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sym/taylor
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
以x自变量的函数y在x=2处的泰勒展开6项：taylor(exp(-x),x,2,'Order',6)
>> syms x y z;
>> taylor(exp(x))
ans =
x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1
>> taylor(exp(-x),x,2,'Order',6)
ans =
exp(-2) - exp(-2)*(x - 2) + (exp(-2)*(x - 2)^2)/2 - (exp(-2)*(x - 2)^3)/6 + (exp(-2)*(x - 2)^4)/24 - (exp(-2)*(x - 2)^5)/120
>> taylor(exp(-x),x,'Order',6)
ans =
- x^5/120 + x^4/24 - x^3/6 + x^2/2 - x + 1
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