gusucode.com > symbolic工具箱matlab源码程序 > symbolic/@sym/taylor.m
function t = taylor(f, varargin)
% TAYLOR(f) is the fifth order Taylor polynomial approximation
% of f about the point x=0 (also known as fifth order
% Maclaurin polynomial), where x is obtained via symvar(f,1).
%
% TAYLOR(f,x) is the fifth order Taylor polynomial approximation
% of f with respect to x about x=0. x can be a vector.
% In case x is a vector, multivariate expansion about x(1)=0,
% x(2)=0,... is used.
%
% TAYLOR(f,x,a) is the fifth order Taylor polynomial approximation
% of f with respect to x about the point a. x and a can be
% vectors. If x is a vector and a is scalar, then a is
% expanded into a vector of the same size as x with all
% components equal to a. If x and a both are vectors, then
% they must have same length.
% In case x and a are vectors, multivariate expansion about
% x(1)=a(1),x(2)=a(2),... is used.
%
% In addition to that, the calls
%
% TAYLOR(f,'PARAM1',val1,'PARAM2',val2,...)
% TAYLOR(f,x,'PARAM1',val1,'PARAM2',val2,...)
% TAYLOR(f,x,a,'PARAM1',val1,'PARAM2',val2,...)
%
% can be used to specify one or more of the following parameter
% name/value pairs:
%
% Parameter Value
%
% 'ExpansionPoint' Compute the Taylor polynomial approximation
% about the point a. a can be a vector. If x is a
% vector, then a has to be of the same length as x.
% If a is scalar and x is a vector, a is expanded into
% a vector of the same length as x with all components
% equal to a. Note that if x is not given as in
% taylor(f,'ExpansionPoint',a), then a must be
% scalar (since x is determined via symvar(f,1)).
% It is always possible to specify the expansion
% point as third argument without explicitly using
% a parameter value pair.
%
% 'Order' Compute the Taylor polynomial approximation with
% order n-1, where n has to be a positive integer. The
% default value n=6 is used.
%
% 'OrderMode' Compute the Taylor polynomial approximation using
% relative or absolute order. 'Absolute' order is the
% truncation order of the computed series. 'Relative'
% order n means the exponents of x in the computed
% series range from some leading order v to the highest
% exponent v + n - 1 (i.e., the exponent of x in the
% Big-Oh term is v + n). In this case, n essentially
% is the "number of x powers" in the computed series
% if the series involves all integer powers of x
%
% Examples:
% syms x y z;
%
% taylor(exp(-x))
% returns x^4/24 - x^5/120 - x^3/6 + x^2/2 - x + 1
%
% taylor(sin(x),x,pi/2,'Order',6)
% returns (pi/2 - x)^4/24 - (pi/2 - x)^2/2 + 1
%
% taylor(sin(x)*cos(y)*exp(x),[x y z],[0 0 0],'Order',4)
% returns x - (x*y^2)/2 + x^2 + x^3/3
%
% taylor(exp(-x),x,'OrderMode','Relative','Order',8)
% returns - x^7/5040 + x^6/720 - x^5/120 + x^4/24 - x^3/6 + ...
% x^2/2 - x + 1
%
% taylor(log(x),x,'ExpansionPoint',1,'Order',4)
% returns x - 1 - 1/2*(x - 1)^2 + 1/3*(x - 1)^3
%
% taylor([exp(x),cos(y)],[x,y],'ExpansionPoint',[1 1],'Order',4)
% returns exp(1) + exp(1)*(x - 1) + (exp(1)*(x - 1)^2)/2 + ...
% (exp(1)*(x - 1)^3)/6'), cos(1) + (sin(1)*(y - 1)^3)/6 - ...
% sin(1)*(y - 1) - (cos(1)*(y - 1)^2)/2
%
% taylor(exp(z)/(x - y),[x,y,z],'ExpansionPoint',[Inf,0,0], ...
% 'OrderMode','Absolute','Order',6)
% returns y^2/x^3 + z^2/(2*x) + z^3/(6*x) + z^4/(24*x) + y/x^2 + ...
% z/x + 1/x + (y*z)/x^2 + (y*z^2)/(2*x^2)
%
% See also SYM/SYMVAR, SYM/SYMSUM, SYM/DIFF, SUBS.
% Copyright 1993-2014 The MathWorks, Inc.
p = inputParser;
p.addRequired('f');
if ~isa(f,'sym'), f = sym(f); end
x = symvar(f,1);
p.addOptional('x', x, @(x) isvector(x) && isAllVars(x));
p.addOptional('a',sym(0));
p.addParameter('Order', 6, @isPositiveInteger);
p.addParameter('ExpansionPoint', sym([]), @(x) ~isempty(x));
p.addParameter('OrderMode', 'absolute', @(x) any(validatestring(x,{'absolute','relative'})));
p.parse(f, varargin{:});
ordermode = validatestring(p.Results.OrderMode, {'absolute','relative'});
if strcmp(ordermode,'relative')
options = ['RelativeOrder = ' int2str(double(p.Results.Order))];
else
options = ['AbsoluteOrder = ' int2str(double(p.Results.Order))];
end
f = p.Results.f;
x = p.Results.x;
a = p.Results.a;
b = p.Results.ExpansionPoint;
if ~isempty(b)
a = b;
end
if ~isa(f,'sym'), f = sym(f); end
if builtin('numel',f) ~= 1, f = normalizesym(f); end
if ~isa(x,'sym'), x = sym(x); end
if builtin('numel',x) ~= 1, x = normalizesym(x); end
if ~isa(a,'sym'), a = sym(a); end
if builtin('numel',a) ~= 1, a = normalizesym(a); end
if isempty(x)
t = f;
return;
end
tSym = mupadmex('symobj::taylor',f.s,x.s,a.s,options);
t = privResolveOutput(tSym, f);