Different output using mldivide with newer matlab version

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Steven H
Steven H 2024-5-15
评论： Steven H 2024-5-21
Problem introduction
I have a simple system of 4 linear equations with 4 unknowns, which can be written as . The content of the matrices is written out below.
The analytical solution to this equation is
Observation
Asking MATLAB R2017b to solve this system using mldivide returns the exact answer. However, MATLAB R2024a returns a slightly different answer for the exact same question, where 3 out of the 4 elements in x are off to the order of 1e-11.
This exact observation can be reproduced using the attached files.
The run in R2017b (err is the difference between the mldivide solution and the analytical solution):
>> test
x =
1.0e+02 *
0.010000000000000
0
-2.865785081173561
0.687432906991598
err =
0
0
0
0
The run in R2024a:
>> test
x =
1.0e+02 *
0.010000000000097
0.000000000000000
-2.865785081173561
0.687432906991647
err =
1.0e-11 *
0.972688596334592
0.004821862926227
0
0.484590145788388
Question
Simply; how come?
Is it simply a numerical error? What is the difference in mldivide between R2017b and R2024a? I read in the release notes that there is an Improved performance with small matrices to mldivide in R2022b. Is this the cause?
And how can I overcome this issue? A lot of my test functions are failing; is there a proper, robust fix for this? Or should I take a close look at my failing test functions and update them if possible? Any other suggestions?
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Star Strider
Star Strider 2024-5-15
Someone who has access to both versions (R2017b and R2024a) will have to respond to you about this. I don°t, so I deleted my answer.

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回答（1 个）

Steven Lord
Steven Lord 2024-5-15
Ran in:
We do not guarantee that the results from mldivide will be exactly the same, down to the last bit, across:
  • different operating systems running the same release of MATLAB (potential differences in system math libraries)
  • different releases of MATLAB running on the same machine (bug fixes in MATLAB or different versions of certain math libraries included in MATLAB, which could be for bug fixes in those libraries or performance improvements or enhanced accuracy)
  • the same release of MATLAB running on the same machine with different math libraries (Intel MKL versus AMD AOCL, for example.)
You can check which libraries you're using with the command version -lapack (or for the BLAS, version -blas) as shown in this Answers post.
"A lot of my test functions are failing; is there a proper, robust fix for this?"
Yes. Don't have your test functions test for exact down to the last bit equality of results unless you have a good reason to expect or require that level of agreement (1+1 must always equal exactly 2 regardless of MATLAB / OS / library version, for example! Another is running the same computation twice in a row in the same MATLAB session as long as no random numbers are involved.)
Instead, specify a tolerance. If you're using the unit testing framework included in MATLAB (specifically verifyEqual) you can specify an AbsTol (absolute tolerance) and/or RelTol (relative tolerance). As an example, should give exactly 0. But since the MATLAB function pi does not return the transcendental value of π but the double precision approximation, it's close to but not exactly equal to 0. So the exact, down-to-the-last-bit test fails.
tc = matlab.unittest.TestCase.forInteractiveUse;
verifyEqual(tc, sin(pi), 0) % fails
Verification failed. --------------------- Framework Diagnostic: --------------------- verifyEqual failed. --> The numeric values are not equal using "isequaln". --> Failure table: Actual Expected Error RelativeError ____________________ ________ ____________________ _____________ 1.22464679914735e-16 0 1.22464679914735e-16 Inf Actual Value: 1.224646799147353e-16 Expected Value: 0 ------------------ Stack Information: ------------------ In /tmp/Editor_utzhl/LiveEditorEvaluationHelperEeditorId.m (LiveEditorEvaluationHelperEeditorId) at 2 In /MATLAB/toolbox/matlab/connector2/interpreter/+connector/+internal/fevalMatlab.p (fevalMatlab) at 0 In /MATLAB/toolbox/matlab/connector2/interpreter/+connector/+internal/fevalJSON.p (fevalJSON) at 0
But if we specify a tolerance, it's "close enough" to 0.
verifyEqual(tc, sin(pi), 0, AbsTol = eps) % passes
Verification passed.
Or in this particular case, to avoid the difference between pi and π you could use sinpi.
verifyEqual(tc, sinpi(1), 0) % no tolerance, passes
Verification passed.
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Steven Lord
Steven Lord 2024-5-16
FYI, from the Wikipedia entry for condition number: "As a rule of thumb, if the condition number , then you may lose up to k digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods."
You might also find section 9 (Norms and Condition Numbers) of the chapter "Linear Equations" in Cleve Moler's Numerical Computing with MATLAB textbook interesting.
Steven H
Steven H 2024-5-21
With your explanation @John D'Errico I am convinced that it's not a matlab issue but a general numerical computation issue. I will find a suitable corresponding solution. If I could somehow accept your answer, I would. The answer of @Steven Lord was helpful, but not as much.

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