vpaintegral

Numerically Integrate Symbolic Expression

Numerically integrate the symbolic expression x^2 from 1 to 2.

syms x
vpaintegral(x^2, 1, 2)

ans =
2.33333

Numerically Integrate Symbolic Function

Numerically integrate the symbolic function y(x) = x² from 1 to 2.

syms y(x)
y(x) = x^2;
vpaintegral(y, 1, 2)

ans =
2.33333

High-Precision Numerical Integration

vpaintegral uses variable-precision arithmetic while the MATLAB^® integral function uses double-precision arithmetic. Using the default values of tolerance, vpaintegral can handle values that cause the MATLAB integral function to overflow or underflow.

Integrate besseli(5,25*u).*exp(-u*25) by using both integral and vpaintegral. The integral function returns NaN and issues a warning while vpaintegral returns the correct result.

syms u x
f = besseli(5,25*x).*exp(-x*25);
fun = @(u)besseli(5,25*u).*exp(-u*25);

usingIntegral = integral(fun, 0, 30)
usingVpaintegral = vpaintegral(f, 0, 30)

Warning: Infinite or Not-a-Number value encountered. 
usingIntegral =
   NaN

usingVpaintegral =
0.688424

Increase Precision Using Tolerances

The digits function does not affect vpaintegral. Instead, increase the precision of vpainteral by decreasing the integration tolerances. Conversely, increase the speed of numerical integration by increasing the tolerances. Control the tolerance used by vpaintegral by changing the relative tolerance RelTol and absolute tolerance AbsTol, which affect the integration through the condition

Numerically integrate besselj(0,x) from 0 to pi, to 32 significant figures by setting RelTol to 10^(-32). Turn off AbsTol by setting it to 0.

syms x
vpaintegral(besselj(0,x), [0 pi], 'RelTol', 1e-32, 'AbsTol', 0)

ans =
1.3475263146739901712314731279612

Using lower tolerance values increases precision at the cost of speed.

Complex Path Integration Using Waypoints

Integrate 1/(2*z-1) over the triangular path from 0 to 1+1i to 1-1i back to 0 by specifying waypoints.

syms z
vpaintegral(1/(2*z-1), [0 0], 'Waypoints', [1+1i 1-1i])

ans =
- 8.67362e-19 - 3.14159i

Reversing the direction of the integral, by changing the order of the waypoints and exchanging the limits, changes the sign of the result.

Multiple Integrals

Perform multiple integration by nesting calls to vpaintegral. Integrate

syms x y
vpaintegral(vpaintegral(x*y, x, [1 3]), y, [-1 2])

ans =
6.0

The limits of integration can be symbolic expressions or functions. Integrate over the triangular region 0 ≤ x ≤ 1 and |y| < x by specifying the limits of the integration over y in terms of x.

vpaintegral(vpaintegral(sin(x-y)/(x-y), y, [-x x]), x, [0 1])

ans =
0.89734