How Can a sym Vector have Elements that are Vectors Themselves?

I ran the following code on 2024a

syms x c1 c2 real
m1c(x) = c2 + c1*x + heaviside(x - 4)*(3*x - 2)*(x - 4);

This line causes an error on 2024b and can't run here

try
    s = simplify(diff(m1c(x),x),'All',true)
catch ME
    ME.message
end

ans = 'Invalid number of arguments.'

so I've uploaded the 2024a result.

load('sfile.mat')

At first glance, the result seems pretty reasonable

s

s = 

but those first two elements do look a bit odd.

As expected, s is a sym and is a 3 x 1 vector

class(s)

ans = 'sym'

size(s)

ans = 1×2

3 1

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But closer inspection of s reveals that its first two elements are themselves 1 x 2 vectors and the third is a scalar (which would show clearly in the command window, but since Answers uses LiveScript I use char here)

char(s)

ans = '[[c1 - 14*heaviside(x - 4) + 6*x*heaviside(x - 4), c1 - 14*heaviside(x - 4) + 6*x*heaviside(x - 4)]; [c1 - 14*heaviside(x - 4) + 6*x*heaviside(x - 4), c1 + 3*heaviside(x - 4)*(x - 4) + heaviside(x - 4)*(3*x - 2)]; c1 + 3*heaviside(x - 4)*(x - 4) + heaviside(x - 4)*(3*x - 2) + dirac(x - 4)*(3*x - 2)*(x - 4)]'

It looks like the first and second elements of the first element of s are the subexpression and expression respectively. Same for the second element of s.

We can verify the dimensions of each element

arrayfun(@(s) size(s),s,'uni',false)

ans = 3x1 cell array
    {[1 2]}
    {[1 2]}
    {[1 1]}

So sym arrays seem more like cell arrays from this perspective, even though sym arrays use paren indexing and not curly brace indexing.

An element can be extracted from s

s1 = s(1)

s1 = 

class(s1)

ans = 'sym'

and we can index into that

s1(1)

ans = 

and operate on the result

s1(2)*2

ans = 

But we can't perform a basic operation on s1 itself

s1*2

Error using * (line 362)
Invalid return value '[2*c1 - 28*heaviside(x - 4) + 12*x*heaviside(x - 4), 2*c1 - 28*heaviside(x - ...'. It must be of type 'Type::MATLABOutput'.

Any idea what's going on here?

Also, any idea why @doc:simplify returns s in this form from the outset? I saw nothing on the doc page that indicated that this behavior is expected.

[Paul]

MathWorks Tech Support confirmed a bug in 2024a.

Answer 1

[Walter Roberson]

I think it is a bug that simplify() returns this form.

I suspect that the result of the simplify() is a piecewise() construction, and what is being exposed is the children() of the piecewise construction.

[Walter Roberson]

What is happening is that the Symbolic Engine is returning an internal MuPAD data structure. Inside MuPAD, there is no problem forming nested components of vectors -- [[a,b],c,[d,[e,f],g]] is a valid internal data structure.