MATLAB function and symbolic function give different answers

% Some values for the plot
A = 0.5
B = 2

% MATLAB and symbolic functions generated two ways
f = @(a, b, x) (a .* (x - b).^2) .* ((x - b) >= 0)
fsym = sym(f); % Uncommenting this line hangs Live Script
syms a b x
fsym2(a, b, x) = (a * (x - b)^2) * ((x - b) >= 0); % Uncommenting this line hangs Live Script

% Plot both functions
figure
X = (0:0.001:3*B);

subplot(1, 3, 1)
plot(X, f(A, B, X), '-r')
title('MATLAB function')

subplot(1, 3, 2)
plot(X, subs(subs(subs(fsym, sym('a'), A), sym('b'), B), sym('x'), X), '-b')
title('Function from sym')

subplot(1, 3, 3)
plot(X, fsym2(A, B, X), '-b')
title('Declared symbolic function')

I am trying to get better at using symbolic functions. I am struggling to understand why the two symbolic subplots do not appear identical (other than color) to the subplot generated by the MATLAB function. Can someone explain why, and what I would do to create a "working" symbolic version of f?

As an aside, both lines 7 and 9 hangs the Live Editor if uncommented...does that suggest a bug in what I am doing?

Answer 1

[David Hill]

A = 0.5;
B = 2;
f = @(a, b, x) (a .* (x - b).^2) .* ((x - b) >= 0);
syms a b x;
fsym1 = piecewise((x-b)>=0,(a * (x - b)^2),(x-b)<0,0); 
figure
X = (0:0.001:3*B);
subplot(1, 3, 1)
plot(X, f(A, B, X), '-r')
title('MATLAB function')
subplot(1, 3, 2)
fsym=subs(fsym1,[a b],[A B]);
fplot(fsym,[0,3*B]);
title('Function from sym')

[James Akula]

Thank you. I am able to make it work symbolically. My question is, why doesn't it work the way I attempted it? Why does the symbolic version produce an equation that gives a different answer?

[David Hill]

I am not an expert here, but matlab (symbolic equation) is not liking your logical expression: (x-b)>=0

syms a b x
fsym1 = piecewise((x-b)>=0,(a * (x - b)^2),(x-b)<0,0);%using piecewise solves the problem
fsym2 = a*(x - b)^2*((x - b)>=0);%not sure why these are not the same.

[Paul]

The result from fsym2 can be explained as follows. However, there might be something odd going on because the Live editor can't render the result, so I'll copy/paste from my local installation.

syms a z
fsym2 = a*(z>=0);

On the right hand side, 'a' is symbolic varibale and 'z >= 0' is an equation. So the the multiplication by 'a' distributes across the equation as it would in a standard mathematical sense. Recall that multiplying an inequality by a negative number reverses the direction of the inequality. Also, the sign of 'a' is unknown absent any additional assumptions, so multiplying through by 'a' has to account for the cases of a <= 0 and a > = 0. The result is

fsym2 = piecewise(0 <= a, 0 <= a*z, a <= 0, a*z <= 0)

The bottom line is that in Symbolic math 'z >= 0' is a math equation, whereas in base Matab 'z >= 0' is an expression to be evaluated to T/F and then converted to double.

[James Akula]

I think this explains it. It is weird behavior though...might catagorize it as a bug. Certainly, the fact that piecewise seems to crash the live editor is a problem!