integrand function.

% Check classes.

if ~(isfloat(x) && isfloat(fx))

error(message('MATLAB:quadgk:UnsupportedClass'));

end

% Check sizes.

if ~isequal(size(x),size(fx))

error(message('MATLAB:quadgk:FxNotSameSizeAsX'));

end

outcls = superiorfloat(x,fx);

outdbl = strcmp(outcls,'double');

% Validate tolerances and apply defaults.

if isempty(RTOL)

if outdbl

RTOL = DEFAULT_DOUBLE_RELTOL;

else

RTOL = DEFAULT_SINGLE_RELTOL;

end

end

if isempty(ATOL)

if outdbl

ATOL = DEFAULT_DOUBLE_ABSTOL;

else

ATOL = DEFAULT_SINGLE_ABSTOL;

end

end

% Make sure that RTOL >= 100*eps(outcls) except when

% using pure absolute error control (ATOL>0 && RTOL==0).

if ~(ATOL > 0 && RTOL == 0) && RTOL < 100*eps(outcls)

RTOL = 100*eps(outcls);

warning(message('MATLAB:quadgk:increasedRelTol', outcls, sprintf( '%g', RTOL )));

end

if outdbl

% Single RTOL or ATOL should not force any single precision

% computations.

RTOL = double(RTOL);

ATOL = double(ATOL);

end

end % finalInputChecks

%==========================================================================

end % quadgk

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

function p = validateAbsTol(x)

if ~(isfloat(x) && isscalar(x) && isreal(x) && x >= 0)

error(message('MATLAB:quadgk:invalidAbsTol'));

end

p = true;

end

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

function p = validateRelTol(x)

if ~(isfloat(x) && isscalar(x) && isreal(x) && x >= 0)

error(message('MATLAB:quadgk:invalidRelTol'));

end

p = true;

end

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

function p = validateWaypoints(x)

if ~(isvector(x) || isequal(x,[]))

error(message('MATLAB:quadgk:WaypointsNotVector'));

elseif any(~isfinite(x))

error(message('MATLAB:quadgk:WaypointsNotFinite'));

end

p = true;

end

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

function p = validateMaxIntervalCount(x)

if ~(isscalar(x) && isreal(x) && x > 0 && floor(x) == x)

error(message('MATLAB:quadgk:invalidMaxIntervalCount'));

end

p = true;

end

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

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