#
#		This maple file verifies that the 14 feedback laws in 
#	"Some remarks on real and complex output feedback" for the system
#	that cannot be controlled with real output feedback are indeed
#	feedback laws for that system
#
interface(quiet=true):
#
#    Below are  14  4 X 2 complex matrices (in conjugate pairs)
#    K1,...,K14
#    
#    There is also a 2 X 6 matrix of degree four real polynomials, F(X),
#    and 8 real points  Poles := (-8,-6,-4,-2,1, 2, 3, 4).
#    	
#    These points are roots of the degree 8 polynomial
#     	                          
#             | F(X) |           The leading monomials of F are:   
#      	      |------|                
#    (1)   det|  |   | =  0         |1 0 0 0 0 0|
#             |Ki| I |              |0 1 0 0 0 0|
#             |  |   |               
#    	
#    Running this file in Maple will verify equation (1) is satisfied
#    for Ki and X, for each i from 1 -- 14 and X in Poles.
#
#
#    We generated these data in the following manner:
#    
#    We first did a random search for matrices of polynomials F(X) and 
#    8 values Poles so that the 14 matrices C_i, D_i satisfying the 
#    equation (1) would be all complex.    This was done using the 
#    software SINGULAR, to generate an eliminant polynomial, whose number
#    of real roots was checked using the realroot routine of MAPLE.
#    This part used exact integer arithmetic in SINGULAR, and Sturm 
#    sequences in Maple.    This constitutes a proof that F(X) and the 
#    8 values in Poles have the the properties:
#    	1) F(X) is non-degenerate.
#       2) The 14 solutions to the system (K1 -- K14) (1) as X ranges over 
#	      Poles are all complex.
#
#	Then we generated (also using exact arithmetic) polynomials
#    Poly_i,j  i=1,2,3,4, j=1,2   whose roots are the components of the
#    matrices K1,...,K14.  These polynomials are stored in the maple file
#    polynomials.maple, which also finds roots with 20 digit precision.
#    To match the roots and form the matrices, we also computed polynomials
#    whose roots were various sums of the coordinates of the matrices.
#
#	Finally, they were assembled into the matrices written below.
#
#	A note about the precision required:
#
#	The end of this file contains code for generating the equations 
#    that the coordinates of the matrices K1 -- K14 must satisfy.  (This 
#    is part of the verifications of equation (1) above.)   These 
#    equations are quadratic in the coordinates with coefficients as
#    large as 2.3 * 10^9.   Since the solutions are all in the range of
#    10^2 to 10^3, this necessitates a great deal of numerical precision
#    (23 digits) in order to verify these are indeed solutions.
#
#	We have not been able to find another example of this phenomena
#     which requires less precision, despite extensive searching.
#
#
#	- Joachim Rosenthal,  Dept. Math, University of Notre Dame.
#	- Frank Sottile, Mathematical Sciences Research Institute
#			(Dept. of Math, University of Toronto)
#		1 June 1997
#
   
with(linalg):
Digits:=23:

K1 := matrix([
[-548.1543631072859+ 539.02172783574002*I,
-2966.220011381735449+1301.806890926492508*I],
[227.99002317474104-195.29675914098226*I,
1189.40572416765385-428.190112835481936*I],
[253.619670619102274-128.997418066861599*I,
1192.66093663708038-127.782426659628597*I],
[-373.4608141108503-376.1870941851628*I,
-907.2715490825303837-2040.657619029875556*I]
]):

K2 := matrix([
[-548.1543631072859- 539.02172783574002*I,
-2966.220011381735449-1301.806890926492508*I],
[ 227.99002317474104+195.29675914098226*I,
1189.40572416765385+428.190112835481936*I],
[ 253.619670619102274+128.997418066861599*I,
1192.66093663708038+127.782426659628597*I],
[-373.4608141108503+376.1870941851628*I, 
-907.2715490825303837+2040.657619029875556*I]
]):
 
K3 := matrix([
[ 182.1974051162797+1524.2891350121054*I,
-3910.9491667600289+3319.9425134666556*I],
[-92.76689536072804-494.390627883840*I,
1206.13014159582817-1171.58923461208352*I],
[ 202.71121387564936- 458.78014215695346*I,
1652.30669576900037-280.820983264097575*I],
[-999.496765955436554-938.918292576740638*I,
771.9810394973421-4516.958140814761213*I]
]):

K4 := matrix([
[ 182.1974051162797-1524.2891350121054*I,
-3910.9491667600289-3319.9425134666556*I],
[-92.76689536072804+494.390627883840*I,
1206.13014159582817+1171.58923461208352*I],
[ 202.71121387564936+ 458.78014215695346*I,
1652.30669576900037+280.820983264097575*I],
[-999.496765955436554+938.918292576740638*I,
771.9810394973421+4516.958140814761213*I]
]):

K5 := matrix([
[ 2792.9110057318105-969.00549705135278*I,
3350.9339523791667-832.762320679797284*I],
[- 338.608141548768 - 31.1420684422097*I, 
-390.733153481711 - 71.9581835765450*I],
[-858.10666480772375+463.34803831698071*I,
-1047.08493981311276+448.52274247532122*I],
[-1736.0182637110866+473.54602107116131*I,
-2069.7786151738302+367.88390311074763*I]
]):

K6 := matrix([
[ 2792.9110057318105+969.00549705135278*I,
3350.9339523791667+832.762320679797284*I],
[ -338.608141548768 + 31.1420684422097*I, 
-390.733153481711 + 71.9581835765450*I],
[-858.10666480772375-463.34803831698071*I,
-1047.08493981311276-448.52274247532122*I],
[-1736.0182637110866-473.54602107116131*I,
-2069.7786151738302-367.88390311074763*I]
]):

K7 := matrix([
[566.14047176252718- 390.1690631954798*I, 
894.7573009772359-213.7664118348474*I],
[ -28.9144418101747 - 8.82325220859399*I, 
- 31.9889032754154 - 25.1286025912621*I],
[-101.611268377237  + 166.198294126534*I,  
- 207.075559094765 + 158.433905818864*I],
[-433.109410705026  + 160.543671922194*I, 
- 618.358581551134 - 8.42746099774335*I]
]):

K8 := matrix([
[566.14047176252718+ 390.1690631954798*I, 
894.7573009772359+213.7664118348474*I],
[ -28.9144418101747 + 8.82325220859399*I,
- 31.9889032754154 + 25.1286025912621*I],
[-101.611268377237  - 166.198294126534*I, 
- 207.075559094765 - 158.433905818864*I],
[-433.109410705026  - 160.543671922194*I, 
- 618.358581551134 + 8.42746099774335*I]
]):

K9 := matrix([
[-1328.31492831596508+780.43146580510958*I, 
2115.8811996413627- 363.25099106004349*I],
[  277.0599315399026-134.0101686258348*I, 
-426.505631447159 + 38.4080785894925*I],
[  242.753288068855 - 128.748683783964*I, 
- 380.517275415650 + 48.1897454846160*I],
[  809.814164981704 - 420.527784827832*I, 
- 1263.86094232894868+149.27131835292291*I]
]):

K10 := matrix([
[-1328.31492831596508-780.43146580510958*I, 
2115.8811996413627+ 363.25099106004349*I],
[  277.0599315399026+134.0101686258348*I, 
-426.505631447159 - 38.4080785894925*I],
[  242.753288068855 + 128.748683783964*I, 
- 380.517275415650 - 48.1897454846160*I],
[  809.814164981704 + 420.527784827832*I,
-1263.86094232894868-149.27131835292291*I]
]):

K11 := matrix([
[-74.07812921055438 - 1186.0867962658997*I, 
481.83814937211068-659.46539248077808*I],
[131.85311577768057+223.6599712395458*I, 
-28.4575338243835 + 176.018708417247*I],
[ 50.0398731323218 + 311.162560564792*I,  
- 110.484321267527 + 186.531966999705*I],
[120.94035205524575+693.23751296762126*I, 
- 241.138619140528 + 419.709352592197*I]
]):

K12 := matrix([
[-74.07812921055438 + 1186.0867962658997*I, 
481.83814937211068+659.46539248077808*I],
[131.85311577768057-223.6599712395458*I, 
-28.4575338243835- 176.018708417247*I],
[ 50.0398731323218 - 311.162560564792*I, 
- 110.484321267527-186.531966999705*I],
[120.94035205524575-693.23751296762126*I, 
- 241.138619140528-419.709352592197 *I]
]):

K13 := matrix([
[-466.3420096818032+2560.3776496553293*I,
-477.06216348936717+1505.4573962873226*I],
[ 206.16217936754085 - 504.1659905544772*I, 
162.819554092696 - 287.715806475160*I],
[ 198.483315335125 - 690.317301079547*I, 
172.179197573658 - 400.2773514799496*I],
[350.2539156691074-1658.3575908118343*I,
337.47012412920796 - 971.424525500586678*I]
]):

K14 := matrix([
[-466.3420096818032-2560.3776496553293*I,
- 477.06216348936717-1505.4573962873226*I],
[ 206.16217936754085 + 504.1659905544772*I,
162.819554092696 + 287.715806475160*I],   
[ 198.483315335125 + 690.317301079547*I,
172.179197573658 + 400.2773514799486*I],
[350.2539156691074+1658.3575908118343*I,
337.47012412920796 + 971.424525500586678*I]
]):

F := x-> transpose(matrix([
[ x^4-16*x^3+ 3*x^2+11*x   ,      6*x^3- 4*x^2- 9*x- 5],
[    -26*x^3+10*x^2+ 7*x+16, x^4+ 3*x^3-   x^2-16*x-13],
[      9*x^3-12*x^2+13*x-17,      8*x^3- 6*x^2+ 5*x+15], 
[    -31*x^3-16*x^2+43*x-23,     26*x^3-14*x^2-11*x+12],
[        x^3-36*x^2+ 8*x-13,     11*x^3+ 5*x^2+11*x+33],
[     23*x^3-   x^2+ 2*x-21,            -7*x^2+11*x+ 5]])):
 
Poles := vector([-8,-6,-4,-2,1, 2, 3, 4]):

for jj from 1 to 14 do
for ii from 1 to 8 do
lprint(det(stack(F(Poles[ii]), concat(K.jj,band([1],4)))),

od;

quit