Validity and Certification
Our proof depends heavily on computer calculations, raising two issues
about its validity:
Calculations. Reproducing elaborate calculations on a large computer
is difficult; particularly when a complicated parallel computer program is involved, everybody should be skeptical
about the reliability of the results; and
We address these issues in turn.
What our program was trying to
compute is an eigenvalue of a matrix. The amount and length of the
computations are irrelevant to the
fact that eventually we have stored
on disk a witness array of floating-point numbers (the “proof”), approximately 450GB in size, which is a good
approximation of the eigenvector
corresponding to λ27. This array provides rigorous bounds on the true eigenvalue λ27, because the relation ( 1)
holds for any vector yold and its successor vector ynew. To check the proof and
evaluate the bounds ( 1), a program
has to read only the approximate eigenvector yold and carry out one iteration (∗). This approach of providing
simple certificates for the result of
complicated computations is the philosophy of “certifying algorithms.” 25
To ensure the correctness of our
checking program, we relied on tradi-
tional methods (such as testing and
code inspection). Some parts of the
program (such as reading the data
from the files) are irrelevant for the
correctness of the result. The main
body of the program consists of a
few simple loops (such as the itera-
tion (∗) and the evaluation of ( 1)). The
only technically challenging part of
the algorithm is the successor com-
putation. For this task, we had two
programs at our disposal that were
written independently by two people
who used different state representa-
tions—and lived in different coun-
tries and did their work several years
apart. We ran two different checking
programs based on these procedures,
giving us additional confidence. We
also tested explicitly that the two suc-
cessor programs yielded the same re-
sults. Both checking programs ran in
a purely sequential manner, and the
running time was approximately 20
Regarding the accuracy of the calculations, one can analyze how the
recurrence (∗) produces ynew from
yold. One finds that each term in the
lower bound ( 1) results from the input data (the approximate eigenvector yold) through at most 26 additions
of positive numbers for computing
ynew[s], plus one division, all in single-precision float. The final minimization was error-free. Since we took care
that no denormalized floating-point
numbers occurred, the magnitude of
the numerical errors was comparable
to the accuracy of floating-point numbers, and the accumulated error was
thus much smaller than the gap we
opened between our new bound and
4. By carefully bounding the floating-point error, we obtained 4.00253176
as a certified lower bound on λ. In
particular, we thus now know that the
leading digit of λ is 4.
We acknowledge the support of the facilities and staff of the Hasso Plattner
Institute Future SOC Lab in Potsdam,
Germany, who let us use their Hewlett
Packard ProLiant DL980 G7 server. A
technical version of this article6 was
presented at the 23rd Annual European
Symposium on Algorithms in Patras,
Greece, September 2015.
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Gill Barequet ( email@example.com) is an associate
professor in and vice dean of the Department of Computer
Science at the Technion - Israel Institute of Technology,
Günter Rote ( firstname.lastname@example.org) is a professor in the
Department of Computer Science at Freie Universität
Mira Shalah ( email@example.com) is a Ph.D.
student, under the supervision of Gill Barequet, in
the Department of Computer Science at the Technion -
Israel Institute of Technology, Haifa, Israel.
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