Why Does polyval Return Zero When the Coefficients are Empty?
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polyval([],1)
polyval(ones(1,0),1)
Do those results follow from a mathematical justification or are they arbitrarily specified?
回答（3 个）
Boils down to applying Horner's rule with internal function that ends up at
function y = iHorner(x, p)
% Apply Horner's method to calculate the value of polynomial p at x.
if isempty(p)
y = zeros(size(x), 'like', x);
return
end
It follow the convention that a missing coefficient in a polynomial is zero such that one can write
y=3x^2 + 1
with the meaning being that of
y=3x^2 + 0x + 1
Without the convention one also would have to start with the x^N where N was some large integer with a zillion terms with zero coefficients.
1 个评论
Even if polyval were to use a more naive polynomial evaluation algorithm than Horner, its operator rules give the same --
compareThem([5,2,4])
compareThem([])
function compareThem(p)
x=7;
n=numel(p);
res1 = polyval(p,x)
res2=x.^(n-1:-1:0)*p(:)
end
For non-empty A and B, we have the general relation,
polyval([A,B],z)=polyval(A,z)*z^numel(B) + polyval(B,z)
Extending this to B=[] would reduce it to,
polyval([A,[]],z) = polyval(A,z)*z^0 + polyval([],z)
polval(A,z) = polyval(A,z)*1 + polyval([],z)
Solving for polyval([],z) then leads to,
polyval([],z)=0
9 个评论
Matt J
28 minutes 前
That is indeed consistent, but I think you could also argue that a polynomial with an empty set of roots has to be a constant, and the constant polynomial with a leading coefficient of one is 1.
Matt J
about 6 hours 前
I don't know if I agree that R is equivalent to the zero-polynomial. That would be like saying, sum([])=0 implies that []=0.
Paul
about 4 hours 前
I would have thought conv(R,R)=1, based on the identity,
conv(A,B)=poly([roots(A);roots(B)])
A=[1,rand(1,3)]; B=[1,rand(1,3)];
poly([roots(A);roots(B)])
conv(A,B)
R = double.empty(1,0);
poly([roots(R);roots(R)])
Paul
about 4 hours 前
I'm not sure anymore. conv has a lot of weird edge behavior, e.g.,
R=double.empty(1,0);
conv(R,nan)
John D'Errico
2026-3-29，16:20
编辑：John D'Errico
2026-3-29，16:23
(This could easily better belong in a discussion than in Answers.)
I could argue that any choice they made was arbitrary, because I'm not sure a null polynomial has what I'd call a good formal mathematical definition. In some sense I might want to argue the prediction from a empty set polynomial should be an empty set. That is, should there be a distinction in the prediction of these two "polynomials", P0 and Pnull?
P0 = 0;
and
Pnull = [];
A meta-rule in MATLAB is that empty generally produces empty in MATLAB. For example:
max([])
Ok, so what does a null set polynomial mean? I threw this into an AI tool, to see if there was something well known I might be missing. Google suggests:
"A polynomial described as the null set polynomial (or more commonly, a polynomial whose set of roots is the null set/empty set, ∅.
This means a polynomial that has no roots(no zeros) within a specified field of numbers, such as the real numbers ℝ."
(I also put the same question to ChatGPT, which essentially concurs with Google, but it also agreed with my intuitive assertion that a null polynomial has no strong formal definition in mathematics.)
In MATLAB of course, we would think of such a polynomial as one which has no roots in the complex plane. Continuing along those lines, that would make a distinction between the zero and null polynomials, where the zero polynomial has infinitely many roots in both the real line and in the complex plane.
Again, I think this argues for a null result from polyval for a null input poly, even though polyval disagrees.
polyval(P0,1),polyval(Pnull,1)
Looking at the answer from @dpb, I first see the argument the polynomial 3*x^2 + 1 has a zero linear coefficient, so when no coefficient is provided, we presume zero. And while I think this makes sense, that seems different from the case where no polynomial is provided at all.
I think a better argument is made when @dpb points out the Matlab convention about higher order coefficients being presumed as zero. That is
P2 = [3 0 1];
P2 is a quadratic polynomial, so that all higher order coefficients are implicitly zero. Thus the coefficient of x^3 is presumed to be zero. If we continue along that line of reasoning, the polynomial P0, with only a zero constant term, has all higher order coefficients as zero.
But now drop away the constant coefficient, and apply the same reasoning. That is, all higher order (unsupplied) coefficients are implicitly zero, which arguably applies even to the constant coefficient. And that suggests the null polynomial would have an implicit zero constant term coefficient. I'd suggest this is mathematically equivalent to the answer from @Matt J, where he extrapolates down to the null polynomial via Horner's rule.
Is the application of Horner's rule a good argument in this? Well, yes, since polyval explicitly does use Horner's rule to evaluate the polynomial.
In the end, as I said, the answer is a little arbitrary, but I think zero as the prediction for polyval([],1) makes sense.
15 个评论
Paul
about 5 hours 前
dpb
about 2 hours 前
I wasn't intendeing the code snippet to be a justification, just showing how/where MATLAB did the deed.
And I don't disagree about the stance on defensive programming regarding results of empty arguments although my practice is to make the check/fixup on the input side rather than waiting to discover something unexpected happened at the output.
A meta-rule in MATLAB is that empty generally produces empty in MATLAB.
That rule applies to binary operators, but seemingly not to iterative reduction operators. So, for example,
[] + []
[] .* []
but, their corresponding iterative extensions give non-empty results,
sum([ [] , [] ])
prod([ [] , [] ])
So, I suppose polyval is classified as an iterative reduction operator.
John D'Errico
about 8 hours 前
编辑：John D'Errico
about 8 hours 前
"So, I suppose polyval is classified as an iterative reduction operator."
My guess is, there was never an explicit classification done of polyval as an operator. Or of exactly how polyval should operate on an empty polynomial. Given that polyval probably goes back to version 1 of MATLAB as written roughly in 1984, and it used Horner's method even then, nobody worried about it then, and nobody ever really worried about it in the 40+ years since. The Horner loop naturally produces zero as a result for empty polynomials. And it is likely true nobody was going to modify the behavior of polyval over that period unless they had some good reason. TMW tries to never damage legacy code if possible, and an arbitrary change to polyval at some point could easily do that for no good reason.
I don't think A*B' would actually fall into the same category of binary operators as times(A,B), since for non-empty vector operands A and B, one is a reduction and the other is not. The real test is,
R = double.empty(1,0);
R.*R
sum(R)
Paul
about 6 hours 前
Steven Lord
about 4 hours 前
Given that polyval probably goes back to version 1 of MATLAB as written roughly in 1984,
polyval was in listed in the MATLAB 1.3 function list quoted in this post on Cleve's blog. I wouldn't be surprised if it was in version 1.0 as well. But it wasn't in Cleve's original "Historic MATLAB" user guide; its name had too many letters for that :)
dpb
about 3 hours 前
I miss the old headers that had original developer's names...it was most interesting to see Cleve and others were the originators.
I also figured this probably went back there but hadn't gone looking specifically.
Steven Lord
about 2 hours 前
Functions written today aren't the product of a single developer. They may be designed by a single developer, but generally that design gets reviewed by (at least) some of that developer's peers and the people responsible for testing and documenting it. [Design reviews are called out as one of the examples for one of our core values, Continuous Improvement and the Pursuit of Excellence.]
For larger features, or those that are likely to be used by lots of our users, those designs may be reviewed by a larger group as well to make sure it fits into "the MATLAB Way" and how users expect MATLAB to work. [I'm a member of one of those larger design review groups, and I use Answers to help me understand user expectations.]
So you're unlikely to see developers' names called out in comments in our code files going forward.
"The Horner loop naturally produces zero as a result for empty polynomials." [but...] "the output of polyval with empty coefficients is not the natural output of the Horner loop,"
I'd venture the two comments are both correct in the context of what each writer was thinking. The empty polynomial is agreed to produce a zero output by either of the justifications here but in implementation the empty vector would produce an empty result in MATLAB without the special-casing.
One was thinking of the underlying polynomial, the other of the details of the implementation.
As an aside, there is no way to code any missing terms in a vector array of coefficients in MATLAB other than by the explicit place-holding zeros in POLYVAL. An extension (not suggesting, just noting) could use a cell array containing an empty cell, but the length would still have to be the power + 1 to have the placeholder in the right location or some other coding scheme. Being from the V1 era, such complexities hadn't yet made the scene and as @John D'Errico suggested, it's highly unlikely Cleve or whoever even thought about anything more involved at the time than having a code path for all combinations.
Steven Lord
about 5 hours 前
According to the History of MATLAB published by the ACM (linked in this blog post) cell arrays were introduced in 1996. PC-MATLAB 1.0 (which we're speculating had polyval) came out in December 1984.
I'm a relatively late adopter with the first Win3.1 release, V4. Don't recall exactly the year; I'm still kicking myself for having tossed the 3.5" diskettes and printed manuals in a fit of housecleaning. :(
Prior for the purpose had access either to mainframe IMSL libraries and CALCOMP graphics via FORTRAN on CDC Cyber76, then VAX. W/ first '286 PC/DOS bought the Boeing poor-man's equivalent of IMSL and the MS F77 compiler had graphics primitives and later built text input files as input for TecPlot for more sophisticated graphics. On DOS also had the PC-SAS statistics package so the limited statistical abilities within MATLAB weren't an issue until the end of the ability to run "real" DOS apps era. That side would still be more painful if still doing in depth statistical analyses; while the capabilities of the Statistics TB have significantly increased, it's still a lot more effort than having a dedicated statistics package to use the TB grafted on top of MATLAB.
When left for the self-employed consulting gig in 2000, the first piece of business was to purchase/transfer the MATLAB license along with the lab test gear needed to continue the personal services contract on my own instead of through an employer.
Walter Roberson
about 4 hours 前
I have a vague memory that we started using MATLAB with V4, but I wasn't paying attention to MATLAB until 5.1 or so.
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