Question

Problem 6 (programming): Consider a two-dimensional function f(x.y) stored in the file f.mat on TritonED. Down load the file and load it into MATLAB. The file includes 3 variables: x, y and f, where f contains 2-dimensional function values at (x, y). You can execute: surf(x, y, f); shad ing interp; view(3); in the command window to visualize the function. Do not put the command in your homework script. In the following exercises, estimate the double integral: maz(ycmax(z) f(x, y) drdy. I(s) J min) Jmin(x) (a, b) Use improved trapezoid method. Integrate in x first, then in y and store the answer in p6a. Repeat the integration, but in y first then in x, and store the answer in p6łb (c, d) Use Simpson's rules. Integrate in x first, then in y and store the answer in p6c. Repeat the integration, in y first then in x, and store the answer in p6d.

Answer 1

Solution:- I have given the required below:-

A) function I1 = TrapezoidalRule2D(a, b, c, d, m, n, func)
h = (b-a)/m;
k = (d-c)/n;
I11 =0;
I12 =0;
I13 =0;
I14 =0;
I15 =0;
for i = 1:m-1
xi = a + i*(h);
I11 = I11 + func(xi, c);
I22 = I22 + func(xi, d);
end
for j = 1:n-1
yj = c + j*(k);
I13 = I13 + func(a, yj);
I14 = I14 + func(b, yj);
end
for j = 1:n-1
for i = 1:m-1
xi = a + i*(h);
yj = c + j*(k);
I15 = I15 + func(xi, yj);
end
end
I1 = (h)*(k)*(0.25)*(func(a,c) + func(b,c) + func(a,d) + func(b,d) + 2*(I11 + I12 + I13 + I14) + 4*(I15));
end

B)

function I2 = SimpsonRule2D(a, b, c, d, m, n, func)

h = (b-a)/2*(m);

k = (d-c)/2*(n);

I21 =0;

I22 =0;

I23 =0;

I24 =0;

I25 =0;

I26 =0;

I27 =0;

I28 =0;

I29 =0;

I210 =0;

I211 =0;

I212 =0;

for j = 1:n

yj = c + (2*(j) - 1)*(k);

I21 = I21 + func(a, yj);

I23 = I23 + func(b, yj);

end

for j = 1:n-1

yj = c + 2*j*(k);

I22 = I22 + func(a, yj);

I24 = I24 + func(b, yj);

end

for i = 1:m

xi = a + (2*(i) - 1)*(h);

I25 = I25 + func(xi, c);

I26 = I26 + func(xi, d);

end

for i = 1:m-1

xi = a + 2*i*(h);

I27 = I27 + func(xi, c);

I28 = I28 + func(xi, d);

end

for j = 1:n

for i = 1:m

xi = a + (2*(i) - 1)*(h);

yj = c + (2*(j) - 1)*(k);

I29 = I29 + func(xi, yj);

end

end

for j = 1:n-1

for i = 1:m

xi = a + (2*(i) - 1)*(h);

yj = a + 2*(j)*(k);

I210 = I210 + func(xi, yj);

end

end

for j = 1:n

for i = 1:m-1

xi = a + 2*(i)*(h);

yj = c + (2*(j) - 1)*(k);

I211 = I211 + func(xi, yj);

end

end

for j = 1:n-1

for i = 1:m-1

xi = a + 2*(i)*(h);

yj = c + 2*(j)*(k);

I212 = I212 + func(xi, yj);

end

end

I2 = (1/9)*(h)*(k)*(func(a,c) + func(b,c) + func(a,d) + func(b,d) + 4*(I21 + I23 + I25 + I26 + I212) + 2*(I22 + I24 + I27 + I28) + 8*(I210 + I211) + 16*(I29));

end

function I1 -TrapezoidalRule2D(a, h = (b-a)/m; k = (d-c)/n; b, d, func) c, m, n, 2 4. 5 I12 -0; I13 -0 I14 -0 6 I150 fori- 1:m-1 9 10 I11 I11 func(xi, c); 122 122 + func (x1, d); 12 13 end for j - 14 1:n-1 yj cj*(k); I13 I13 func (a, yj); I14 I14func (b, yj); 15 16 17 18 for j 1:n-1 fori- 1:m-1 19 20 X1 = a + 1"(h); yj = c + J"(k); I15 I15func(xi, yj); end 21 23 24 25 end I1-(h)(k) (0.25)unc(a,c) func (b,c) func (a,d)func(b,d) 2 (111I12I13 I14) 4*(115)); 26 27en

function 12-SimpsonRulc2D(a, b, c,d, m, n, func) 21 -; 122 -8; 123 -8; 124 -; 25-; 126 -8; 27 -; 128 -; 129-; 1218 - 211 -e; 1212 -e; for j-1:n yj - c (2"()- 1) (k) 121 121func(a, yj) 123 123 func(b, yj) end for j -1:n-1 yj - 2"j"(k) 122 122 func(a, yj) 124 124func(b, yj) end for i 1:m (2*(1) 1)"(h); ㄨㄧ -a + . 125-125 func(xi, c) 126 126func(xi, d); end for i 1:-1 27-127 func(xi, c) 128 12func(xi, d); end for j-1:n for i 1m yj - c (2*()- 1) (k) 129 129fune(xi, yj); end for j-1:n-1 for i 1m xi-a (21) (h); yj -a2"() (k); 1 1218fune(xi, yj) end for j - 1:n for i -1:m-1 xi-a 2(i) (h); yj - c (2*()- 1) (k) 211 1211fune(xi, yj) end for j - 1:n-1 for i -1:m-1 xi -2 () (h); yj c 2"() (k); 212 1212fune(xi, yj) end 12 - (1/9)*(h) (k) (func(a,cfunc(b,c) func(a,d) func(, 4121123125+126 1212) 2 (122 124 127128) 8 (1218121116 (129)); ond

I hope it solves your problem if you have any doubt please ask in the comments and if you liked the solution please upvote. Thanks.