intgrat2

Purpose

Integrates the following double integral, using user-defined functions f, g1 and g2 and scalars a and b:

\[\int_{a}^{b}\int_{g_2(x)}^{g_1(x)} f(x,y)dydx\]

Format

y = intgrat2(&f, xl, gl)
Parameters:
  • &f (scalar) – pointer to the procedure containing the function to be integrated.
  • xl (2x1 or 2xN matrix) – the limits of x. These must be scalar limits.
  • gl (2x1 or 2xN matrix) –

    Function pointers to functions defining the limits of y.

    For xl and gl, the first row is the upper limit and the second row is the lower limit. N integrations are computed.

Returns:

y (Nx1 vector) – of the estimated integral(s) of \(f(x, y)\), evaluated between the limits given by xl and gl.

Global Input

_intord scalar, the order of the integration. The larger _intord, the more precise the final result will be. _intord may be set to 2, 3, 4, 6, 8, 12, 16, 20, 24, 32, 40. Default = 12.

Examples

proc (1) = f(x, y);
  retp((cos(x) + 1).*(sin(y) + 1));
endp;

proc (1) = g1(x);
   retp(sqrt(1 - x^2));
endp;

proc (1) = g2(x);
   retp(0);
endp;

// Limits
xl = 1|-1;

// Create vector of function pointers
g0 = &g1|&g2;

// Order of integration
_intord = 40;

// Integrate
y = intgrat2(&f, xl, g0);

This will integrate the function

f(x, y) = (cos(x) + 1)(sin(y) + 1)

over the upper half of the unit circle. Note the use of the .* operator instead of just * in the definition of \(f(x, y)\). This allows f to return a vector or matrix of function values.

Remarks

The user-defined functions specified by f and gl must either

  1. Return a scalar constant

          - or -

  2. Return a vector of function values. intgrat2() will pass to user-defined functions a vector or matrix for x and y and expect a vector or matrix to be returned. Use .* and ./ instead of * and /.

Source

intgrat.src

Globals

_intord, _intq12, _intq16, _intq2, _intq20, _intq24, _intq3, _intq32, _intq4, _intq40, _intq6, _intq8