I traced a bug in a larger program back to this problem. I am getting larger than expected errors for changing the order of multiplicative operations using numpy. Here is a self-contained example (the actual code at the end is short, but I copy in explicit matrices to make it run):
import numpy
M = numpy.array(
[[ 1.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 0.00000000e+00, 4.24142041e-01, 1.49421321e-08, 7.87081062e-09, 2.58494178e-08, -5.09534880e-02, -1.09351587e-08, 5.95629388e-10, 1.34707444e-08, 0.00000000e+00, -4.88069109e-01, 1.84060546e-08, 2.76013885e-08, -1.40665727e-08, 5.86332400e-02, 1.87260208e-08, 1.59941364e-08, 1.04757263e-08],
[ 0.00000000e+00, 1.49421321e-08, 5.26397502e-16, 2.77281383e-16, 9.10651097e-16, -1.79504429e-09, -3.85235532e-16, 2.09834728e-17, 4.74561875e-16, 0.00000000e+00, -1.71942236e-08, 6.48428290e-16, 9.72371404e-16, -4.95552353e-16, 2.06559485e-09, 6.59700404e-16, 5.63458643e-16, 3.69050154e-16],
[ 0.00000000e+00, 7.87081062e-09, 2.77281383e-16, 1.46058758e-16, 4.79688058e-16, -9.45544689e-10, -2.02923914e-16, 1.10531040e-17, 2.49976819e-16, 0.00000000e+00, -9.05710624e-09, 3.41561448e-16, 5.12199406e-16, -2.61033613e-16, 1.08805797e-09, 3.47499066e-16, 2.96803444e-16, 1.94398219e-16],
[ 0.00000000e+00, 2.58494178e-08, 9.10651097e-16, 4.79688058e-16, 1.57539771e-15, -3.10537007e-09, -6.66445336e-16, 3.63007470e-17, 8.20977096e-16, 0.00000000e+00, -2.97454652e-08, 1.12176052e-15, 1.68217190e-15, -8.57289960e-16, 3.57341403e-09, 1.14126092e-15, 9.74765703e-16, 6.38445141e-16],
[ 0.00000000e+00, -5.09534880e-02, -1.79504429e-09, -9.45544689e-10, -3.10537007e-09, 6.12119924e-03, 1.31367425e-09, -7.15547905e-11, -1.61828196e-09, 0.00000000e+00, 5.86332433e-02, -2.21117595e-09, -3.31583970e-09, 1.68986064e-09, -7.04379147e-03, -2.24961448e-09, -1.92142480e-09, -1.25848122e-09],
[ 0.00000000e+00, -1.09351587e-08, -3.85235532e-16, -2.02923914e-16, -6.66445336e-16, 1.31367425e-09, 2.81928419e-16, -1.53564166e-17, -3.47300464e-16, 0.00000000e+00, 1.25833156e-08, -4.74541799e-16, -7.11614349e-16, 3.62662007e-16, -1.51167232e-09, -4.82791114e-16, -4.12358131e-16, -2.70083410e-16],
[ 0.00000000e+00, 5.95629388e-10, 2.09834728e-17, 1.10531040e-17, 3.63007470e-17, -7.15547905e-11, -1.53564166e-17, 8.36451786e-19, 1.89171798e-17, 0.00000000e+00, -6.85403182e-10, 2.58479141e-17, 3.87610671e-17, -1.97539108e-17, 8.23395879e-11, 2.62972477e-17, 2.24608192e-17, 1.47112284e-17],
[ 0.00000000e+00, 1.34707444e-08, 4.74561875e-16, 2.49976819e-16, 8.20977096e-16, -1.61828196e-09, -3.47300464e-16, 1.89171798e-17, 4.27830628e-16, 0.00000000e+00, -1.55010671e-08, 5.84575999e-16, 8.76619656e-16, -4.46754122e-16, 1.86219076e-09, 5.94738120e-16, 5.07973517e-16, 3.32708899e-16],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 1.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
[ 0.00000000e+00, -4.88069109e-01, -1.71942236e-08, -9.05710624e-09, -2.97454652e-08, 5.86332433e-02, 1.25833156e-08, -6.85403182e-10, -1.55010671e-08, 0.00000000e+00, 5.61631322e-01, -2.11802316e-08, -3.17614944e-08, 1.61866991e-08, -6.74704944e-02, -2.15484234e-08, -1.84047870e-08, -1.20546371e-08],
[ 0.00000000e+00, 1.84060546e-08, 6.48428290e-16, 3.41561448e-16, 1.12176052e-15, -2.21117595e-09, -4.74541799e-16, 2.58479141e-17, 5.84575999e-16, 0.00000000e+00, -2.11802316e-08, 7.98748561e-16, 1.19778898e-15, -6.10432541e-16, 2.54444623e-09, 8.12633805e-16, 6.94081037e-16, 4.54604285e-16],
[ 0.00000000e+00, 2.76013885e-08, 9.72371404e-16, 5.12199406e-16, 1.68217190e-15, -3.31583970e-09, -7.11614349e-16, 3.87610671e-17, 8.76619656e-16, 0.00000000e+00, -3.17614944e-08, 1.19778898e-15, 1.79618282e-15, -9.15393662e-16, 3.81560581e-09, 1.21861104e-15, 1.04083144e-15, 6.81716412e-16],
[ 0.00000000e+00, -1.40665727e-08, -4.95552353e-16, -2.61033613e-16, -8.57289960e-16, 1.68986064e-09, 3.62662007e-16, -1.97539108e-17, -4.46754122e-16, 0.00000000e+00, 1.61866991e-08, -6.10432541e-16, -9.15393662e-16, 4.66514627e-16, -1.94455784e-09, -6.21044146e-16, -5.30441833e-16, -3.47425038e-16],
[ 0.00000000e+00, 5.86332400e-02, 2.06559485e-09, 1.08805797e-09, 3.57341403e-09, -7.04379147e-03, -1.51167232e-09, 8.23395879e-11, 1.86219076e-09, 0.00000000e+00, -6.74704944e-02, 2.54444623e-09, 3.81560581e-09, -1.94455784e-09, 8.10543757e-03, 2.58867824e-09, 2.21102355e-09, 1.44816055e-09],
[ 0.00000000e+00, 1.87260208e-08, 6.59700404e-16, 3.47499066e-16, 1.14126092e-15, -2.24961448e-09, -4.82791114e-16, 2.62972477e-17, 5.94738120e-16, 0.00000000e+00, -2.15484234e-08, 8.12633805e-16, 1.21861104e-15, -6.21044146e-16, 2.58867824e-09, 8.26760426e-16, 7.06146767e-16, 4.62507011e-16],
[ 0.00000000e+00, 1.59941364e-08, 5.63458643e-16, 2.96803444e-16, 9.74765703e-16, -1.92142480e-09, -4.12358131e-16, 2.24608192e-17, 5.07973517e-16, 0.00000000e+00, -1.84047870e-08, 6.94081037e-16, 1.04083144e-15, -5.30441833e-16, 2.21102355e-09, 7.06146767e-16, 6.03129082e-16, 3.95033217e-16],
[ 0.00000000e+00, 1.04757263e-08, 3.69050154e-16, 1.94398219e-16, 6.38445141e-16, -1.25848122e-09, -2.70083410e-16, 1.47112284e-17, 3.32708899e-16, 0.00000000e+00, -1.20546371e-08, 4.54604285e-16, 6.81716412e-16, -3.47425038e-16, 1.44816055e-09, 4.62507011e-16, 3.95033217e-16, 2.58736061e-16]]
)
A = numpy.array(
[[-0.00000000e+00, -7.49420654e-01, 2.36645133e-08, 2.83214009e-09, -6.52157725e-08, 9.00302357e-02, 2.07762711e-08, -3.93769333e-08, -3.96766769e-08, 0.00000000e+00, -6.51261884e-01, 2.35425627e-08, 3.46140156e-08, -1.89780598e-08, 7.82380887e-02, 2.51868179e-08, 2.13865316e-08, 1.44744812e-08],
[ 3.05311332e-16, 3.63603065e-02, 1.77351557e-01, 1.52401227e-02, -1.31437146e-06, 3.02666853e-01, -5.56572048e-07, -1.60534678e-06, -4.65237579e-07, 4.42851120e-16, 1.11596525e-01, -8.16894013e-07, 9.17216520e-07, -2.96530450e-07, 9.28940840e-01, 3.66607356e-07, -9.91357211e-09, -2.27619767e-07],
[-1.35525272e-20, -3.78275577e-05, 2.02944409e-04, -1.63077293e-05, 3.96391244e-01, -3.14375043e-04, 2.19758080e-01, 4.39817640e-01, 5.14025579e-01, 5.28070693e-20, 7.95260108e-06, 2.07648708e-01, 2.80669103e-01, -2.70743918e-02, 6.62922300e-05, -3.97663420e-01, -9.72844144e-02, 2.16082768e-01],
[ 3.17637355e-22, 3.02225698e-07, -2.20579495e-06, 1.36032858e-05, 1.89121855e-01, 2.46318863e-06, 3.29601248e-01, 5.12358908e-01, -6.11878058e-01, -4.09855638e-21, 5.33328462e-08, -1.13839294e-01, -3.51810943e-01, 1.99126758e-01, 8.07593444e-07, -1.95429714e-01, 6.34840080e-02, -9.20601029e-03],
[ 2.11758237e-22, 2.93821317e-07, -1.25434775e-06, 5.01041209e-07, -2.41225155e-01, 2.43924254e-06, -2.10410277e-01, 5.85509545e-01, -1.53207165e-01, 2.17127990e-22, -6.54394382e-08, 4.61741591e-02, 3.94747797e-01, -1.31509913e-01, -6.95848293e-07, 4.33611439e-01, -3.91114733e-01, -1.22481544e-01],
[-3.17637355e-22, 1.30494444e-07, -6.28897449e-07, -5.69416759e-07, -4.89409426e-01, 8.16591863e-07, 2.25530164e-01, 1.36440113e-01, 1.56457139e-01, 6.09145504e-22, -1.46265910e-07, -7.06195107e-01, 1.76837517e-01, -1.84706615e-01, -1.10677502e-06, -2.65190566e-01, 1.75625222e-01, 3.53552474e-02],
[-1.32348898e-23, 9.76074136e-09, -6.64706365e-08, 8.97102007e-08, 1.04453697e-02, 3.77554871e-08, 1.06780632e-01, -5.51530507e-02, -2.67072454e-02, 8.34309184e-23, 9.76785235e-09, -1.66380632e-01, 8.89817377e-02, 3.52794263e-01, -1.01989239e-07, 3.79383206e-01, -2.21311259e-02, 8.24771602e-01],
[-2.64697796e-23, -3.81071470e-09, -1.52470457e-08, 2.31867964e-07, 1.61167689e-01, 1.21476062e-08, 4.62920198e-01, -6.30833114e-02, 1.55154734e-01, -6.81907803e-23, 3.36355606e-09, -1.04616530e-01, 3.04210903e-01, 4.74962450e-01, -3.26436018e-08, 3.66884819e-01, 1.89001475e-01, -4.81947218e-01],
[ 3.30872245e-23, -3.82486279e-09, 4.57705305e-08, -1.53730823e-07, -1.09868035e-01, 9.49535030e-08, -6.11367953e-01, 2.09616196e-01, 6.92500565e-03, -1.51367999e-23, -1.37726681e-08, 2.39736500e-02, 1.68012777e-01, 5.33544789e-01, -7.45547417e-08, -2.35647324e-01, 4.47745356e-01, -2.63191932e-02],
[ 0.00000000e+00, 2.95812127e-07, -4.47630753e-07, -6.75579326e-08, -5.05647373e-01, 2.12597159e-06, 3.29042472e-01, 1.47754834e-01, 4.59445954e-02, -4.42071975e-22, 7.64987025e-09, 5.88363526e-01, -7.18829924e-02, -1.25810121e-01, -3.22069856e-07, 1.14059812e-01, 4.67535857e-01, 1.15511821e-01],
[-1.58818678e-22, -9.75341754e-08, 8.33486809e-07, -7.58797257e-07, -4.09124432e-01, -1.22408647e-06, 1.92397506e-01, -2.35640312e-01, -1.54747228e-01, 4.33075192e-22, -5.97477948e-08, 2.27856071e-01, 1.87299705e-01, 4.18606298e-01, -3.80248666e-07, -4.18737252e-01, -5.27560777e-01, -1.53387616e-02],
[-3.17637355e-22, -2.73493680e-07, 1.20952799e-06, -1.15938669e-07, 2.19496103e-01, -2.38765220e-06, 5.63854677e-02, -2.33009173e-01, -5.12386378e-01, -1.42993918e-21, 1.90409524e-08, 7.60732865e-02, 6.64392889e-01, -2.87091516e-01, -3.51089068e-07, -1.51359887e-01, 2.62855799e-01, 1.00893426e-01],
[ 0.00000000e+00, -9.82865059e-02, 5.33893628e-01, -8.64663760e-02, -1.52855583e-04, -8.18146794e-01, -8.26226436e-05, -1.62722671e-04, -1.97549957e-04, 1.91248595e-16, 2.01206803e-02, -7.98176370e-05, -1.09471810e-04, 1.05532690e-05, 1.67486593e-01, 1.53257887e-04, 3.75957532e-05, -8.28268558e-05],
[ 3.46944695e-18, 9.86836968e-03, -3.21503728e-02, -9.96041972e-01, 9.89973139e-06, 8.21452559e-02, 7.87619622e-06, 1.42732304e-05, 4.19866736e-07, 1.08567995e-16, -5.53252953e-04, 2.12382866e-06, 5.52438245e-08, 2.06607768e-06, -4.60524623e-03, -8.61743850e-06, -7.93397037e-07, 3.40561040e-06],
[ 0.00000000e+00, 5.60976585e-02, 8.26117626e-01, 1.38453289e-02, 1.66897114e-06, 4.66962542e-01, 5.21437516e-07, 1.01948728e-06, 6.95013869e-07, -3.76249560e-17, -3.69824778e-02, 1.99982217e-08, 2.01540087e-07, 1.03736187e-07, -3.07846242e-01, -1.07869609e-06, -3.39760103e-07, 4.32423200e-07]]
)
B = numpy.array(
[[ 0.00000000e+00, -1.00000000e+00, 1.06205003e-05],
[ 6.51261884e-01, 2.02210886e-17, 2.58415811e-17],
[ 2.29433542e-08, 1.34184985e-16, -2.74111890e-17],
[ 1.20854772e-08, 7.60639281e-18, 2.96257867e-16],
[ 3.96912801e-08, 5.98385402e-23, -1.93603571e-17],
[-7.82380933e-02, 1.68322376e-16, 2.15108010e-16],
[-1.67907242e-08, -1.39105759e-21, -1.31411424e-16],
[ 9.14577381e-10, -9.94043901e-22, -1.02202642e-16],
[ 2.06840670e-08, 2.09868713e-21, 2.04206049e-16],
[ 0.00000000e+00, 1.06205003e-05, 1.00000000e+00],
[-7.49420658e-01, -1.59224140e-17, 4.48478893e-17],
[ 2.82621401e-08, 3.37439644e-21, 3.18670444e-16],
[ 4.23813971e-08, -8.80703978e-22, -8.05157812e-17],
[-2.15989497e-08, -1.22983410e-22, -1.35186018e-17],
[ 9.00302037e-02, -1.32539872e-16, 3.73318253e-16],
[ 2.87534420e-08, 1.37971615e-21, 1.40919484e-16],
[ 2.45586865e-08, 5.08971450e-22, 5.55657960e-17],
[ 1.60852746e-08, 1.11154797e-23, -1.54979207e-18]]
)
Z = A @ M @ B
print("original matrix\n", Z, '\n')
norm = max(max(row) for row in abs(Z))
print("largest element\n", norm, '\n')
Z1 = Z / norm
print("matrix normalized one way\n", Z1, '\n')
Z2 = A @ (M/norm) @ B
print("matrix normalized another way\n", Z1, '\n')
print("unexpected deviation \n", Z1-Z2, '\n')
Below is the output. I expect the matrices normalized in the two different "ways" to be the same to within machine tolerance (roughly 1e-16), relative to the largest element, which is ~1. So why am I getting errors on the order of 1e-7? I checked that all the dtypes are float64. Thanks for any help!
original matrix
[[-2.70976453e-10 2.77813727e-27 -2.66468642e-27]
[ 3.14803157e-10 -3.05306629e-16 4.42854363e-16]
[ 2.40140485e-14 1.35530880e-20 5.28069254e-20]
[ 4.02675253e-16 -3.17680884e-22 -4.09855301e-21]
[-5.36313469e-16 -2.11755931e-22 2.17130239e-22]
[ 3.12972364e-16 3.17643824e-22 6.09142131e-22]
[-2.99648801e-17 1.32357759e-23 8.34307778e-23]
[-1.00370207e-16 2.64690554e-23 -6.81910614e-23]
[ 4.76605198e-18 -3.30873853e-23 -1.51364485e-23]
[-4.00415286e-16 -4.69502499e-27 -4.42071975e-22]
[-1.67417098e-17 1.58823277e-22 4.33073505e-22]
[ 5.36721887e-16 3.17622168e-22 -1.42994255e-21]
[ 2.42670193e-11 2.03115564e-21 1.91248595e-16]
[-6.86052564e-12 -3.46829390e-18 1.08568032e-16]
[-8.03322460e-11 -3.99595601e-22 -3.76249560e-17]]
largest element
3.148031574328576e-10
matrix normalized one way
[[-8.60780609e-01 8.82499813e-18 -8.46461149e-18]
[ 1.00000000e+00 -9.69833439e-07 1.40676595e-06]
[ 7.62827434e-05 4.30525797e-11 1.67745857e-10]
[ 1.27913346e-06 -1.00914135e-12 -1.30194152e-11]
[-1.70364704e-06 -6.72661395e-13 6.89733358e-13]
[ 9.94184324e-07 1.00902363e-12 1.93499371e-12]
[-9.51860850e-08 4.20446097e-14 2.65025226e-13]
[-3.18834816e-07 8.40812894e-14 -2.16614922e-13]
[ 1.51397846e-08 -1.05104998e-13 -4.80822639e-14]
[-1.27195448e-06 -1.49141610e-17 -1.40428063e-12]
[-5.31815181e-08 5.04516152e-13 1.37569619e-12]
[ 1.70494442e-06 1.00895484e-12 -4.54233866e-12]
[ 7.70863275e-02 6.45214508e-12 6.07518033e-07]
[-2.17930649e-02 -1.10173415e-08 3.44875931e-07]
[-2.55182466e-01 -1.26935068e-12 -1.19518992e-07]]
matrix normalized another way
[[-8.60780609e-01 8.82499813e-18 -8.46461149e-18]
[ 1.00000000e+00 -9.69833439e-07 1.40676595e-06]
[ 7.62827434e-05 4.30525797e-11 1.67745857e-10]
[ 1.27913346e-06 -1.00914135e-12 -1.30194152e-11]
[-1.70364704e-06 -6.72661395e-13 6.89733358e-13]
[ 9.94184324e-07 1.00902363e-12 1.93499371e-12]
[-9.51860850e-08 4.20446097e-14 2.65025226e-13]
[-3.18834816e-07 8.40812894e-14 -2.16614922e-13]
[ 1.51397846e-08 -1.05104998e-13 -4.80822639e-14]
[-1.27195448e-06 -1.49141610e-17 -1.40428063e-12]
[-5.31815181e-08 5.04516152e-13 1.37569619e-12]
[ 1.70494442e-06 1.00895484e-12 -4.54233866e-12]
[ 7.70863275e-02 6.45214508e-12 6.07518033e-07]
[-2.17930649e-02 -1.10173415e-08 3.44875931e-07]
[-2.55182466e-01 -1.26935068e-12 -1.19518992e-07]]
unexpected deviation
[[-1.85496826e-07 -3.37450100e-24 -4.46082542e-24]
[-2.90512590e-08 0.00000000e+00 -2.11758237e-22]
[ 3.36542580e-12 6.46234854e-27 0.00000000e+00]
[-1.03382940e-13 2.01948392e-28 -1.61558713e-27]
[-6.32965290e-14 0.00000000e+00 -2.01948392e-28]
[ 7.05435233e-15 2.01948392e-28 -4.03896783e-28]
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[ 1.07272085e-08 2.81919955e-25 0.00000000e+00]
[ 2.16769630e-10 1.65436123e-24 5.29395592e-23]
[ 4.15586621e-09 -5.77572400e-26 0.00000000e+00]]
