preconditioning.tex
Paper VIII: Packet Operator Algorithms, Exact Compilation, and GPU Realization
preconditioning.tex
This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.
1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \draw[black!13,line width=.25pt] (2.020588,0.7) -- (2.020588,3.95);
4 \draw[] (2.020588,0.64) -- (2.020588,0.76);
5 \node[font=\small,align=center,font=\scriptsize] at (2.020588,0.44) {-1};
6 \draw[black!13,line width=.25pt] (3.55,0.7) -- (3.55,3.95);
7 \draw[] (3.55,0.64) -- (3.55,0.76);
8 \node[font=\small,align=center,font=\scriptsize] at (3.55,0.44) {0};
9 \draw[black!13,line width=.25pt] (5.079412,0.7) -- (5.079412,3.95);
10 \draw[] (5.079412,0.64) -- (5.079412,0.76);
11 \node[font=\small,align=center,font=\scriptsize] at (5.079412,0.44) {1};
12 \draw[black!13,line width=.25pt] (0.95,1.369118) -- (6.15,1.369118);
13 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.79,1.369118) {-1};
14 \draw[black!13,line width=.25pt] (0.95,2.325) -- (6.15,2.325);
15 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.79,2.325) {0};
16 \draw[black!13,line width=.25pt] (0.95,3.280882) -- (6.15,3.280882);
17 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.79,3.280882) {1};
18 \draw[->,line width=.6pt] (0.95,0.7) -- (6.27,0.7);
19 \draw[->,line width=.6pt] (0.95,0.7) -- (0.95,4.07);
20 \draw[line width=1.1pt] plot coordinates {(4.080778,1.22935) (4.052485,1.22493) (4.023407,1.222227) (3.99359,1.221246) (3.963081,1.221989) (3.931926,1.224454) (3.900175,1.228637) (3.867878,1.234532) (3.835084,1.24213) (3.801845,1.251418) (3.768213,1.262383) (3.73424,1.275007) (3.699979,1.28927) (3.665484,1.30515) (3.630809,1.322623) (3.596008,1.341661) (3.561135,1.362234) (3.526244,1.384311) (3.491391,1.407856) (3.456629,1.432834) (3.422013,1.459204) (3.387597,1.486926) (3.353434,1.515957) (3.319578,1.546251) (3.286082,1.577761) (3.252998,1.610437) (3.220378,1.64423) (3.188273,1.679085) (3.156732,1.714949) (3.125805,1.751765) (3.095541,1.789476) (3.065986,1.828024) (3.037187,1.867347) (3.009189,1.907385) (2.982035,1.948075) (2.955767,1.989354) (2.930428,2.031157) (2.906056,2.073418) (2.88269,2.116072) (2.860365,2.159053) (2.839117,2.202293) (2.81898,2.245724) (2.799983,2.289279) (2.782158,2.33289) (2.765531,2.376488) (2.75013,2.420006) (2.735977,2.463376) (2.723095,2.506529) (2.711505,2.5494) (2.701223,2.59192) (2.692267,2.634023) (2.68465,2.675643) (2.678385,2.716716) (2.67348,2.757178) (2.669944,2.796964) (2.667782,2.836014) (2.666997,2.874266) (2.667591,2.91166) (2.669563,2.948139) (2.672909,2.983644) (2.677625,3.018121) (2.683704,3.051516) (2.691134,3.083776) (2.699906,3.114851) (2.710005,3.144694) (2.721416,3.173256) (2.73412,3.200494) (2.748099,3.226365) (2.763329,3.250829) (2.779787,3.273847) (2.797449,3.295384) (2.816285,3.315405) (2.836267,3.33388) (2.857363,3.35078) (2.879541,3.366078) (2.902765,3.379751) (2.927001,3.391776) (2.952209,3.402137) (2.97835,3.410815) (3.005384,3.417798) (3.033268,3.423075) (3.061959,3.426637) (3.091412,3.428479) (3.121581,3.428598) (3.152419,3.426994) (3.183878,3.423669) (3.215908,3.418629) (3.24846,3.411881) (3.281483,3.403437) (3.314925,3.393308) (3.348734,3.381512) (3.382858,3.368065) (3.417242,3.352991) (3.451834,3.336311) (3.486579,3.318052) (3.521423,3.298242) (3.556312,3.276913) (3.59119,3.254098) (3.626005,3.229832) (3.660701,3.204153) (3.695224,3.177101) (3.72952,3.148719) (3.763536,3.119051) (3.797218,3.088143) (3.830515,3.056044) (3.863373,3.022803) (3.895742,2.988472) (3.927572,2.953106) (3.958811,2.916759) (3.989413,2.879488) (4.019328,2.841351) (4.048511,2.802408) (4.076915,2.762719) (4.104497,2.722347) (4.131212,2.681355) (4.157021,2.639806) (4.181881,2.597766) (4.205755,2.5553) (4.228605,2.512474) (4.250395,2.469355) (4.271092,2.426012) (4.290663,2.38251) (4.309078,2.338919) (4.326307,2.295305) (4.342324,2.251739) (4.357104,2.208286) (4.370624,2.165016) (4.382862,2.121996) (4.393801,2.079292) (4.403421,2.036973) (4.411709,1.995103) (4.418652,1.953748) (4.424238,1.912973) (4.42846,1.872841) (4.431309,1.833415) (4.432783,1.794756) (4.432878,1.756926) (4.431595,1.719982) (4.428935,1.683984) (4.424903,1.648986) (4.419505,1.615043) (4.412749,1.582209) (4.404647,1.550535) (4.395209,1.52007) (4.384452,1.490862) (4.372392,1.462956) (4.359049,1.436397) (4.344441,1.411224) (4.328594,1.387479) (4.311531,1.365197) (4.293278,1.344414) (4.273865,1.325162) (4.253322,1.307471) (4.231681,1.291369) (4.208975,1.276881) (4.185241,1.264029) (4.160515,1.252834) (4.134835,1.243313) (4.108242,1.235481) (4.080778,1.22935)};
21 \node[font=\small,align=center] at (3.55,4.3) {before: $x^TAx=1$};
22 \draw[black!13,line width=.25pt] (8.720588,0.7) -- (8.720588,3.95);
23 \draw[] (8.720588,0.64) -- (8.720588,0.76);
24 \node[font=\small,align=center,font=\scriptsize] at (8.720588,0.44) {-1};
25 \draw[black!13,line width=.25pt] (10.25,0.7) -- (10.25,3.95);
26 \draw[] (10.25,0.64) -- (10.25,0.76);
27 \node[font=\small,align=center,font=\scriptsize] at (10.25,0.44) {0};
28 \draw[black!13,line width=.25pt] (11.779412,0.7) -- (11.779412,3.95);
29 \draw[] (11.779412,0.64) -- (11.779412,0.76);
30 \node[font=\small,align=center,font=\scriptsize] at (11.779412,0.44) {1};
31 \draw[black!13,line width=.25pt] (7.65,1.369118) -- (12.85,1.369118);
32 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (7.49,1.369118) {-1};
33 \draw[black!13,line width=.25pt] (7.65,2.325) -- (12.85,2.325);
34 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (7.49,2.325) {0};
35 \draw[black!13,line width=.25pt] (7.65,3.280882) -- (12.85,3.280882);
36 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (7.49,3.280882) {1};
37 \draw[->,line width=.6pt] (7.65,0.7) -- (12.97,0.7);
38 \draw[->,line width=.6pt] (7.65,0.7) -- (7.65,4.07);
39 \draw[line width=1.1pt] plot coordinates {(8.720588,3.280882) (8.756667,3.301939) (8.795077,3.32147) (8.835759,3.339446) (8.878649,3.355837) (8.92368,3.370619) (8.970783,3.383768) (9.019882,3.395264) (9.070902,3.405089) (9.123764,3.413228) (9.178383,3.419668) (9.234676,3.424398) (9.292554,3.427412) (9.351928,3.428704) (9.412703,3.428273) (9.474786,3.42612) (9.538079,3.422247) (9.602484,3.416661) (9.667899,3.40937) (9.734224,3.400387) (9.801354,3.389724) (9.869184,3.377399) (9.93761,3.363431) (10.006522,3.347841) (10.075815,3.330654) (10.14538,3.311897) (10.215109,3.291599) (10.284891,3.269792) (10.35462,3.24651) (10.424185,3.221789) (10.493478,3.195667) (10.56239,3.168187) (10.630816,3.139389) (10.698646,3.10932) (10.765776,3.078027) (10.832101,3.045558) (10.897516,3.011963) (10.961921,2.977296) (11.025214,2.941611) (11.087297,2.904963) (11.148072,2.867409) (11.207446,2.829008) (11.265324,2.78982) (11.321617,2.749907) (11.376236,2.70933) (11.429098,2.668153) (11.480118,2.626441) (11.529217,2.584257) (11.57632,2.541669) (11.621351,2.498743) (11.664241,2.455545) (11.704923,2.412143) (11.743333,2.368606) (11.779412,2.325) (11.813102,2.281394) (11.844352,2.237857) (11.873113,2.194455) (11.899339,2.151257) (11.92299,2.108331) (11.944029,2.065743) (11.962423,2.023559) (11.978143,1.981847) (11.991165,1.94067) (12.001468,1.900093) (12.009037,1.86018) (12.013859,1.820992) (12.015926,1.782591) (12.015237,1.745037) (12.011791,1.708389) (12.005595,1.672704) (11.996657,1.638037) (11.984993,1.604442) (11.970619,1.571973) (11.953559,1.54068) (11.933838,1.510611) (11.911489,1.481813) (11.886546,1.454333) (11.859047,1.428211) (11.829036,1.40349) (11.796559,1.380208) (11.761668,1.358401) (11.724416,1.338103) (11.684862,1.319346) (11.643068,1.302159) (11.599099,1.286569) (11.553023,1.272601) (11.504913,1.260276) (11.454843,1.249613) (11.402892,1.24063) (11.349141,1.233339) (11.293674,1.227753) (11.236577,1.22388) (11.17794,1.221727) (11.117854,1.221296) (11.056413,1.222588) (10.993713,1.225602) (10.929851,1.230332) (10.864928,1.236772) (10.799045,1.244911) (10.732305,1.254736) (10.664812,1.266232) (10.596671,1.279381) (10.527988,1.294163) (10.458872,1.310554) (10.389429,1.32853) (10.319769,1.348061) (10.25,1.369118) (10.180231,1.391667) (10.110571,1.415673) (10.041128,1.441099) (9.972012,1.467906) (9.903329,1.49605) (9.835188,1.525489) (9.767695,1.556176) (9.700955,1.588064) (9.635072,1.621102) (9.570149,1.65524) (9.506287,1.690423) (9.443587,1.726597) (9.382146,1.763705) (9.32206,1.801689) (9.263423,1.840491) (9.206326,1.880049) (9.150859,1.920302) (9.097108,1.961187) (9.045157,2.00264) (8.995087,2.044596) (8.946977,2.08699) (8.900901,2.129756) (8.856932,2.172826) (8.815138,2.216135) (8.775584,2.259613) (8.738332,2.303193) (8.703441,2.346807) (8.670964,2.390387) (8.640953,2.433865) (8.613454,2.477174) (8.588511,2.520244) (8.566162,2.56301) (8.546441,2.605404) (8.529381,2.64736) (8.515007,2.688813) (8.503343,2.729698) (8.494405,2.769951) (8.488209,2.809509) (8.484763,2.848311) (8.484074,2.886295) (8.486141,2.923403) (8.490963,2.959577) (8.498532,2.99476) (8.508835,3.028898) (8.521857,3.061936) (8.537577,3.093824) (8.555971,3.124511) (8.57701,3.15395) (8.600661,3.182094) (8.626887,3.208901) (8.655648,3.234327) (8.686898,3.258333) (8.720588,3.280882)};
40 \node[font=\small,align=center] at (10.25,4.3) {after: $y^T(I+E)y=1$};
41 \node[font=\small,align=center,font=\scriptsize] at (7,-0.1) {Exact normalized eigenvalues: $1/2,3/2$. Hence $\kappa_2=3$.};
42 \end{tikzpicture}
Other PDFs for this work
Original file and source record
This copy matches the recorded source file. Its publication here does not change the work’s review status.
SHA-256: 286bedf11de1d96755c07813a150f2f73a4383472e61995963fab7b9105074a2
Back to Paper VIII: Packet Operator Algorithms, Exact Compilation, and GPU Realization