Web Supplement for the Optimal Disk Inclusions to ERK-Stability Regions Project (2017):
Description. The goal of this project to systematically investigate and characterize the computational problems related to optimal disk inclusions for the m-stage p-th order ERK stabilty regions.
In the current form it is a web repository which contains a list of Results and Benchmark Problems for symbolic computation.
This page is also thought as a web supplement of researh articles.
Results (Table I):
$p=m\, (m=1,\dots,4)$ (optimal disk inclusion)
$m=1$:
$P_{1a}=r-1$, ${\rm root}\, 1$ $(r^{(0,1)}={\rm Root[}P_{1a},1{\rm]}=1)$
$m=2$:
$P_{2a}=r-1$, ${\rm root}\, 1$ $(r^{(0,2)}={\rm Root[}P_{2a},1{\rm]}=1)$
$m=3$:
$P_{3a}=2r^3-3r^2+3r-3$, ${\rm root}\,1$ $(r^{(0,3)}={\rm Root[}P_{3a},1{\rm ]}\approx1.2564)$
$m=4$:
$P_{4a}=r^3-2r^2+3r-3$, ${\rm root}\,1$ $(r^{(0,4)}={\rm Root[}P_{4a},1{\rm ]}\approx1.3926)$
Results (Table II):
$p=m\, (m=1,\dots,4)$ (optimal left half-disk inclusion)
$m=1$:
$P_{1b}=\rho$, ${\rm root}\, 1$ $(\rho^{(0,1)}={\rm Root[}P_{1b},1{\rm]}=0)$
$m=2$:
$P_{2b}=\rho$, ${\rm root}\, 1$ $(\rho^{(0,2)}={\rm Root[}P_{2b},1{\rm]}=0)$
$m=3$:
$P_{3b}=\rho^2-3$, ${\rm root}\, 2$ $(\rho^{(0,3)}={\rm Root[}P_{3b},2{\rm]}\approx1.7321)$
$m=4$:
$P_{4b}=\rho^{24}-24\rho^{22}+288\rho^{20}-2496\rho^{18}
+14400\rho^{16}-69120\rho^{14}+506880\rho^{12}-3981312\rho^{10}
+\\28532736\rho^{8}-122093568\rho^6+111476736\rho^4+573308928\rho^2-1847328768,$
$\rm{root}\, 4$ $(\rho^{(0,4)}={\rm Root[}P_{4b},4{\rm]}\approx2.6156)$
Results (Table III):
$p=m-1\, (m=2,\dots,7)$ (optimal disk inclusion with one parameter)
$m=2$:
$P_{2a}=r-2$, ${\rm root}\, 1$ $(r^{(1,2)}={\rm Root[}P_{2a},1{\rm]}=2)$
$m=3$:
$P_{3a}=r-2$, ${\rm root}\,1$ $(r^{(1,3)}={\rm Root[}P_{3a},1{\rm]}=2)$
$m=4$:
$P_{4a}=r^{10}+12r^9-186r^8+1072r^7-3747r^6+9030r^5-15783r^4+20196r^3-18495r^2+11232r-3456$,
${\rm root}\,2$ $(r^{(1,4)}={\rm Root[}P_{4a},2{\rm]}\approx2.0731)$
$m=5$:
$P_{5a}=16r^{14}-64r^{13}-888r^{12}+11840r^{11}-74159r^{10}
+313140r^9-989268r^8+2447448r^7-4844952r^6\\
+7727040r^5-9877536r^4+9929088r^3-7516800r^2+3877632-1036800,$ ${\rm root}\, 2$ $(r^{(1,5)}={\rm Root[}P_{5a},2]\approx2.2295)$
$m=6$:
$P_{6a}=13r^{27}-3750r^{26}+133575r^{25}-2286320r^{24}+21837375r^{23}
-101908374r^{22}-244534635r^{21}
+8747537580r^{20}-86953900320r^{19}\\
+585650144880r^{18}-3076570274940r^{17}
+13322953402200r^{16}-48955620177900r^{15}+155290826100600r^{14}
-429832183413600r^{13}\\
+1045027671422400r^{12}-2239652384640000r^{11}+4235658591456000r^{10}
-7060047805296000r^9+10333631803680000r^8-13196225613600000r^7\\
+14555248292160000r^6-13657660335360000r^5+10655427916800000r^4
-6669118281600000r^3+3154444819200000r^2-1006761830400000r\\
+163258675200000,$ ${\rm root}\, 2$ $(r^{(1,6)}={\rm Root[}P_{6a},2{\rm ]}\approx2.3766)$
$m=7$:
$P_{7a}=384r^{35}-13568r^{34}+354048r^{33}-4376064r^{32}+1059640r^{31}+759044496r^{30}
-12699118092r^{29}+118729402032r^{28}-699929926056r^{27}\\
+1922024936160r^{26}
+10846447471935r^{25}-195309091721880r^{24}+1641106178614080r^{23}
-10232765389767240r^{22}+52457461045048800r^{21}\\
-230691398369450400r^{20}
+890137563868512000r^{19}-3054736914341155200r^{18}+9404755403533228800r^{17}
-26122860409715136000r^{16}\\
+65692525167620928000r^{15}
-149848386946456320000r^{14}+310212988127964288000r^{13}
-582418748526884352000r^{12}\\
+989793787059002880000r^{11}
-1517710206185671680000r^{10}+2089863955265333760000r^9
-2567343491912355840000r^8\\
+2788677723085516800000r^{7}
-2645695536168038400000r^6+2155255889583513600000r^5
-1471027377899520000000r^4\\
+810569164050432000000r^3
-339319732912128000000r^2+96280533909504000000r-13937211703296000000,$ ${\rm root}\, 3$ $(r^{(1,7)}={\rm Root[}P_{7a},3]\approx2.5554)$
Results (Table IV):
$p=m-1\, (m=2,\dots7)$ (optimal half-disk inclusion with one parameter)
$m=2$:
$P_{2b}=\rho-1$, ${\rm root}\, 1$ $(\rho^{(1,2)}={\rm Root[}P_{2b},1{\rm]}=1)$
$m=3$:
$P_{3b}=\rho^6-10\rho^4+65\rho^2-160$, ${\rm root}\,2$ $(\rho^{(1,3)}={\rm Root[}P_{3b},2{\rm]}\approx1.9692)$
$m=4$:
$P_{4b}=\rho^{26}-48\rho^{24}+1098\rho^{22}-15788\rho^{20}+162621\rho^{18}
-1345932\rho^{16}+ 9913500\rho^{14}-66381120\rho^{12}\\+373227588\rho^{10}
-1584756576\rho^8
+4658967072\rho^{6}-8735589504\rho^4+9251184960\rho^2-4156489728,$
${\rm root}\,9$ $(\rho^{(1,4)}={\rm Root[}P_{4b},9{\rm]}\approx2.7671)$
$m=5$:
$P_{5b}=\rho^{40}-12\rho^{38}+48\rho^{36}-1216\rho^{34}+66816\rho^{32}-677376\rho^{30}
+3548160\rho^{28}-16441344\rho^{26}+2768560128\rho^{24}
-210027773952\rho^{22}\\
+3337250734080\rho^{20}-25247888179200\rho^{18}
-7285015314432\rho^{16}+4375764680048640\rho^{14}-11394247399833600\rho^{12}
-99501281088897024\rho^{10}\\
-358312421313478656\rho^8+20414789140768358400\rho^6
-347378817879320297472\rho^4-1782603667891814400000\rho^2+23776267862016000000000,$
${\rm root}\, 4$ $(\rho^{(1,5)}={\rm Root[}P_{5b},4]\approx3.1708)$
$m=6$:
$P_{6b}=\rho^5-20\rho^3+120\rho-120$,
${\rm root}\,1$ $(\rho^{(1,6)}={\rm Root[}P_{6b},1{\rm]}\approx1.4913$
$m=7$:
$P_{7b}=\rho^6-30\rho^4+360\rho^2-1440$,
${\rm root}\,2$ $(\rho^{(1,7)}={\rm Root[}P_{7b},2{\rm]}\approx2.7517$
Results (Table V):
$p=m-2\, (m=3,\dots,6)$ (optimal disk inclusion with two parameters)
$m=3$:
$P_{3a}=r-3$, ${\rm root}\, 1$ $(r^{(2,3)}={\rm Root[}P_{3a},1{\rm]}=3)$
$m=4$:
$P_{4a}=r-3$, ${\rm root}\, 1$ $(r^{(2,4)}={\rm Root[}P_{4a},1{\rm]}=3)$
$m=5$:
$P_{5a}=8r^9+84r^8-1470r^7+8527r^6-28527r^5+61554r^4-88167r^3+80910r^2-36225r-1125$,
${\rm root}\, 3$ $(r^{(2,5)}={\rm Root[}P_{5a},3{\rm]}\approx2.9488)$
$m=6$:
$P_{6a}=r^7-12r^6+72r^5-276r^4+720r^3-1296r^2+1512r-900$,
${\rm root}\, 1$ $(r^{(2,6)}={\rm Root[}P_{6a},1{\rm]}\approx3.0598)$
Results (Table VI):
$p=m-2\, (m=3,...,5)$ (optimal half-disk inclusion with two parameters)
$m=3$:
$P_{3b}=\rho^6-10\rho^4+65\rho^2-160$, ${\rm root}\,2$ $(\rho^{(2,3)}={\rm Root[}P_{3b},2{\rm]}\approx1.9692)$
$m=4$:
$P_{4b}=9\rho^{28}-336\rho^{26}+6496\rho^{24}-82432\rho^{22}+839552\rho^{20}-7891968\rho^{18}+63852544\rho^{16}-362717184\rho^{14}
+1012895744\rho^{12}\\+119275520\rho^{10}-2973630464\rho^8-5912395776\rho^6-3647995904\rho^4+1468006400\rho^2+1627389952,$ ${\rm root}\, 7$ $(\rho^{(2,4)}={\rm Root[}P_{4b},7{\rm]}\approx2.7975)$
$m=5$:
$P_{5b}=\rho^{46}$,
${\rm root}\, 12$ $(\rho^{(2,5)}={\rm Root[}P_{5b},12{\rm]}\approx3.5602)$
Full Description of the Solutions
Results (Table III):
p=m (m=2,...,7) (optimal disk inclusion with one parameter)
m=2:
P2r=r-1, root 1 (r(1,2)=Root[P2r,1]=1)
P2α=2α-1, root 1 (α(1,2)=Root[P2α,1]=1/2)
Optimal Radius For ERK Stab Region with RQE
Description.
Code: QE2001
Name: MaxDiskRad03
Vars: 3
Formula: (Ax)(Ay)[((x+r)^2+y^2 [|P3[x+y i]|^2<=1]]
Output:
SW/HW: Mma9/Qepcad/Corei7-920
Best Time (confirmed): xxx sec [M9quine]
Remarks.
qepcad xxx sec
proj-lev-factors 1 17(12)
$p=m-1\, (m=2,...,7)$ (optimal disk inclusion with one parameter)
Robert Vajda