why does my code take so long to run?

I am trying to compute v' * f(A) * v, where f is a given function, A matrix, and v is a unite vector.

My code takes along time doing run and did not get the result. Could please help me to fix this problem?

% want to compute v'*f(A)*v
syms f(x)

N=1000;
Ns=1:N;
R=1./(Ns);
A=sym(toeplitz(R));% the input matrix
v=ones(N,1);
v=v/norm(v);
f(x) = atan(sqrt(x)); % the function f(x)=arctan(sqrt(x))/x
B = funm(A,f); % the resulting matrix
y=A\v;
exact=v'*B*y

[Walter Roberson]

Original code:

% want to compute v'*f(A)*v
syms f(x)
N=1000;
Ns=1:N;
R=1./(Ns);
A=sym(toeplitz(R));% the input matrix
v=ones(N,1);
v=v/norm(v);
f(x) = atan(sqrt(x)); % the function f(x)=arctan(sqrt(x))/x
B = funm(A,f); % the resulting matrix
size(A)
size(v)

[Rena Berman]

(Answers Dev) Restored edit

[Walter Roberson]

Omar, I had my computer working on your question for several hours, gave it about an hour of my attention. It would not be fair to me to delete the question after I took the time to respond.

[Rik]

@Omar, Please explain why you want your question removed.

I also made a capture of the Google cache from before you removed your question here.

[Rena Berman]

(Answers Dev) Restored edit

Answer 1

[Hiro Yoshino]

It looks that your problem is not that complicated. Why don't you write your code without using symbolic expressions? I bet it is faster.

[Omar B.]

How can I do that?

[Walter Roberson]

There is a challenge that the numeric funm is different than the symbolic funm .

For the symbolic funm, you pass in a symbolic function or symbolic expression, and funm automatically takes derivatives in order to build a taylor series (or however it is represented). No control is offered over the number of terms used, and all the derivatives are likely to be exact derivatives.

If we look at even just the sqrt() part, as a matrix function, sqrtm(), that mathematically involves a closed form svd, sqrt of the diagonal, and reconstruct -- all as closed form expressions. For a 1000 x 1000 matrix, that is going to take a very long time.

For the numeric funm, you pass in a function handle to a function that must accept a vector and must also accept a derivative number to calculate and evaluate the input at. But you do not have diff() available to help because it is a numeric function handle you are working on, not a symbolic function. So you need to have a formula to calculate the N'th derivative.... Or you need to slip into the symbolic world, probably memoizing as you go.

[Walter Roberson]

Even for N=10 the symbolic version takes much much much too long.