How to change the heaviside function -- and how to use the new function with symbolic objects?

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condor
condor 2011-4-9
Hi.
I created this function (it is equal to the heaviside function):
function Y = fzine( X )
Y = zeros(size(X));
Y(X > 0) = 1;
eng = symengine;
if strcmp(eng.kind,'maple')
Y(X == 0) = nan;
else
Y(X == 0) = 0;
end
Now, why if
syms x
I get:
>> heaviside(x)
ans =
heaviside(x)
BUT
>> fzine(x) ??? Error using ==> sym.sym>notimplemented at 2621 Function 'gt' is not implemented for MuPAD symbolic objects.
Error in ==> sym.sym>sym.gt at 801 notimplemented('gt');
Error in ==> fzine at 7 Y(X > 0) = 1;
The two functions (fzine and heaviside) share the same code...?

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condor
condor 2011-4-10
Yes...
and using heaviside is very fast, I wrote also this (very fast):
function Y = fzine( X )
%FZINE Summary of this function goes here
% Restituisce 0 quando X<0 oppure X==0
% Restituisce 1 quando X>0
Y = heaviside(X);
if Y==0.5
Y=0;
end
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Andrew Newell
Andrew Newell 2011-4-10
The answers are out of sequence now, @Walter. This one is in reply to my suggestion that @condor do the calculation numerically.
Andrew Newell
Andrew Newell 2011-4-10
@condor, are we done? If so, it would really help sort out this mess if you accepted the most useful answer and voted for any others that you found useful.

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更多回答（13 个）

Andrew Newell
Andrew Newell 2011-4-9
The symbolic toolbox cannot evaluate X > 0 for a symbolic variable X. You can make a small change in your function so it can handle a symbolic variable:
function Y = fzine( X )
Y = zeros(size(X));
Y(double(X) > 0) = 1;
eng = symengine;
if strcmp(eng.kind,'maple')
Y(X == 0) = nan;
else
Y(X == 0) = 0;
end
Now if you try
fzine(sym(1))
you get
ans =
1
(you can also input a double variable). It still won't handle fzine(x); but notice that
heaviside(x)
just returns the name of the function. It isn't trying to actually evaluate heaviside(x).
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condor
condor 2011-4-10
Yes Walter....very very good! It seems working.
Just a question..why do I get 2 results:
>> fzine = @(x) subs('piecewise([x > 0,1],[Otherwise,0])','x',x);
>> syms MSP
>> costi=fzine(MSP-10)*2+fzine(MSP-50)*2.5+fzine(MSP-100)*3+15
costi =
piecewise([100 < MSP, 45/2], [not 10 < MSP, 15], [MSP in Dom::Interval(10, [50]), 17], [MSP in Dom::Interval(50, [100]), 39/2])
>> delta=1.3*costi+costi-MSP
delta =
piecewise([100 < MSP, 207/4 - MSP], [not 10 < MSP, 69/2 - MSP], [MSP in Dom::Interval(10, [50]), 391/10 - MSP], [MSP in Dom::Interval(50, [100]), 897/20 - MSP])
>> double(solve(delta))
ans =
39.1000
34.5000

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Andrew Newell
Andrew Newell 2011-4-10
Here is a numerical version of what you wish to do. First, the function:
function y = heavy0(x)
y = 0*x;
y(x>0) = 1;
Now solve with it:
costi= @(x) heavy0(x-10)*2+heavy0(x-50)*2.5+heavy0(x-100)*3+15;
delta = @(x) 1.3*costi(x)+costi(x)-x;
fzero(delta,0)
ans =
39.1000
This also works for problems where the solution is the zero point of a heaviside function:
costi = @(x) heavy0(x-10)+x-10;
fzero(costi,1)
ans =
10
Paulo Silva
Paulo Silva 2011-4-9
good question, I'm also clueless about why that happens
Walter Roberson
Walter Roberson 2011-4-9
Your fzine is probably not in the private directory of the symbolic functions, so probably you are not getting the correct methods invoked for all items.
condor
condor 2011-4-9
So what you mean with private directory (where is it?)... where should I put the fzine?
condor
condor 2011-4-9
ok I did:
>> which heaviside C:\Program Files\MATLAB\R2010b\toolbox\symbolic\symbolic\heaviside.m
so I cut and pasted the fzine in that folder; now:
>> syms x >> fzine(x) ??? Undefined function or method 'fzine' for input arguments of type 'sym'.
IT doesn't work, help please.
condor
condor 2011-4-9
>> rehash toolbox >> syms x >> heaviside(x)
ans =
heaviside(x)
>> fzine(x) ??? Error using ==> sym.sym>notimplemented at 2621 Function 'gt' is not implemented for MuPAD symbolic objects.
Error in ==> sym.sym>sym.gt at 801 notimplemented('gt');
Error in ==> fzine at 7 Y(X > 0) = 1;
Still doesn't work...
condor
condor 2011-4-9
Andrew you say:
"The symbolic toolbox cannot evaluate X > 0 for a symbolic variable X."
Why this doesn;t work with heaviside? It doens't evaluate the function but it accept a symbolic object as input so you can solve an equation that contains heaviside(x)...
??
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condor
condor 2011-4-9
An example:
>> syms x
>> delta=heaviside(x)+15-12*x
delta =
heaviside(x) - 12*x + 15
>> solve(delta)
ans =
4/3
>> delta=fzine(x)+15-12*x
??? Error using ==> sym.sym>notimplemented at 2621
Function 'gt' is not implemented for MuPAD symbolic objects.
Error in ==> sym.sym>sym.gt at 801
notimplemented('gt');
Error in ==> fzine at 8
Y(X > 0) = 1;

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condor
condor 2011-4-10
Here where the error gets generated:
function X = gt(A,B)
%GT Symbolic greater-than.
A = sym(A);
B = sym(B);
if isa(A.s,'maplesym')
X = A.s > B.s;
else
notimplemented('gt');
end
end
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Walter Roberson
Walter Roberson 2011-4-10
Odd... your path says 2010b, which is restricted to using MuPad and cannot use Maple. I wonder if the internals of the symbolic toolbox still say Maple in places ??
condor
condor 2011-4-10
You are right Walter...and I don't know why! But heaviside (I don't know how) is able to bypass that check above...I mean that with heaviside
isa(A.s,'maplesym') returns 1 otherwise I would get the same error!
If I am able to discover how to bypass that I could replicatye it in my function fzine....

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Andrew Newell
Andrew Newell 2011-4-10
Here is a different way you could approach this problem:
fzine = (x + abs(x))/2;
delta = fzine+15+12*x;
simplify(solve(delta))
ans =
-5/4
In fact, you could define your function this way:
function Y = fzine( X )
Y = (X + abs(X))/2;
Y(X~=0) = Y/X;
This now behaves almost exactly like the Heaviside function. Even the derivative works:
>> fd = simplify(diff(fzine(x)))
fd =
-(abs(x) - x*sign(x))/(2*x^2)
>> subs(fd,-2)
ans =
0
>> subs(fd,2)
ans =
0
The only difference is for x=0:
>> subs(fd,0)
ans =
NaN
(the derivative of heaviside gives Inf).
EDIT: Another difference occurs if you try this:
y = fzine(x);
subs(y,0)
ans =
NaN
EDIT: I think the problem with the original function is not just the greater than sign. It is the conditional commands, which make it impossible for the function to return an explicit symbolic expression. For more general cases, you could replace a lot of if/then conditions by expressions involving ... (drum roll) heaviside.
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condor
condor 2011-4-10
I think the best solution up to know is the one that Walter suggested (it seems to be a little bit slow but that is ok for my use...):
fzine = @(x) subs('piecewise([x > 0,1],[Otherwise,0])','x',x);
however I still don't understand:
1) Why using the fzine defined as above and solving and equation as:
>> syms MSP
>> costi=fzine(MSP-10)*2+fzine(MSP-50)*2.5+fzine(MSP-100)*3+15
costi =
piecewise([100 < MSP, 45/2], [not 10 < MSP, 15], [MSP in Dom::Interval(10, [50]), 17], [MSP in Dom::Interval(50, [100]), 39/2])
>> delta=1.3*costi+costi-MSP
delta =
piecewise([100 < MSP, 207/4 - MSP], [not 10 < MSP, 69/2 - MSP], [MSP in Dom::Interval(10, [50]), 391/10 - MSP], [MSP in Dom::Interval(50, [100]), 897/20 - MSP])
>> double(solve(delta))
ans =
39.1000
34.5000
returns 2 solutions?? The first is the correct one...the second is what we would have if in costi all the fzine are = zero so we would have: costi = 15 and costi*1.3+costi=34.5 BUT MSP cannot be 34.5 otherwise costi would not be 15.... !?!
2) I still don't understand why heaviside works with symbolic objects while another function that uses the same code does not...
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Walter Roberson
Walter Roberson 2011-4-10
1) I don't know. I used the piecewise version in Maple and it gave only the first (391/10) solution.
2) I do not have the Symbolic Toolbox so I cannot check or test the heaviside source code myself.

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Andrew Newell
Andrew Newell 2011-4-10
@Condor, I think you are mistaken that the function heaviside uses the same code as the other functions we have considered. heaviside is an interface to MuPAD code, which creates a MuPAD object that can be manipulated according to the rules in MuPAD. For example, operators like diff and int can act on heaviside, while they cannot on Walter's version of fzine. My function behaves a little more like a symbolic object, but I have described some shortcomings.
You can easily create a function that gives the correct numerical answer for a numerical input. But you want it to do more than that. You want it to create that MuPAD object even though we don't even know the properties of MuPAD objects.
It's an interesting exercise seeing where attempts to recreate heaviside fall short, but do we really need to go further with this? Why not just use heaviside?
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Andrew Newell
Andrew Newell 2011-4-10
Then you would be better off treating this as a numerical problem in Matlab itself instead of MuPAD. Implementing a numerical version of heaviside is trivial, and you can use FZERO to find the solution.

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Konstantinos
Konstantinos 2013-5-18
编辑：Walter Roberson 2013-5-18
The way I found so you can do your job is for example:
I wanted to find the Z transform of the following:
syms n z
x2 = n*heaviside(n) + (6-2*n)*heaviside(n-6) + (n-6)*heaviside(n-6);
ztrans(x2, n, z)
and it would give me a wrong value because of the heaviside(n-6). So I fixed it by instead using:
x2 = n*heaviside(n) + (6-2*n)*heaviside(n-5.5) + (n-6)*heaviside(n-5.5);
Basically I guess using some double inbetween your discrete values will fix all the problems. I could have also changed heaviside(n) to heaviside(n+0.5) but there wasn't a point in this particular example. Hope this helps.

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