Note: Use only in the MuPAD Notebook Interface. This functionality does not run in MATLAB. |
Taylor series expansions approximate an arbitrary expression with a polynomial. The number of terms in a series expansion determines the accuracy of the approximation. The number of terms in a series expansion depends on the truncation order of the expansion. By default, MuPAD^{®} computes the first six terms of series expansions:
taylor(exp(x), x)
The number of terms includes the terms with coefficients equal
to zero. For example, the taylor series expansion of cos(x)
includes
the terms 0 x, 0 x^{3},
and 0 x^{5}. MuPAD computes
these terms, but does not display them:
taylor(cos(x), x)
Suppose, you want to approximate an exponential function with
the polynomial expression around x = 0
. Use the
third parameter in taylor
to
specify the order of series expansion. For example, compute the series
expansions approx1
specifying the truncation order 3
.
Compare the result with the series expansion computed for the default
order:
exact := exp(x): approx1 := taylor(exp(x), x, 3); approx2 := taylor(exp(x), x)
Plot the exact expression, exact
, and its
taylor series expansions, approx1
and approx2
,
in the same coordinate system. The series expansion with more terms
approximates the expression exp(x)
better:
plot( plot::Function2d(exact, x = -PI..PI, Legend = "exp(x)", Color = RGB::Red), plot::Function2d(approx2, x = -PI..PI, Legend = "approximation of exp(x), up to O(x^6)", Color = RGB::Blue), plot::Function2d(approx1, x = -PI..PI, Legend = "approximation of exp(x), up to O(x^3)", Color = RGB::Green) )
There are two ways to change the truncation order for series expansions:
Locally by passing the truncation order as the third
parameter to taylor
.
By using this parameter, you specify the truncation order for a particular
series expansion. All other series expansions use the default order.
The parameter is available for the following commands: taylor
, mtaylor
, and series
. For more information,
see the help pages for these commands.
Globally by using the environment variable ORDER
.
When you change this variable, all series expansions use the new truncation
order.
To change the truncation order for a particular series expansion,
pass the new order as a third parameter to taylor
:
taylor(exp(x), x, 10)
To change the default truncation order for all series expansions,
modify the environment variable ORDER
:
ORDER := 7: taylor(exp(x), x)
The following computations use the new value of ORDER
:
taylor(sqrt(1 - x), x)
delete ORDER: taylor(sqrt(1 - x), x)