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Fix contraction constant in inv for interval matrices - #793

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OlivierHnt merged 1 commit into
JuliaIntervals:masterfrom
jan-bouwe:fix-inv-opnorm
Nov 8, 2026
Merged

Fix contraction constant in inv for interval matrices#793
OlivierHnt merged 1 commit into
JuliaIntervals:masterfrom
jan-bouwe:fix-inv-opnorm

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@jan-bouwe

@jan-bouwe jan-bouwe commented Nov 8, 2026

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inv uses the right residual F = A * approx_A⁻¹ - I. Then E = A⁻¹ - approx_A⁻¹ satisfies E = -approx_A⁻¹ * F - E * F, so the contraction constant must be the largest column sum opnorm(F, 1), not the largest row sum opnorm(F, Inf). With the row sum the enclosure can miss the true inverse:

A = [interval(0.75, 1.25) interval(-0.5, 0.5); interval(0.0) interval(1.0, 3.0)]
in_interval.([4//3 -2//3; 0 1], inv(A))   # [1 0; 1 1] before this fix

This PR uses opnorm(F, 1) and adds this example as a test.

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Benchmark Results (Julia v1)

Time benchmarks
master 1c7453a... master / 1c7453a...
BigFloat MPFI/basics/* 22.1 ± 3.8 μs 23 ± 3.7 μs 0.96 ± 0.23
BigFloat MPFI/basics/+ 12.8 ± 3.9 μs 12.6 ± 3.8 μs 1.02 ± 0.44
BigFloat MPFI/basics/- 14 ± 3.7 μs 13 ± 3.8 μs 1.08 ± 0.42
BigFloat MPFI/basics// 24.7 ± 8.5 μs 24.5 ± 7.6 μs 1.01 ± 0.47
BigFloat MPFI/basics/abs 9.81 ± 3.6 μs 9.84 ± 3.6 μs 0.997 ± 0.51
BigFloat MPFI/basics/acos 0.667 ± 0.011 ms 0.675 ± 0.01 ms 0.989 ± 0.022
BigFloat MPFI/basics/asin 0.656 ± 0.011 ms 0.673 ± 0.011 ms 0.975 ± 0.023
BigFloat MPFI/basics/atan 2.09 ± 0.0088 ms 2.09 ± 0.011 ms 1 ± 0.0068
BigFloat MPFI/basics/cos 0.239 ± 0.0095 ms 0.241 ± 0.0099 ms 0.988 ± 0.056
BigFloat MPFI/basics/cosh 0.427 ± 0.013 ms 0.428 ± 0.013 ms 0.997 ± 0.042
BigFloat MPFI/basics/exp 0.5 ± 0.012 ms 0.501 ± 0.013 ms 0.997 ± 0.034
BigFloat MPFI/basics/inv 17.3 ± 3.6 μs 17.4 ± 3.7 μs 0.996 ± 0.3
BigFloat MPFI/basics/log 0.467 ± 0.011 ms 0.472 ± 0.011 ms 0.99 ± 0.034
BigFloat MPFI/basics/sin 0.339 ± 0.012 ms 0.342 ± 0.012 ms 0.991 ± 0.048
BigFloat MPFI/basics/sinh 0.568 ± 0.011 ms 0.568 ± 0.011 ms 0.999 ± 0.027
BigFloat MPFI/basics/sqrt 22.7 ± 3.8 μs 22.8 ± 3.7 μs 0.998 ± 0.23
BigFloat MPFI/basics/tan 0.272 ± 0.01 ms 0.274 ± 0.01 ms 0.994 ± 0.052
BigFloat MPFI/basics/tanh 0.581 ± 0.011 ms 0.583 ± 0.011 ms 0.995 ± 0.026
BigFloat bareinterval/basics/* 0.0398 ± 0.0026 ms 0.0397 ± 0.0028 ms 1 ± 0.097
BigFloat bareinterval/basics/+ 23.3 ± 3.6 μs 22.3 ± 3.6 μs 1.04 ± 0.23
BigFloat bareinterval/basics/- 23.5 ± 3.7 μs 23.2 ± 3.7 μs 1.02 ± 0.23
BigFloat bareinterval/basics// 19.9 ± 5.4 μs 20.1 ± 5.2 μs 0.99 ± 0.37
BigFloat bareinterval/basics/abs 16.1 ± 3.6 μs 15.8 ± 3.5 μs 1.01 ± 0.32
BigFloat bareinterval/basics/acos 0.703 ± 0.015 ms 0.709 ± 0.014 ms 0.992 ± 0.029
BigFloat bareinterval/basics/asin 0.695 ± 0.014 ms 0.714 ± 0.015 ms 0.973 ± 0.028
BigFloat bareinterval/basics/atan 2.11 ± 0.0098 ms 2.09 ± 0.013 ms 1.01 ± 0.0076
BigFloat bareinterval/basics/cos 0.432 ± 0.1 ms 0.434 ± 0.1 ms 0.996 ± 0.33
BigFloat bareinterval/basics/cosh 0.479 ± 0.013 ms 0.484 ± 0.014 ms 0.991 ± 0.04
BigFloat bareinterval/basics/exp 0.511 ± 0.012 ms 0.51 ± 0.012 ms 1 ± 0.034
BigFloat bareinterval/basics/inv 27.1 ± 3.7 μs 26.7 ± 3.6 μs 1.02 ± 0.19
BigFloat bareinterval/basics/log 0.501 ± 0.012 ms 0.506 ± 0.012 ms 0.99 ± 0.034
BigFloat bareinterval/basics/sin 0.57 ± 0.1 ms 0.573 ± 0.1 ms 0.995 ± 0.25
BigFloat bareinterval/basics/sinh 0.578 ± 0.012 ms 0.577 ± 0.012 ms 1 ± 0.029
BigFloat bareinterval/basics/sqrt 0.0322 ± 0.0026 ms 0.0325 ± 0.0025 ms 0.99 ± 0.11
BigFloat bareinterval/basics/tan 0.337 ± 0.061 ms 0.338 ± 0.061 ms 0.997 ± 0.26
BigFloat bareinterval/basics/tanh 0.592 ± 0.012 ms 0.593 ± 0.012 ms 0.999 ± 0.029
BigFloat interval/basics/* 0.0442 ± 0.013 ms 0.0429 ± 0.012 ms 1.03 ± 0.42
BigFloat interval/basics/+ 18.5 ± 9.1 μs 17.1 ± 8.9 μs 1.08 ± 0.78
BigFloat interval/basics/- 17.5 ± 0.42 μs 17.7 ± 0.38 μs 0.989 ± 0.032
BigFloat interval/basics// 21.1 ± 0.46 μs 20.6 ± 0.71 μs 1.02 ± 0.042
BigFloat interval/basics/abs 15.9 ± 8.7 μs 16.1 ± 8.7 μs 0.993 ± 0.76
BigFloat interval/basics/acos 0.696 ± 0.038 ms 0.698 ± 0.039 ms 0.997 ± 0.078
BigFloat interval/basics/asin 0.682 ± 0.011 ms 0.702 ± 0.012 ms 0.972 ± 0.022
BigFloat interval/basics/atan 2.07 ± 0.012 ms 2.09 ± 0.01 ms 0.991 ± 0.0074
BigFloat interval/basics/cos 0.335 ± 0.013 ms 0.336 ± 0.013 ms 0.998 ± 0.055
BigFloat interval/basics/cosh 0.462 ± 0.013 ms 0.463 ± 0.012 ms 0.997 ± 0.037
BigFloat interval/basics/exp 0.517 ± 0.01 ms 0.516 ± 0.01 ms 1 ± 0.028
BigFloat interval/basics/inv 22.3 ± 0.44 μs 21.7 ± 0.44 μs 1.03 ± 0.029
BigFloat interval/basics/log 0.497 ± 0.018 ms 0.499 ± 0.014 ms 0.996 ± 0.044
BigFloat interval/basics/sin 0.485 ± 0.015 ms 0.49 ± 0.014 ms 0.989 ± 0.042
BigFloat interval/basics/sinh 0.585 ± 0.011 ms 0.585 ± 0.011 ms 1 ± 0.027
BigFloat interval/basics/sqrt 0.0393 ± 0.012 ms 0.0389 ± 0.011 ms 1.01 ± 0.41
BigFloat interval/basics/tan 0.279 ± 0.013 ms 0.281 ± 0.013 ms 0.995 ± 0.065
BigFloat interval/basics/tanh 0.598 ± 0.01 ms 0.601 ± 0.01 ms 0.995 ± 0.024
bareinterval/basics/* 1.62 ± 0.051 μs 1.61 ± 0.03 μs 1.01 ± 0.037
bareinterval/basics/+ 0.511 ± 0.031 μs 0.521 ± 0.03 μs 0.981 ± 0.082
bareinterval/basics/- 0.551 ± 0.039 μs 0.561 ± 0.011 μs 0.982 ± 0.072
bareinterval/basics// 1.28 ± 0.05 μs 1.28 ± 0.04 μs 0.999 ± 0.05
bareinterval/basics/abs 0.16 ± 0.01 μs 0.161 ± 0.01 μs 0.994 ± 0.088
bareinterval/basics/acos 2.71 ± 0.03 μs 2.71 ± 0.029 μs 1 ± 0.015
bareinterval/basics/asin 2.98 ± 0.011 μs 2.98 ± 0.02 μs 1 ± 0.0077
bareinterval/basics/atan 13.3 ± 0.26 μs 13.5 ± 0.23 μs 0.982 ± 0.026
bareinterval/basics/cos 0.0913 ± 0.032 ms 0.0922 ± 0.033 ms 0.99 ± 0.49
bareinterval/basics/cosh 7.17 ± 0.05 μs 7.16 ± 0.049 μs 1 ± 0.0098
bareinterval/basics/exp 4.23 ± 0.05 μs 4.26 ± 0.05 μs 0.993 ± 0.017
bareinterval/basics/inv 1.05 ± 0.02 μs 1.03 ± 0.031 μs 1.02 ± 0.036
bareinterval/basics/log 1.93 ± 0.02 μs 1.95 ± 0.02 μs 0.99 ± 0.014
bareinterval/basics/sin 0.0896 ± 0.0073 ms 0.09 ± 0.014 ms 0.995 ± 0.17
bareinterval/basics/sinh 10 ± 0.06 μs 9.96 ± 0.059 μs 1.01 ± 0.0085
bareinterval/basics/sqrt 1.17 ± 0.021 μs 1.24 ± 0.021 μs 0.943 ± 0.023
bareinterval/basics/tan 0.0506 ± 0.0049 ms 0.0506 ± 0.0046 ms 1 ± 0.13
bareinterval/basics/tanh 0.305 ± 0.011 ms 0.309 ± 0.01 ms 0.986 ± 0.049
interval/basics/* 2.77 ± 0.06 μs 2.75 ± 0.08 μs 1 ± 0.036
interval/basics/+ 1.34 ± 0.079 μs 1.33 ± 0.061 μs 1.01 ± 0.075
interval/basics/- 1.35 ± 0.031 μs 1.35 ± 0.039 μs 1 ± 0.037
interval/basics// 1.82 ± 0.58 μs 1.82 ± 0.99 μs 1 ± 0.63
interval/basics/abs 0.341 ± 0.11 μs 0.351 ± 0.099 μs 0.972 ± 0.42
interval/basics/acos 3.32 ± 0.081 μs 3.36 ± 0.05 μs 0.988 ± 0.028
interval/basics/asin 3.6 ± 0.04 μs 3.55 ± 0.04 μs 1.01 ± 0.016
interval/basics/atan 12 ± 0.16 μs 13.3 ± 0.22 μs 0.904 ± 0.019
interval/basics/cos 0.0907 ± 0.031 ms 0.0913 ± 0.032 ms 0.994 ± 0.49
interval/basics/cosh 8.91 ± 0.079 μs 8.91 ± 0.071 μs 1 ± 0.012
interval/basics/exp 4.82 ± 0.051 μs 4.84 ± 0.08 μs 0.996 ± 0.02
interval/basics/inv 1.56 ± 0.03 μs 1.57 ± 0.06 μs 0.994 ± 0.042
interval/basics/log 2.54 ± 0.04 μs 2.54 ± 0.051 μs 1 ± 0.026
interval/basics/sin 0.0895 ± 0.031 ms 0.0906 ± 0.031 ms 0.988 ± 0.48
interval/basics/sinh 10.6 ± 0.09 μs 10.7 ± 0.083 μs 0.987 ± 0.011
interval/basics/sqrt 1.62 ± 0.17 μs 1.63 ± 0.04 μs 0.994 ± 0.11
interval/basics/tan 0.0508 ± 0.0039 ms 0.0513 ± 0.0037 ms 0.991 ± 0.1
interval/basics/tanh 0.302 ± 0.013 ms 0.305 ± 0.013 ms 0.992 ± 0.06
time_to_load 0.105 ± 0.00073 s 0.107 ± 0.00041 s 0.979 ± 0.0078
Memory benchmarks
master 1c7453a... master / 1c7453a...
BigFloat MPFI/basics/* 0.332 k allocs: 17.7 kB 0.332 k allocs: 17.7 kB 1
BigFloat MPFI/basics/+ 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/- 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics// 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/abs 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/acos 1.14 k allocs: 0.0407 MB 1.14 k allocs: 0.0407 MB 1
BigFloat MPFI/basics/asin 1.08 k allocs: 0.0417 MB 1.08 k allocs: 0.0417 MB 1
BigFloat MPFI/basics/atan 2.14 k allocs: 0.0543 MB 2.14 k allocs: 0.0543 MB 1
BigFloat MPFI/basics/cos 1.56 k allocs: 0.0425 MB 1.56 k allocs: 0.0425 MB 1
BigFloat MPFI/basics/cosh 0.759 k allocs: 0.0313 MB 0.759 k allocs: 0.0313 MB 1
BigFloat MPFI/basics/exp 0.902 k allocs: 0.036 MB 0.902 k allocs: 0.036 MB 1
BigFloat MPFI/basics/inv 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/log 0.407 k allocs: 20.6 kB 0.407 k allocs: 20.6 kB 1
BigFloat MPFI/basics/sin 1.91 k allocs: 0.0536 MB 1.91 k allocs: 0.0536 MB 1
BigFloat MPFI/basics/sinh 0.915 k allocs: 0.0369 MB 0.915 k allocs: 0.0369 MB 1
BigFloat MPFI/basics/sqrt 0.302 k allocs: 16.5 kB 0.302 k allocs: 16.5 kB 1
BigFloat MPFI/basics/tan 1.65 k allocs: 0.0493 MB 1.65 k allocs: 0.0492 MB 1
BigFloat MPFI/basics/tanh 0.908 k allocs: 0.0362 MB 0.908 k allocs: 0.0362 MB 1
BigFloat bareinterval/basics/* 0.262 k allocs: 26 kB 0.262 k allocs: 26 kB 1
BigFloat bareinterval/basics/+ 0.202 k allocs: 20.4 kB 0.202 k allocs: 20.4 kB 1
BigFloat bareinterval/basics/- 0.202 k allocs: 20.4 kB 0.202 k allocs: 20.4 kB 1
BigFloat bareinterval/basics// 0.1 k allocs: 10.8 kB 0.1 k allocs: 10.8 kB 1
BigFloat bareinterval/basics/abs 0.199 k allocs: 20.1 kB 0.199 k allocs: 20.1 kB 1
BigFloat bareinterval/basics/acos 1.27 k allocs: 0.0692 MB 1.27 k allocs: 0.0692 MB 1
BigFloat bareinterval/basics/asin 1.17 k allocs: 0.0659 MB 1.17 k allocs: 0.0659 MB 1
BigFloat bareinterval/basics/atan 2.04 k allocs: 0.058 MB 2.04 k allocs: 0.058 MB 1
BigFloat bareinterval/basics/cos 2.58 k allocs: 0.196 MB 2.58 k allocs: 0.196 MB 1
BigFloat bareinterval/basics/cosh 1.01 k allocs: 0.0671 MB 1.01 k allocs: 0.0671 MB 1
BigFloat bareinterval/basics/exp 0.802 k allocs: 0.0397 MB 0.802 k allocs: 0.0397 MB 1
BigFloat bareinterval/basics/inv 0.198 k allocs: 20 kB 0.198 k allocs: 20 kB 1
BigFloat bareinterval/basics/log 0.616 k allocs: 0.0522 MB 0.616 k allocs: 0.0522 MB 1
BigFloat bareinterval/basics/sin 3.02 k allocs: 0.209 MB 3.02 k allocs: 0.209 MB 1
BigFloat bareinterval/basics/sinh 0.815 k allocs: 0.0407 MB 0.815 k allocs: 0.0407 MB 1
BigFloat bareinterval/basics/sqrt 0.26 k allocs: 25.8 kB 0.26 k allocs: 25.8 kB 1
BigFloat bareinterval/basics/tan 1.73 k allocs: 0.129 MB 1.73 k allocs: 0.129 MB 1
BigFloat bareinterval/basics/tanh 0.808 k allocs: 0.0399 MB 0.808 k allocs: 0.0399 MB 1
BigFloat interval/basics/* 0.263 k allocs: 26.8 kB 0.263 k allocs: 26.8 kB 1
BigFloat interval/basics/+ 0.203 k allocs: 21.2 kB 0.203 k allocs: 21.2 kB 1
BigFloat interval/basics/- 0.203 k allocs: 21.2 kB 0.203 k allocs: 21.2 kB 1
BigFloat interval/basics// 0.101 k allocs: 11.6 kB 0.101 k allocs: 11.6 kB 1
BigFloat interval/basics/abs 0.2 k allocs: 20.9 kB 0.2 k allocs: 20.9 kB 1
BigFloat interval/basics/acos 1.71 k allocs: 0.11 MB 1.71 k allocs: 0.11 MB 1
BigFloat interval/basics/asin 1.47 k allocs: 0.0944 MB 1.47 k allocs: 0.0943 MB 1
BigFloat interval/basics/atan 2.04 k allocs: 0.0592 MB 2.04 k allocs: 0.0594 MB 0.997
BigFloat interval/basics/cos 2.58 k allocs: 0.197 MB 2.58 k allocs: 0.197 MB 1
BigFloat interval/basics/cosh 1.01 k allocs: 0.0681 MB 1.01 k allocs: 0.0681 MB 1
BigFloat interval/basics/exp 0.803 k allocs: 0.0407 MB 0.803 k allocs: 0.0407 MB 1
BigFloat interval/basics/inv 0.199 k allocs: 20.8 kB 0.199 k allocs: 20.8 kB 1
BigFloat interval/basics/log 0.917 k allocs: 0.0805 MB 0.917 k allocs: 0.0805 MB 1
BigFloat interval/basics/sin 3.02 k allocs: 0.209 MB 3.02 k allocs: 0.209 MB 1
BigFloat interval/basics/sinh 0.816 k allocs: 0.0417 MB 0.816 k allocs: 0.0417 MB 0.999
BigFloat interval/basics/sqrt 0.714 k allocs: 0.0675 MB 0.714 k allocs: 0.0675 MB 1
BigFloat interval/basics/tan 1.73 k allocs: 0.13 MB 1.73 k allocs: 0.13 MB 1
BigFloat interval/basics/tanh 0.809 k allocs: 0.0411 MB 0.809 k allocs: 0.0409 MB 1
bareinterval/basics/* 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/+ 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/- 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics// 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/abs 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/acos 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/asin 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/atan 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/cos 0.806 k allocs: 0.0623 MB 0.806 k allocs: 0.0623 MB 1
bareinterval/basics/cosh 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/exp 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/inv 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/log 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/sin 0.806 k allocs: 0.0623 MB 0.806 k allocs: 0.0623 MB 1
bareinterval/basics/sinh 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/sqrt 2 allocs: 1.62 kB 2 allocs: 1.62 kB 1
bareinterval/basics/tan 0.416 k allocs: 0.0328 MB 0.416 k allocs: 0.0328 MB 1
bareinterval/basics/tanh 1.03 k allocs: 0.0359 MB 1.03 k allocs: 0.0359 MB 1
interval/basics/* 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/+ 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/- 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics// 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/abs 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/acos 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/asin 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/atan 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/cos 0.807 k allocs: 0.0631 MB 0.807 k allocs: 0.0631 MB 1
interval/basics/cosh 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/exp 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/inv 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/log 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/sin 0.807 k allocs: 0.0631 MB 0.807 k allocs: 0.0631 MB 1
interval/basics/sinh 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/sqrt 3 allocs: 2.45 kB 3 allocs: 2.45 kB 1
interval/basics/tan 0.417 k allocs: 0.0336 MB 0.417 k allocs: 0.0336 MB 1
interval/basics/tanh 1.03 k allocs: 0.0367 MB 1.03 k allocs: 0.0367 MB 1
time_to_load 0.2 k allocs: 11.8 kB 0.2 k allocs: 11.8 kB 1

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Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 88.41%. Comparing base (cdf05f8) to head (1c7453a).
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@@            Coverage Diff             @@
##           master     #793      +/-   ##
==========================================
- Coverage   94.66%   88.41%   -6.26%     
==========================================
  Files          33       33              
  Lines        3376     3350      -26     
==========================================
- Hits         3196     2962     -234     
- Misses        180      388     +208

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@OlivierHnt

OlivierHnt commented Nov 8, 2026

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Thanks for reporting this issue. LGTM.

@OlivierHnt
OlivierHnt merged commit a30de47 into JuliaIntervals:master Nov 8, 2026
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@jan-bouwe @codecov-commenter @OlivierHnt