8 × 8: three exact determinants agree to the digit, and the float one does not
At its defaults it draws 8 × 8: three exact determinants agree to the digit, and the float one does not. det A = -8650618 on a random matrix of order 8, computed three ways with no rounding anywhere: fraction-free elimination, elimination over the rationals, and elimination modulo 2 word-sized primes put back together by the Chinese remainder theorem. The three differ by exactly zero — as integers, not to a tolerance. The floating-point LU of the same matrix returns -8.650618·10⁶, a relative error of 2.153·10⁻¹⁶, which is what a determinant of 24 bits does to a significand of 53. The modular route's entries never exceed 26 bits, which is why the number of primes has to be decided in advance: Hadamard's bound says 33 bits are needed and 2 primes supply them.
det-routes is one function in lib/figures/integer.js —
exact arithmetic — no residual to print, and a cost measured in the length of the numbers. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
det A = -8650618 on a random matrix of order 8, computed three ways with no rounding anywhere: fraction-free elimination, elimination over the rationals, and elimination modulo 2 word-sized primes put back together by the Chinese remainder theorem. The three differ by exactly zero — as integers, not to a tolerance. The floating-point LU of the same matrix returns -8.650618·10⁶, a relative error of 2.153·10⁻¹⁶, which is what a determinant of 24 bits does to a significand of 53. The modular route's entries never exceed 26 bits, which is why the number of primes has to be decided in advance: Hadamard's bound says 33 bits are needed and 2 primes supply them.
family: "random", n: 10
The arguments are the ones A prime that divides the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
det A = -77985307738 on a random matrix of order 10, computed three ways with no rounding anywhere: fraction-free elimination, elimination over the rationals, and elimination modulo 2 word-sized primes put back together by the Chinese remainder theorem. The three differ by exactly zero — as integers, not to a tolerance. The floating-point LU of the same matrix returns -7.79853077·10¹⁰, a relative error of 3.913·10⁻¹⁶, which is what a determinant of 37 bits does to a significand of 53. The modular route's entries never exceed 26 bits, which is why the number of primes has to be decided in advance: Hadamard's bound says 43 bits are needed and 2 primes supply them.
family: "random", n: 12
The arguments are the ones An answer with no error in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
det A = 647452350646 on a random matrix of order 12, computed three ways with no rounding anywhere: fraction-free elimination, elimination over the rationals, and elimination modulo 3 word-sized primes put back together by the Chinese remainder theorem. The three differ by exactly zero — as integers, not to a tolerance. The floating-point LU of the same matrix returns 6.47452351·10¹¹, a relative error of 3.394·10⁻¹⁵, which is what a determinant of 40 bits does to a significand of 53. The modular route's entries never exceed 26 bits, which is why the number of primes has to be decided in advance: Hadamard's bound says 53 bits are needed and 3 primes supply them.
family: "wide", n: 8
The arguments are the ones An answer with no error in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
det A = 3458508270104059228258152 on a wide matrix of order 8, computed three ways with no rounding anywhere: fraction-free elimination, elimination over the rationals, and elimination modulo 4 word-sized primes put back together by the Chinese remainder theorem. The three differ by exactly zero — as integers, not to a tolerance. The floating-point LU of the same matrix returns 3.45850827·10²⁴, a relative error of 3.105·10⁻¹⁶, which is what a determinant of 82 bits does to a significand of 53. The modular route's entries never exceed 26 bits, which is why the number of primes has to be decided in advance: Hadamard's bound says 87 bits are needed and 4 primes supply them.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
7 distinct claims across 4 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a family the library builds
a size all three routes are affordable at
an integer matrix has an integer determinant
and with the modular route through the CRT
enough primes for Hadamard's bound on this determinant
every Bareiss division is exact
fraction-free agrees with rational elimination
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A prime that divides the answer
A modular elimination reports a singular matrix and is telling the truth — over the field with p elements the matrix is singular. Over the rationals it is not. Nothing in the residue distinguishes the two cases, no quantity is small enough to be suspicious, and the wrong answer is a correct computation of a different question.
Exact arithmetic, and what it costs insteadAn answer with no error in it
An integer matrix eliminated over the rationals rounds nothing, so the forward error is zero, the residual is the zero vector, and the identity this site is built on has no terms left. The cost does not vanish with the error. It moves into the length of the numbers, where three correct routes differ by four orders of magnitude.
Exact arithmetic, and what it costs insteadHow many primes the answer needs
Work modulo a word-sized prime and no intermediate can exceed twenty-six bits, whatever the matrix does. The catch is that the answer must be reassembled from several such computations, and the number of them has to be fixed before the first one runs — by a theorem about how large a determinant can be, not by trying more until it settles.