Matlab integral not giving right result as trapz function

I have the following code

x = 0:0.1:1;
y = x.*x;
dy = gradient(y);
dx = gradient(x);
dx_fun = @(z) spline(x,dx,z);
dy_fun = @(z) spline(x,dy,z);
integral(dy_fun,0,1)
trapz(dy_fun(x))

the result obtained by the integral function does not match the result obtained by trapz function. Both bounds of the intergation are the same for both function. Value of trapz(dy_fun(x)) = y(end,1)

Answer 1

[Ameer Hamza]

Try this

integral(dy_fun,0,1)
trapz(dy_fun(x).*dx) % <== multiply the function value with dx

[Manushanker Balasubramanian]

Thanks for your reply. I modified the code as

x = 0:0.1:1;

y = x.*x;

dy = gradient(y);

dx = gradient(x);

dx_fun = @(z) spline(x,dx,z);

dy_fun = @(z) spline(x,dy,z);

for i=1:length(x)

yval(i,1)=integral(dy_fun,0,x(i),'RelTol',1e-10,'AbsTol',1e-12);

yval(i,1)=yval(i,1)/dx(i);

end

cumtrapz(dy)

The value of yval is [ 0

0.013638392857143

0.043333333333333

0.093415178571508

0.163392857142853

0.253400297619018

0.363392857142850

0.493415178572472

0.643333333334451

0.813638392858353

1.000000000000000]

and cumtrapz(dy) is [0 0.015000000000000 0.045000000000000 0.095000000000000 0.165000000000000 0.255000000000000 0.365000000000000 0.495000000000000 0.645000000000000 0.815000000000000 1.000000000000000]

The final value of integration matches but I have issues with intermediate values

[Ameer Hamza]

integral() and cumtrapz() are not similar functions. cumtapz is the implementation of simple trapezoidal rule with fixed step-size, whereas integral() is a more advanced function that uses adaptive step-size. Therefore, the result can differ, and the value returned by integral will be more accurate.

[Ameer Hamza]

Try decreasing fixed-step size for cumtrapz() function, and you will see that output is similar

x = 0:0.01:1;

[Steven Lord]

Or even easier, trapz(x, dy_fn(x)).