As I think more on this, it seems that the context matters.

In one example,

x = a:dx:(a + b);
y = trapz(x, f(x))

When repeated with b approaching zero, the integration range decreases, as does the number of elements in x.  Then y would approach 0, unless there's an asymptote in f at a.  This might explain the choice by Mathworks to return 0 in some cases.

In another example,

x = linspace(a, a + b, n);
y = trapz(x, f(x))

When repeated with n approaching 0, the numerical integration becomes progressively more erroneous until it becomes 0 when n = 1, and quite undefined when n = 0.  This is even though in theory one might instead choose to reduce the trapezoidal technique to just b * f(x+b/2) when n = 1 (but Octave cannot because trapz no longer has the integration bounds with n = 1).

also worth noting that octave currently allows a superset of matlab allowed inputs to trapz.  e.g. it allows different domains for X:

Octave:

>> x = reshape(0:15,4,4)
x =

    0    4    8   12
    1    5    9   13
    2    6   10   14
    3    7   11   15

>> y = magic(4)
y =

   16    2    3   13
    5   11   10    8
    9    7    6   12
    4   14   15    1

>> trapz(x(:),y(:))
ans = 127.50

>> trapz(x(:,1),y)
ans =

   24   26   25   27

>> trapz(x,y)
ans =

   24   26   25   27

whereas matlab:

>> trapz(x(:),y(:))
ans =
  127.5000
>> trapz(x(:,1),y)
ans =
    24    26    25    27
>> trapz(x,y)
Error using trapz
X must be a vector.

so, some of that will play into octave having a different error set and possibly handling empties for input 1 differently.

I guess this report really boils down to whether Octave's current empty input handling is:

 1. compatible for valid matlab inputs.  so far seems like it is.

 2. consistent/useful for any other non-error producing input empty input.  not sure about this.

 3. do any error messages make sense. not sure here either. (e.g., 'x and y must be have same shape' when trapz(x(:,1),y) is perfectly valid.)

Disregard my comment (#1) -- I used b=[].

Dmitri.
--

checking matlab compatibility for different types of empty inputs:

matlab 2022b:

>> a = []
a =
     []
>> b = 1:0
b =
  1×0 empty double row vector
>> c = ones(0,1)
c =
  0×1 empty double column vector
>> d = ones(0,0,0)
d =
  0×0×0 empty double array
>> trapz(a,a)
Error using trapz
X must be a vector.
>> trapz(b,a)
ans =
     0
>> trapz(a,c)
Error using trapz
X must be a vector.
>> trapz(c,a)
ans =
     0
>> trapz(b,c)
ans =
     0
>> trapz(c,b)
ans =
     0
>> trapz(d,a)
Error using trapz
X must be a vector.
>> trapz(d,b)
Error using trapz
X must be a vector.
>> trapz(d,c)
Error using trapz
X must be a vector.
>> trapz(d,d)
Error using trapz
X must be a vector.
>> trapz(a,d)
Error using trapz
X must be a vector.
>> trapz(b,d)
ans =
  1×0×0 empty double array
>> trapz(c,d)
ans =
  1×0×0 empty double array

same commands in Octave 8.1.0:

>> a = []
a = [](0x0)
>> b = 1:0
b = [](1x0)
>> c = ones(0,1)
c = [](0x1)
>> d = ones(0,0,0)
d = [](0x0x0)
>> trapz(a,a)
ans = 0
>> trapz(a,b)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
>> trapz(b,a)
ans = 0
>> trapz(a,c)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
>> trapz(c,a)
ans = 0
>> trapz(b,c)
ans = 0
>> trapz(c,b)
error: trapz: length of X and length of Y along DIM must match
error: called from
    trapz at line 117 column 7
>> trapz(d,a)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
>> trapz(d,b)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
>> trapz(d,c)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
>> trapz(d,d)
ans = [](1x0x0)
>> trapz(a,d)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
>> trapz(b,d)
ans = [](1x0x0)
>> trapz(c,d)
ans = [](1x0x0)

Octave tries to produce compatible output for inputs that don't result in errors. Note that [], ones(0,1), and ones(1,0) are not necessarily equivalent, and they sometimes can and should produce different outputs for certain functions.  In this case, however, matlab produces errors for a number of input combinations. And judging by brevity in some of the messages (e.g., "Error using trapz" sans any detail), this one has probably been around for a long time without being revisited by mathworks like some of the other functions we've recently been trying to bring back into empty-input compatibility.

Anyway, for inputs where matlab isn't producing an error,  we appear to be producing compatible output.  Maybe we could be more consistent on some of the others either in error message or by producing an actual output, but I'm not seeing a specific problem here.

The trapz function sometimes gives errors when the input parameters are empty.  This depends on specifically how the parameters are defined.  The below shows this for inputs that are 0x0 and 1x0.  I would expect the trapz result to be 0 in all cases.

octave:1> version
ans = 7.1.0
octave:2> a = []
a = [](0x0)
octave:3> b = 1:0
b = [](1x0)
octave:4> trapz(a,a)
ans = 0
octave:5> trapz(a,b)
error: trapz: X and Y must have same shape
error: called from
    trapz at line 126 column 7
octave:6> trapz(b,a)
ans = 0
octave:7> trapz(b,b)
error: trapz: length of X and length of Y along DIM must match
error: called from
    trapz at line 117 column 7