pade

Pade approximant

Syntax

pade(f,var)

pade(f,var,a)

pade(___,Name,Value)

Description

pade(f,var) returns the third-order Padé approximant of the expression f at var = 0. For details, see Padé Approximant.

If you do not specify var, then pade uses the default variable determined by symvar(f,1).

pade(f,var,a) returns the third-order Padé approximant of expression f at the point var = a.

pade(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments. You can specify Name,Value after the input arguments in any of the previous syntaxes.

Examples

Find Padé Approximant for Symbolic Expressions

Find the Padé approximant of sin(x). By default, pade returns a third-order Padé approximant.

syms x
pade(sin(x))

ans =
-(x*(7*x^2 - 60))/(3*(x^2 + 20))

Specify Expansion Variable

If you do not specify the expansion variable, symvar selects it. Find the Padé approximant of sin(x) + cos(y). The symvar function chooses x as the expansion variable.

syms x y
pade(sin(x) + cos(y))

ans =
(- 7*x^3 + 3*cos(y)*x^2 + 60*x + 60*cos(y))/(3*(x^2 + 20))

Specify the expansion variable as y. The pade function returns the Padé approximant with respect to y.

pade(sin(x) + cos(y),y)

ans =
(12*sin(x) + y^2*sin(x) - 5*y^2 + 12)/(y^2 + 12)

Approximate Value of Function at Particular Point

Find the value of tan(3*pi/4). Use pade to find the Padé approximant for tan(x) and substitute into it using subs to find tan(3*pi/4).

syms x
f = tan(x);
P = pade(f);
y = subs(P,x,3*pi/4)

y =
(pi*((9*pi^2)/16 - 15))/(4*((9*pi^2)/8 - 5))

Use vpa to convert y into a numeric value.

vpa(y)

ans =
-1.2158518789569086447244881326842

Increase Accuracy of Padé Approximant

You can increase the accuracy of the Padé approximant by increasing the order. If the expansion point is a pole or a zero, the accuracy can also be increased by setting OrderMode to relative. The OrderMode option has no effect if the expansion point is not a pole or zero.

Find the Padé approximant of tan(x) using pade with an expansion point of 0 and Order of [1 1]. Find the value of tan(1/5) by substituting into the Padé approximant using subs, and use vpa to convert 1/5 into a numeric value.

syms x
p11 = pade(tan(x),x,0,'Order',[1 1])
p11 = subs(p11,x,vpa(1/5))

p11 =
x
p11 =
0.2

Find the approximation error by subtracting p11 from the actual value of tan(1/5).

y = tan(vpa(1/5));
error = y - p11

error =
0.0027100355086724833213582716475345

Increase the accuracy of the Padé approximant by increasing the order using Order. Set Order to [2 2], and find the error.

p22 = pade(tan(x),x,0,'Order',[2 2])
p22 = subs(p22,x,vpa(1/5));
error = y - p22

p22 =
-(3*x)/(x^2 - 3)
error =
0.0000073328059697806186555689448317799

The accuracy increases with increasing order.

If the expansion point is a pole or zero, the accuracy of the Padé approximant decreases. Setting the OrderMode option to relative compensates for the decreased accuracy. For details, see Padé Approximant. Because the tan function has a zero at 0, setting OrderMode to relative increases accuracy. This option has no effect if the expansion point is not a pole or zero.

p22Rel = pade(tan(x),x,0,'Order',[2 2],'OrderMode','relative')
p22Rel = subs(p22Rel,x,vpa(1/5));
error = y - p22Rel

p22Rel =
(x*(x^2 - 15))/(3*(2*x^2 - 5))
error =
0.0000000084084014806113311713765317725998

The accuracy increases if the expansion point is a pole or zero and OrderMode is set to relative.

Plot Accuracy of Padé Approximant

Plot the difference between exp(x) and its Padé approximants of orders [1 1] through [4 4]. Use axis to focus on the region of interest. The plot shows that accuracy increases with increasing order of the Padé approximant.

syms x
expr = exp(x);

hold on
grid on

for i = 1:4
    fplot(expr - pade(expr,'Order',i))
end

axis([-4 4 -4 4])
legend('Order [1,1]','Order [2,2]','Order [3,3]','Order [4,4]',...
                                            'Location','Best')
title('Difference Between exp(x) and its Pade Approximant')
ylabel('Error')

Input Arguments

f — Input to approximate

symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input to approximate, specified as a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

var — Expansion variable

symbolic variable

Expansion variable, specified as a symbolic variable. If you do not specify var, then pade uses the default variable determined by symvar(f,1).

a — Expansion point

number | symbolic number | symbolic variable | symbolic function | symbolic expression

Expansion point, specified as a number, or a symbolic number, variable, function, or expression. The expansion point cannot depend on the expansion variable. You also can specify the expansion point as a Name,Value pair argument. If you specify the expansion point both ways, then the Name,Value pair argument takes precedence.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: pade(f,'Order',[2 2]) returns the Padé approximant of f of order m = 2 and n = 2.

'ExpansionPoint' — Expansion point

number | symbolic number | symbolic variable | symbolic function | symbolic expression

Expansion point, specified as a number, or a symbolic number, variable, function, or expression. The expansion point cannot depend on the expansion variable. You can also specify the expansion point using the input argument a. If you specify the expansion point both ways, then the Name,Value pair argument takes precedence.

'Order' — Order of Padé approximant

integer | vector of two integers | symbolic integer | symbolic vector of two integers

Order of the Padé approximant, specified as an integer, a vector of two integers, or a symbolic integer, or vector of two integers. If you specify a single integer, then the integer specifies both the numerator order m and denominator order n producing a Padé approximant with m = n. If you specify a vector of two integers, then the first integer specifies m and the second integer specifies n. By default, pade returns a Padé approximant with m = n = 3.

'OrderMode' — Flag that selects absolute or relative order for Padé approximant

'absolute' (default) | 'relative'

Flag that selects absolute or relative order for Padé approximant, specified as 'absolute' or 'relative'. The default value of 'absolute' uses the standard definition of the Padé approximant. If you set 'OrderMode' to 'relative', it only has an effect when there is a pole or a zero at the expansion point a. In this case, to increase accuracy, pade multiplies the numerator by (var - a)p where p is the multiplicity of the zero or pole at the expansion point. For details, see Padé Approximant.

More About

Padé Approximant

By default, pade approximates the function f(x) using the standard form of the Padé approximant of order [m, n] around x = x₀ which is

$$\frac{a_0+a_1\left(x-x_0\right)+\mathrm{...}+a_m{\left(x-x_0\right)}^m}{1+b_1\left(x-x_0\right)+\mathrm{...}+b_n{\left(x-x_0\right)}^n}.$$

When OrderMode is relative, and a pole or zero exists at the expansion point x = x₀, the pade function uses this form of the Padé approximant

$$\frac{{\left(x-x_0\right)}^p\left(a_0+a_1\left(x-x_0\right)+\mathrm{...}+a_m{\left(x-x_0\right)}^m\right)}{1+b_1\left(x-x_0\right)+\mathrm{...}+b_n{\left(x-x_0\right)}^n}.$$

The parameters p and a₀ are given by the leading order term f = a₀ (x - x₀)^p + O((x - x₀)^{p + 1}) of the series expansion of f around x = x₀. Thus, p is the multiplicity of the pole or zero at x₀.

Tips

• If you use both the third argument a and ExpansionPoint to specify the expansion point, the value specified via ExpansionPoint prevails.

Algorithms

• The parameters a₁,…,b_n are chosen such that the series expansion of the Padé approximant coincides with the series expansion of f to the maximal possible order.

• The expansion points ±∞ and ±i∞ are not allowed.

• When pade cannot find the Padé approximant, it returns the function call.

• For pade to return the Padé approximant, a Taylor or Laurent series expansion of f must exist at the expansion point.