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Compute a quadrature of

$\int_c^d\int_a^b f(x,y)dxdy$

using the Simpson rule and estimate the error.

So the Simpson rule says

$S(f) = (b-a)/6(f(a)+4f((a+b)/2) +f(b))$

So i get

$\int_c^d(b-a)/6(f(a)+4f((a+b)/2) +f(b))dy$

Is that even correct? How do I go on?

  • Are you looking for the general theory or is there a specific $f(x, y)$ and limits? – Moo Jan 30 '17 at 23:18
  • General theory, there is a part exercise with a specific function, but I think I'll be able to do this then –  Jan 30 '17 at 23:20
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    Try: http://jmahaffy.sdsu.edu/courses/s10/math541/lectures/pdf/week09/lecture.pdf and http://www.math.usm.edu/lambers/mat460/fall09/lecture32.pdf – Moo Jan 30 '17 at 23:27

2 Answers2

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Something to get you started. Define an operator: $$ I_x(f) := \int_a^b f(x) \, dx $$ We can see that $I_x(f + g) = I_x(f) + I_x(g)$ and if $\alpha$ does not depend on $x$, $I_x(\alpha f) = \alpha I_x(f)$. In other words the operator is linear.

If it acts on $f(x, y)$ it returns a function of $y$ only.

By analogy define another operator $$ I_y(f) := \int_c^d f(y) \, dy $$

Applying $I_x$ and $I_y$ consequently we get a double integral: $$ I_y(I_x(f)) := \int_c^d \left[ \int_a^b f(x, y) \, dx \right] dy . $$

Now define a "Simpson's rule" operator: $$ S_x(f) := (f(a) + 4 f((a+b)/2) + f(b)) (b - a)/6 $$ It is also linear and if it acts on $f(x, y)$ it "kills" dependency on $x$. The same for $S_y$: $$ S_y(f) := (f(c) + 4 f((c+d)/2) + f(d)) (d - c)/6 $$

By analogy with integral let compute $S_y(S_x(f(x, y)))$ and call it Simpson's rule for double integral. Apply $S_x$ to f(x, y): $$ S_x(f(x, y)) = (f(a, y) + 4 f((a+b)/2, y) + f(b, y)) (b - a)/6 $$ and apply $S_y$ to the result using linearity $$ S_y(S_x(f(x, y)) = (S_y(f(a, y)) + 4 S_y(f((a+b)/2, y)) + S_y(f(b, y))) (b - a)/6 $$ expand all three terms with $S_y$ $$ S_y(S_x(f(x, y)) = \\ \left(16f\left(\frac{b+a}{2},\frac{d+c}{2}\right)+4f\left(\frac{b+a}{2},d\right)+4f\left(\frac{b+a}{2},c\right)+4f\left(b,\frac{d+c}{2}\right) +f\left(b,d\right)+f\left(b,c\right)+4f\left(a,\frac{d+c}{2}\right)+f\left(a,d\right)+f\left(a,c\right)\right) \frac{\left(b-a\right)\left(d-c\right)}{36} $$

slitvinov
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0

A C implementation for applying Simpson's Rule towards solving double integrals can be found here if you are interested. Simpson integration technique for evaluating double integrals

It can be also represented in the following form:

$$ S_x(y_j) = f(x_0, y_j) + f(x_n,y_j) + 4\sum_{i = 1}^{(N_x-2)/2} f(x_{2i-1},y_j) + 2\sum_{i = 1}^{(N_x-2)/2} f(x_{2i},y_j) $$

$$ S = \frac{h_xh_y}{9}[(S_x(y_0) +S_x(y_n) + 4\sum_{j = 1}^{(N_y-2)/2} S_x(y_{2j-1}) + 2\sum_{j = 1}^{(N_y-2)/2} S_x(y_{2j}) $$

Where $$ h_x = \frac{UpperLimit_x - LowerLimit_x}{N_x} $$

and $$ h_y = \frac{UpperLimit_y - LowerLimit_y}{N_y} $$

Alexander
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