Hello everyone, today I have I problem which I could not solve because of the limitation of my matlab skills. I am sure Matlab can do this . So time to learn something new - I hope you can teach me how to :D I will try to explain what I want to do first :) My function is F(s,x_i). The function is defined on three intervals for s which are known (s<s_a, s_a<=s<=s_b, s_b<s). X is a set of known parameters. We have two known x called x_1 and x_2 which are different. The values of the intervals s_a and s_b are known but different for both x parameter sets. S is what we are looking for. To solve this I have the condition of the sum of F(s,x_1) and F(s,x_2) to match a known value F_total. So F_total = F(s,x_1) + F(s,x_2) is what we want so solve to get our s. The simple and stupid idea was to increase s from minimum value till we reach F_total. But this was much too slow, so I need a more efficient way to get the solution :( I hope you can help me, thank you very much in advance! Marius

solve a function on intervalls

Walter Roberson 2020-6-26

(s<s_a, s_a<=s<=s_b, s_b<s)

For any given x_i, those three values are constants in x_i, but you do not know where the s_a, s_b breakpoints are, and your goal is effectively to find those s_a and s_b values? I know you say that the goal is to find s, but if the values are independent of s (but dependent on x_i) within each of the intervals, then determining the s is pretty much the same as finding the two breakpoints.

Or are the s_a and s_b breakpoints known, and within each interval the value is dependent upon s as well as x_i ? If so then how do you know that there are not multiple s values that lead to the same output?

Marius Brettner 2020-6-26

编辑：Marius Brettner 2020-6-26

First, thank you for your answer! I will try to make it more clear.

The interval boundaries s_a and s_b are known because I can calculate them out of the x_i parameters. So s_a and s_b depend on x_i which is why in the F_total = F(s,x_1) + F(s,x_2) problem F(s,x_1) can be on a diffrent interval then F(s,x_2).

For all parameters the graph of F(s) is rising to an maximum (first interval) and then falling to an constant (lower then maximum) value (interval two). The third interval for s>s_b is just a constant F. The F_total is limited, so I know that I can find a solution s. In some cases there might be two solutions for s but thats no problem because I only care for the smaller s.

Maybe I did not explain it very well. I think a simple "example" would help:

Known are x_1, x_2 and the intervals s_a_1, s_b_1, s_a_2, s_b_2 which seperate the function F. The target for F_total is known as well. Now I started with a low s and calculated F(s,x_1) and F(s,x_2). Because I did this "stupid" iteration I knew on which interval I was and how F(s,x) was defined due to this. Last I checked if the sum of F(s,x_1) and F(s,x_2) matched the F_total target. If not I repeated for a bigger s until F_total was reached.

Walter Roberson 2020-6-26

It looks to me as if want you are trying to say is that the integral of F(s,x_i) is known and the question is to determine s given x_i such that the integral from some fixed value to s produces the desired value?

Marius Brettner 2020-6-26

Yes, I think you got what I want!

Most easy solution would be something like write F_total = F(s,x_1) + F(s,x_2) to something like s(F_total,x_1,x_2). But I think this is not possible because before knowing s I can´t know the interval and so the current definition for F(s,x_i). So we need something like a iteration?

Walter Roberson 2020-6-26

编辑：Walter Roberson 2020-6-26

在 MATLAB Online 中打开

Use fzero() or fsolve(), integral() of some function from known lower bound to (parameter) s, and subtract from that the known value of the integral.

for example,

known = 2;
fzero(@(X) integral(@(x) tan(x).^2 + exp(x), -1, X) - known, 0)

Marius Brettner 2020-6-26

Thank you very much! fzero seems to be what I was looking for!

One thing I don´t get: just how do I manage the diffrent intervals of my function. Since it´s a broken rational function on the different intervals the function is definded by a diffrent expression. And when defining the function (handle) which I put in fzero I need already to know in which interval s will be, but I don´t know at that moment.

For example for

s<s_a: F(s)=F1(s)

s_a<=s<=s_b: F(s)=F2(s)

s_b < s: F(s)=F3(s)

How do I get this in the function of fzero whithout knowing s?

Thank you very much again!!

Walter Roberson 2020-6-26

在 MATLAB Online 中打开

F = @(s) (s<s_a) .* F1(s) + (s_a <= s & s < s_b) .* F2(s) + (s_b < s) .* F3(s);

However, this will fail if any of the functions could return infinity or nan when invoked "when they shouldn't be". For example if F2 included 1/s and that wasn't supposed to be a problem because s_a and s_b where chosen such that the range for F2 excluded s = 0, then there would be a problem.

For cases that can include infinity or nan, you need more complicated phrasings such as

H = {@F1, @F2, @F3};
F = @(s) H{cumsum([1, s_a <=s, s_b <s])}(s)

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