Differentiation consistency in matlab

Question

I'm trying to write a program in matlab that checks how consistent the definition of the derivative becomes:

(f(x+h)-f(x))/h ~= f'(x)

when h is small enough. Thus far i have this:

function [errList] = diffConsistency(f,df,x,iMax,h0)
h=h0;
for i=1:iMax
    leftSide = (f(x+h) - f(x)) / h;
    rightSide = df(x);
    errList = abs(leftSide - rightSide);
    h = h*10^(-1);
end

I then use f=@(x)sin(x) and df=@(x)cosx, I'm new to using function handles so this might be wrong completely. iMax is set to 10 and h0 = 1, x=rand(10)

Could anyone check if this is even remotely correct. Especially the use of the function handles inside the diffConsistency function and use of the rand.

Should i define x differently, leftside rightside are correct? etc

Any feedback would help. Thanks in advance

How have you tested this so far? Does it work for sin(x) and cos(x)? Does it work for cases where the function don't match (try it for f = sin(x) and df = sin(x), say)? — EdR, May 01 '18 at 12:38
It does not work as i hoped, since the error fluctuates too much and does not converge to something — p.late, May 01 '18 at 12:46
Could you edit the question then to give an example of the output for various values of `h`? — EdR, May 01 '18 at 12:49

score 2 · Accepted Answer · answered May 01 '18 at 13:10

2

You use some specific data that obscures the result. You input 10x10 random numbers, and output a 10x10 matrix of errors, but this is only for the last i, as you overwrite errList every iteration!

change the function to:

function [errList] = diffConsistency(f,df,x,iMax,h0)
h=h0;
for i=1:iMax
    leftSide = (f(x+h) - f(x)) / h;
    rightSide = df(x);
    errList(i) = abs(leftSide - rightSide);
    h = h*10^(-1);
end

and if you call it as :

err=diffConsistency(@sin,@cos,rand,10,1)

and plot(err), you can clearly see how the error gets reduced each smaller h.

answered May 01 '18 at 13:10

Ander Biguri

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Done. I do however have one more question. When I set iMax to 100 my error decreases until a certain value (around 10^-8), but then it starts increasing again. I do not know how to explain this. Is it a flaw in the code, number loss or something else entirely? – p.late May 01 '18 at 13:45
2

@p.late dangerous place you are getting into, careful careful! This has to do with computers, and floating point arithmetic. You can not store arbitrarily small numbers in a computer, as it has limited memory. When you reach 1e-12 you are starting to get into dangerous area where accuracy of the computation is not reliable anymore. You are saying you are trying to compute 1e-100 (`h`). Read more about why this happens [here](https://stackoverflow.com/questions/686439/why-is-24-0000-not-equal-to-24-0000-in-matlab) – Ander Biguri May 01 '18 at 13:55
Ah I see. What do you think would be an appropriate value to give iMax in this case? By the way I adapted my code with the following break statement: – p.late May 01 '18 at 16:14
if h < 2^(-52) break end – p.late May 01 '18 at 16:15

Differentiation consistency in matlab

1 Answers1