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')