intgrat2 ============================================== Purpose ---------------- Integrates the following double integral, using user-defined functions *f*, *g1* and *g2* and scalars *a* and *b*: .. math:: \int_{a}^{b}\int_{g_2(x)}^{g_1(x)} f(x,y)dydx Format ---------------- .. function:: y = intgrat2(&f, xl, gl) :param &f: pointer to the procedure containing the function to be integrated. :type &f: scalar :param xl: the limits of *x*. These must be scalar limits. :type xl: 2x1 or 2xN matrix :param gl: 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. :type gl: 2x1 or 2xN matrix :return y: of the estimated integral(s) of :math:f(x, y), evaluated between the limits given by *xl* and *gl*. :rtype y: Nx1 vector 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 :math: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 #. Return a scalar constant       - or - #. Return a vector of function values. :func: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* .. seealso:: Functions :func:intgrat3, :func:intquad1, :func:intquad2, :func:intquad3, :func:intsimp