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perf: Tile 8×8 covariance matrix multiplication (#1181)
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Currently, we are multiplying an 8x8 covariance matrix with an 8x8 transport matrix, and we see that Eigen is failing to optimize this properly, because it is calling a generalized GEMM method rather than an optimized small matrix method. In order to resolve this, we change the code to use a tiled multiplication method which splits the matrices into 4x4 sub-matrices which can be multiplied and added to achieve the desired effect. This has two advantages:

  1. It allows Eigen to use its hand-rolled optimized 4x4 matrix multiplication methods.
  2. It allows us to perform some trickery with matrix identities to reduce the number of floating point operations.


Co-authored-by: Andreas Stefl <487211+andiwand@users.noreply.github.com>
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@stephenswat @andiwand
stephenswat and andiwand committed Mar 4, 2024
1 parent a1b40bc commit 6182aef
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37 changes: 35 additions & 2 deletions Core/include/Acts/Propagator/EigenStepper.ipp
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Original file line number Diff line number Diff line change
Expand Up @@ -233,8 +233,41 @@ Acts::Result<double> Acts::EigenStepper<E, A>::step(
return EigenStepperError::StepInvalid;
}

// for moment, only update the transport part
state.stepping.jacTransport = D * state.stepping.jacTransport;
// See the documentation of Acts::blockedMult for a description of blocked
// matrix multiplication. However, we can go one further. Let's assume that
// some of these sub-matrices are zero matrices 0₈ and identity matrices
// I₈, namely:
//
// D₁₁ = I₈, J₁₁ = I₈, D₂₁ = 0₈, J₂₁ = 0₈
//
// Which gives:
//
// K₁₁ = I₈ * I₈ + D₁₂ * 0₈ = I₈
// K₁₂ = I₈ * J₁₂ + D₁₂ * J₂₂ = J₁₂ + D₁₂ * J₂₂
// K₂₁ = 0₈ * I₈ + D₂₂ * 0₈ = 0₈
// K₂₂ = 0₈ * J₁₂ + D₂₂ * J₂₂ = D₂₂ * J₂₂
//
// Furthermore, we're constructing K in place of J, and since
// K₁₁ = I₈ = J₁₁ and K₂₁ = 0₈ = D₂₁, we don't actually need to touch those
// sub-matrices at all!
if ((D.topLeftCorner<4, 4>().isIdentity()) &&
(D.bottomLeftCorner<4, 4>().isZero()) &&
(state.stepping.jacTransport.template topLeftCorner<4, 4>()
.isIdentity()) &&
(state.stepping.jacTransport.template bottomLeftCorner<4, 4>()
.isZero())) {
state.stepping.jacTransport.template topRightCorner<4, 4>() +=
D.topRightCorner<4, 4>() *
state.stepping.jacTransport.template bottomRightCorner<4, 4>();
state.stepping.jacTransport.template bottomRightCorner<4, 4>() =
(D.bottomRightCorner<4, 4>() *
state.stepping.jacTransport.template bottomRightCorner<4, 4>())
.eval();
} else {
// For safety purposes, we provide a full matrix multiplication as a
// backup strategy.
state.stepping.jacTransport = D * state.stepping.jacTransport;
}
} else {
if (!state.stepping.extension.finalize(state, *this, navigator, h)) {
return EigenStepperError::StepInvalid;
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44 changes: 22 additions & 22 deletions Examples/Python/tests/root_file_hashes.txt
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Original file line number Diff line number Diff line change
Expand Up @@ -29,9 +29,9 @@ test_truth_tracking_kalman[odd-0.0]__performance_track_finder.root: 39aec6316cce
test_truth_tracking_kalman[odd-1000.0]__trackstates_fitter.root: 72c79be1458c4f9c9a1661778c900f0875d257f2d391c4183a698825448919a1
test_truth_tracking_kalman[odd-1000.0]__tracksummary_fitter.root: 3d424dec9b172f253c8c4ffbda470f678fd1081a3d36dcfea517ab0f94995ae4
test_truth_tracking_kalman[odd-1000.0]__performance_track_finder.root: 39aec6316cceb90e314e16b02947faa691c18f57c3a851a25e547a8fc05a4593
test_truth_tracking_gsf[generic]__trackstates_gsf.root: b1ff88d67cb89cc76850e242aa5ad3bcfd041bf5e4a62e614a878bb974967242
test_truth_tracking_gsf[generic]__trackstates_gsf.root: e971da3dcc1997bc9f8a74e0f75a46d4b92c9cf6286b0e05dc8a1bd34df75444
test_truth_tracking_gsf[generic]__tracksummary_gsf.root: ec3ef893f7138392bcdce96322e2dd31951737271b72ac9dd3282fc187700d03
test_truth_tracking_gsf[odd]__trackstates_gsf.root: 4117e585b59e5332bc54d1928096d8d2477a9f5b9cacb2f010b4df25106ad768
test_truth_tracking_gsf[odd]__trackstates_gsf.root: c28b23358cf88fb40eeb3c47db9af62462cc7cdd9d70dec13d0615f136065191
test_truth_tracking_gsf[odd]__tracksummary_gsf.root: 3425c0974943e0dac06dfedceaf1e9e3bec3bdc6ff73588d849a234de017e230
test_particle_gun__particles.root: 7eec62018b6944fea565dad75aa41ef87d1f2737b2a814fbab189817ac8180fe
test_material_mapping__material-map_tracks.root: 6e1441c418ff0b17983c2d0483248cc1dee6b77b09d0ca9d03c742c9d1373630
Expand All @@ -44,22 +44,22 @@ test_digitization_example_input[smeared]__particles.root: 7eec62018b6944fea565da
test_digitization_example_input[smeared]__measurements.root: 0c168d371d0130c68d1ee44bd77eeeb3cf702a77c2afbf12bed8354b61a29262
test_digitization_example_input[geometric]__particles.root: 7eec62018b6944fea565dad75aa41ef87d1f2737b2a814fbab189817ac8180fe
test_digitization_example_input[geometric]__measurements.root: 0c6d88b4de3ee7365103b8f0d6be6b4db3d7b7f2a59d3db58a1e5f89fa8130b3
test_ckf_tracks_example[generic-full_seeding]__trackstates_ckf.root: aa727b8c2c30a3d2ea155917abad1a77071da16e361c6ada500405839b697310
test_ckf_tracks_example[generic-full_seeding]__tracksummary_ckf.root: e122dd3afe20a1ebf870dab5a42ea154c3b5eedc13c60e2e99efc6bb8a417c21
test_ckf_tracks_example[generic-full_seeding]__trackstates_ckf.root: 52fe86b966e4bfe05039d17025d15e360f25ef5041abe6729e02e8e64e0d2fb2
test_ckf_tracks_example[generic-full_seeding]__tracksummary_ckf.root: 29d861eb98477288adebf347be91aeed63942351bfba59cf504cda4ebf6ced78
test_ckf_tracks_example[generic-full_seeding]__performance_seeding_trees.root: 0e0676ffafdb27112fbda50d1cf627859fa745760f98073261dcf6db3f2f991e
test_ckf_tracks_example[generic-truth_estimated]__trackstates_ckf.root: 50eb0539a059b5e7c51ad4953629e82d8b08159d179181ca39570f910be47168
test_ckf_tracks_example[generic-truth_estimated]__tracksummary_ckf.root: a474c6fbd5af9d43d973783288f980984df694a8eb8dd3ebe9e5c8df4bbd7924
test_ckf_tracks_example[generic-truth_estimated]__trackstates_ckf.root: 40e52eb4b31997c6d2c41e13d10cddf2921224f5bcb1f58a16b47ea79b973234
test_ckf_tracks_example[generic-truth_estimated]__tracksummary_ckf.root: 09a470d45be79b33910e81a39400d7bf617d05ddc33fc85ad3a6043597813143
test_ckf_tracks_example[generic-truth_estimated]__performance_seeding.root: 1facb05c066221f6361b61f015cdf0918e94d9f3fce2269ec7b6a4dffeb2bc7e
test_ckf_tracks_example[generic-truth_smeared]__trackstates_ckf.root: 9c7dc2b10e4d52a565e2ecd72077c4c90862a0c6ef3b4bad0b5dbfb50d8b91df
test_ckf_tracks_example[generic-truth_smeared]__tracksummary_ckf.root: 987c47fc59851e6307715ceea261aa650aef352876febdd228778c4d030a8653
test_ckf_tracks_example[odd-full_seeding]__trackstates_ckf.root: 1340a741753a4a1d6fed0de1867282e055e76fc5762655698d50c9978ba87219
test_ckf_tracks_example[odd-full_seeding]__tracksummary_ckf.root: a033bf22d4cc4c715fb948610151fcfeddc601a6ce427d21d4885fd7c1c971d4
test_ckf_tracks_example[generic-truth_smeared]__trackstates_ckf.root: a94c3f7d2495f6bdb4259e051d35dc020ed81f42a98ecae4dec4513bc99aad09
test_ckf_tracks_example[generic-truth_smeared]__tracksummary_ckf.root: ee5714ee81d5f3f6cf87e1fe792bfdd633566bab96fa82f2d65843193c463789
test_ckf_tracks_example[odd-full_seeding]__trackstates_ckf.root: 7149db7a21da9dd51ccb11bf58002bcd3201989fef1483946ac4adeca42391ab
test_ckf_tracks_example[odd-full_seeding]__tracksummary_ckf.root: 50e44670cc6691adf1a127e92bc79568bfb96cf272aafab473a990f88373a0db
test_ckf_tracks_example[odd-full_seeding]__performance_seeding_trees.root: 43c58577aafe07645e5660c4f43904efadf91d8cda45c5c04c248bbe0f59814f
test_ckf_tracks_example[odd-truth_estimated]__trackstates_ckf.root: 991050c7e3a2bf83f129e5725eac5a8e00a21c162199b29b69db38405fea0a34
test_ckf_tracks_example[odd-truth_estimated]__tracksummary_ckf.root: 5590db074051ab5726fcc2d30d9391f61d9beee2083a51375540d2cf9d38211c
test_ckf_tracks_example[odd-truth_estimated]__trackstates_ckf.root: ca9be15cf58f36a4e43142dc093471bddf9ff115cd2465f51e8c3ed23cf2bc67
test_ckf_tracks_example[odd-truth_estimated]__tracksummary_ckf.root: c69d5b9ba378d162df5726c2abc5c7159ef15de8c33e18575ace15cd66cfe2d2
test_ckf_tracks_example[odd-truth_estimated]__performance_seeding.root: 1a36b7017e59f1c08602ef3c2cb0483c51df248f112e3780c66594110719c575
test_ckf_tracks_example[odd-truth_smeared]__trackstates_ckf.root: 3ac3b12d50c2882d5e03fadb82884b1897dddda6366cece6a3ac433c4e156d25
test_ckf_tracks_example[odd-truth_smeared]__tracksummary_ckf.root: 3ec2e3dc3dc6dc2ed4dc112d668256f6ee0672c8e04baf7895ae473ad78b5870
test_ckf_tracks_example[odd-truth_smeared]__trackstates_ckf.root: 03fdef333c5451ccecc129c6325367d7145f34a0788a6bbad148c844f3c3a835
test_ckf_tracks_example[odd-truth_smeared]__tracksummary_ckf.root: 0aeb9a337af75bfb25509cb18e585e1567c185d8b16c4fbd8ff2eeec16cfdf7d
test_vertex_fitting_reading[Truth-False-100]__performance_vertexing.root: 76ef6084d758dfdfc0151ddec2170e12d73394424e3dac4ffe46f0f339ec8293
test_vertex_fitting_reading[Iterative-False-100]__performance_vertexing.root: 60372210c830a04f95ceb78c6c68a9b0de217746ff59e8e73053750c837b57eb
test_vertex_fitting_reading[Iterative-True-100]__performance_vertexing.root: e34f217d524a5051dbb04a811d3407df3ebe2cc4bb7f54f6bda0847dbd7b52c3
Expand All @@ -85,19 +85,19 @@ test_exatrkx[cpu-torch]__performance_track_finding.root: e0875db5eb3aa6b46ad4baf
test_exatrkx[gpu-onnx]__performance_track_finding.root: 4845dc9f62e287e20c80d479e246eb69d01b279964cfdff83543bea0ea9afbed
test_exatrkx[gpu-torch]__performance_track_finding.root: e0875db5eb3aa6b46ad4baf846d39b37b68f1efd3436448cc75919b637f8d8d9
test_ML_Ambiguity_Solver__performance_ambiML.root: 284ff5c3a08c0b810938e4ac2f8ba8fe2babb17d4c202b624ed69fff731a9006
test_truth_tracking_kalman[generic-False-0.0]__trackstates_fitter.root: b598a07612c58866563ae862e651d9f42c13249d83d7176a103a565fa60f9cb3
test_truth_tracking_kalman[generic-False-0.0]__tracksummary_fitter.root: 11b2e2a50343c636fa977175a30220805412d3200e164ae4c3f439fe2087fb88
test_truth_tracking_kalman[generic-False-0.0]__trackstates_fitter.root: f258c7ae1cd64efd48b4840d19890f08afdc02a692ea392d050cdb818b6d70c2
test_truth_tracking_kalman[generic-False-0.0]__tracksummary_fitter.root: 385a763e16f55d20570c5bfc15f51c87e39f4ca8e5b012a3333ce59a91cd5d34
test_truth_tracking_kalman[generic-False-1000.0]__trackstates_fitter.root: bdc70d86ba6e717307e32c147401dce80f76dc9f544e7ed0f26ebf2f9e5c3057
test_truth_tracking_kalman[generic-False-1000.0]__tracksummary_fitter.root: fc82abfc4e3016cda806e743a270bf78b6d4cc404cd52145ea1eabed85d32feb
test_truth_tracking_kalman[generic-True-0.0]__trackstates_fitter.root: b598a07612c58866563ae862e651d9f42c13249d83d7176a103a565fa60f9cb3
test_truth_tracking_kalman[generic-True-0.0]__tracksummary_fitter.root: 11b2e2a50343c636fa977175a30220805412d3200e164ae4c3f439fe2087fb88
test_truth_tracking_kalman[generic-True-0.0]__trackstates_fitter.root: f258c7ae1cd64efd48b4840d19890f08afdc02a692ea392d050cdb818b6d70c2
test_truth_tracking_kalman[generic-True-0.0]__tracksummary_fitter.root: 385a763e16f55d20570c5bfc15f51c87e39f4ca8e5b012a3333ce59a91cd5d34
test_truth_tracking_kalman[generic-True-1000.0]__trackstates_fitter.root: bdc70d86ba6e717307e32c147401dce80f76dc9f544e7ed0f26ebf2f9e5c3057
test_truth_tracking_kalman[generic-True-1000.0]__tracksummary_fitter.root: fc82abfc4e3016cda806e743a270bf78b6d4cc404cd52145ea1eabed85d32feb
test_truth_tracking_kalman[odd-False-0.0]__trackstates_fitter.root: b9d96ea97b967c4fb40edc93b00fc307b80fc509309625c8e2ccc136c3e1a40a
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test_truth_tracking_kalman[odd-False-0.0]__trackstates_fitter.root: d0a9203c917be9caacd1c6c4244538e2038e2472b4707b61bb5c0bb1c5e2a3ad
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test_truth_tracking_kalman[odd-False-1000.0]__trackstates_fitter.root: b858d09a75e852a7c23f2f3a512de8c334c593e700319d4f935e844ec00c379f
test_truth_tracking_kalman[odd-False-1000.0]__tracksummary_fitter.root: 3d424dec9b172f253c8c4ffbda470f678fd1081a3d36dcfea517ab0f94995ae4
test_truth_tracking_kalman[odd-True-0.0]__trackstates_fitter.root: b9d96ea97b967c4fb40edc93b00fc307b80fc509309625c8e2ccc136c3e1a40a
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test_truth_tracking_kalman[odd-True-1000.0]__tracksummary_fitter.root: 3d424dec9b172f253c8c4ffbda470f678fd1081a3d36dcfea517ab0f94995ae4

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