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ZigangPan`NonlinearCholeskyFactorization`
approximatenonlinearCholeskyfactorization  |
| {δ,transformation}=approximatenonlinearCholeskyfactorization[Vh,xc,m] Vh m xc δ transformation transformation | |
Examples
(1)
Basic Examples
(1)
In[1]:=
xc={x1,x2};
In[2]:=
$Assumptions=((x1|x2)∈Reals);
In[3]:=
fe[x1_,x2_]:={x2+x1^2,0}
In[4]:=
ge[x1_,x2_]:={{0},{1}}
In[5]:=
he[x1_,x2_]:={{1},{0}}
In[6]:=
qe[x1_,x2_]:=x1^2+x2^2
In[7]:=
γ=4;
In[8]:=
f=Apply[fe,xc];
In[9]:=
g=Apply[ge,xc];
In[10]:=
h=Apply[he,xc];
In[11]:=
r={{1}};
In[12]:=
q=Apply[qe,xc];
In[13]:=
s=(g.Inverse[r].Transpose[g]-h.Transpose[h]/γ^2)/4;
In[14]:=
Va2=
[f,s,q,xc,2]
 | approximateHJIequation |
Out[14]=
1.8717007154895465283161760209428058465234317466386+2.2081249720274031853679747747788583689816484446750x1x2+1.8122664250438800684138278298792941843986802781164
2
x1
2
x2
In[15]:=
ans2=
[Va2,xc〚2;;1;;-1〛,2]
| approximatenonlinearCholeskyfactorization |
Out[15]=
{{Function[{zl1,zl2},1.8122664250438800684138278298792941843986802781164],Function[{zl2},1.199087706619594793644145819209993217660935279861]},{0.6092164323945741774428587979317335225960949003534x1+x2,x1}}
In[16]:=
Expand[ans2〚1,2〛[ans2〚2,2〛]ans2〚2,2〛^2+Apply[ans2〚1,1〛,ans2〚2〛]ans2〚2,1〛^2]
Out[16]=
1.871700715489546528316176020942805846523431746639+2.208124972027403185367974774778858368981648444675x1x2+1.8122664250438800684138278298792941843986802781164
2
x1
2
x2
In[17]:=
Va3=
[f,s,q,xc,3]
| approximateHJIequation |
Out[17]=
1.8717007154895465283161760209428058465234317466386+2.9823690239504571962366257455014334394864580442933+2.2081249720274031853679747747788583689816484446750x1x2+4.338563859297960000755192639715877875550556325191x2+1.8122664250438800684138278298792941843986802781164+2.2304196799928599937349927998131939360983499770050x1+0.4385534995646503487291876620616039596984551071135
2
x1
3
x1
2
x1
2
x2
2
x2
3
x2
In[18]:=
ans3=
[Va3,xc〚2;;1;;-1〛,3]
| approximatenonlinearCholeskyfactorization |
Out[18]=
{{Function[{zl1,zl2},1.8122664250438800684138278298792941843986803+0.43855349956465034872918766206160395969845511zl1+0.035817634011408625176799268919195892560559865+1.4288976847360648183277113387288026751349561zl2+0.20746885937998485426249974839900903435667047zl1zl2+0.33798815095213994392965860274844986944628551],Function[{zl2},1.199087706619594793644145819209993217660935279861+1.067892781801172349157710451607757259573851706524zl2+0.320980115262227963478037937658955889844380060473]},{0.6092164323945741774428587979317335225960949003534x1+0.5819363974883601406717074564766337573201032185841+x2,x1}}
2
zl1
2
zl2
2
zl2
2
x1
In[19]:=
Expand[ans3〚1,2〛[ans3〚2,2〛]ans3〚2,2〛^2+Apply[ans3〚1,1〛,ans3〚2〛]ans3〚2,1〛^2]
Out[19]=
1.87170071548954652831617602094280584652343175+2.9823690239504571962366257455014334394864580+2.4093109341358512427748164434945991814695382+1.1482627441892133887846779101637246235578247+0.35630712991692577343164675256828556660572478+0.058087511374447238146562242750070333526399888+0.0041077024710645444168090206679713724478038198+2.2081249720274031853679747747788583689816484x1x2+4.3385638592979600007551926397158778755505563x2+3.2711380689228034562005563417133472933410175x2+1.3730730794979610615504746020835711060384671x2+0.29945288673377283282357152168601505017480520x2+0.028234717668758338507387288344215577246833122x2+1.8122664250438800684138278298792941843986803+2.2304196799928599937349927998131939360983500x1+1.5625603388184540802550090641919489141129645+0.51458009505198971979255875285746019440734918+0.072777844255711177836120474272129653209378257+0.43855349956465034872918766206160395969845511+0.29475162421696454787144351249520610368675734x1+0.083374339612622788057797216697737654585227473+0.035817634011408625176799268919195892560559865
2
x1
3
x1
4
x1
5
x1
6
x1
7
x1
8
x1
2
x1
3
x1
4
x1
5
x1
6
x1
2
x2
2
x2
2
x1
2
x2
3
x1
2
x2
4
x1
2
x2
3
x2
3
x2
2
x1
3
x2
4
x2
""