Speed up powering a matrix by a large exponent when its minimal polynomial is small - #6495
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Speed up powering a matrix by a large exponent when its minimal polynomial is small#6495fingolfin wants to merge 1 commit into
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…omial is small Powering a matrix by a large exponent reduces x^n modulo the characteristic polynomial, after a base change making the matrix block companion. If the minimal polynomial has degree e rather than d, both the Log2(n) polynomial multiplications and the final evaluation work with degree e, which is worth a factor of 2 to 60. Computing the minimal polynomial of a dense matrix costs about as much as the whole characteristic polynomial method, so probe first with a single spun random vector, abandoned once its order polynomial grows past the bound where the minimal polynomial stops being worth using. That order polynomial divides the minimal polynomial, so abandoning settles the question, and the probe does not show up in the timings. The break even points were determined over a grid of dimensions from 60 to 2000, representations GF(2), GF(5), GF(251), GF(257) and GF(3^10), minimal polynomial degrees from d/30 up to d, and exponents from 2^20 to 2^1000. They depend mostly on the representation: 6*e <= d for compressed matrices, 2*e <= d for the rest, where matrix multiplication is far more expensive than polynomial arithmetic. The choice is within a factor of 1.07 of the best of the three methods everywhere measured, and at least 1.5 times faster than before on a third of the cells. Matrix objects without access to their rows, such as those in IsGenericMatrixRep, now work here at all: the helper evaluating a polynomial at a matrix tested mutability by looking at the first row. With the preceding commit they reach the generic methods for the characteristic and minimal polynomial again, so powering them uses the polynomial method rather than failing with "row access unsupported". For this, Matrix_OrderPolynomialInner gained an optional degree bound, and the helpers of POW_MAT_INT moved out of its body to be shared. This commit was prepared with assistance from the AI tool Claude Code (benchmarking, analysis and drafting of the implementation and tests). Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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Codecov Report❌ Patch coverage is
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## master #6495 +/- ##
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+ Coverage 79.03% 79.07% +0.03%
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Files 685 685
Lines 293804 293856 +52
Branches 8664 8641 -23
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+ Hits 232214 232368 +154
+ Misses 59788 59686 -102
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Powering a matrix by a large exponent reduces
x^nmodulo the characteristic polynomial. Doing it modulo the minimal polynomial instead, when that has degreeerather thand, is worth a factor of 2 to 60.The minimal polynomial of a dense matrix costs about as much as the whole method, so a single spun random vector probes for it first, abandoned once its order polynomial passes the break even point. Depending on
e, the reduced polynomial is then evaluated at the matrix itself or at its block companion form.Break even points measured over dimensions 60 to 2000,
GF(2),GF(5),GF(251),GF(257),GF(3^10), minimal polynomial degreesd/30tod, and exponents2^20to2^1000:6*e <= dfor compressed matrices,2*e <= dotherwise, where matrix multiplication is far more expensive than polynomial arithmetic. The choice is within a factor of 1.07 of the best of the three methods on every measured cell, and at least 1.5 times faster than before on a third of them.This also fixes
m^nfor matrix objects without access to their rows, such as those inIsGenericMatrixRep, which failed withrow access unsupported.AI disclosure: benchmarking, analysis and drafting were done with the AI tool Claude Code; the commit carries a corresponding
Co-authored-by:line.