The program was run twice with two different sets of a, b, c, d, e data. The output for the first run is Section One below; there were six solutions of System D, none of which were solutions for System B. The output for the second run is Section Two below; there were two System D solutions and both satisfied System B.
The program was written with a view towards testing the polynomial reduction method; thus, program output is simplistic in design and rather minimal.
Every time a solution is found for System D, the i, j, k loop indices are displayed, then followed by the x1pflt, x2pflt, x3pflt values (i.e., the solution). The next two data displayed are solution_error and basic_set_error. solution_error shows how well the solution satisfies System D (the "polynomial system"). basic_set_error shows how well the solution satisfies System B (the "basic set") and in some cases this will be a large number; thus, such a solution is extraneous with respect to System B.
SECTION ONE
Maxima 5.18.0 http://maxima.sourceforge.net
Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
"Use fn1:simplode([maxpath, file-name-in-dbl-quotes])\; etc., to create in
fn1 a file name"
(%i1) /* Three surface program Test Case 1.wpd November 10, 2011 (3:03AM) */
/* begin one time code */
a[1]:2$ a[2]:6$ a[3]:1$
b[1]:-9$ b[2]:4$ b[3]:-6$
c[1]:-3$ c[2]:8$ c[3]:-9$
d[1]:4$ d[2]:5$ d[3]:7$
e1 :2$ e2 :2.8$ e3 :5.4$
--Balance of program listing (and associated statement numbers) deleted--
(%i32)
(%o33) "Equations eq1, eq2, eq3 follow:"
(%i34)
(%o34)
20*x3^2+12*x3+x2*(-96*x3-144)+x1*(-144*x3+192*x2-696)+48*x2^2+128*x1^2
-1839
(%i35)
(%o35) (102900*x3^2+x1*(385000*x3+110000*x2+2392800)+x2*(70000*x3+656000)
+1512000*x3-9600*x2^2+282900*x1^2+3623216)
/625
(%i36)
(%o36) (623100*x3^2+x1*(560000*x3-105000*x2+1136900)+2526200*x3
+x2*(-240000*x3-260300)+5600*x2^2+105600*x1^2+526761)
/625
(%i37)
(%i38)
(%i39)
"x1eq follows"
4.1330677279253044E+113*x1^16+3.4816177638165769E+115*x1^15
+9.5207986699337984E+116*x1^14
+1.088049147573676E+118*x1^13
+4.0372335247515427E+118*x1^12
-2.8088674285340226E+119*x1^11
-3.6721720106422438E+120*x1^10
-1.4704989543772044E+121*x1^9
-6.5480770692093099E+120*x1^8
+1.4992496645586712E+122*x1^7
+4.9798669198946339E+122*x1^6
+2.4535454592141987E+122*x1^5
-1.7705595453732661E+123*x1^4
-3.3755178557825277E+123*x1^3
+1.6329215110953E+120*x1^2
+4.4160212507051009E+123*x1
+2.323589376673053E+123
(%i40)
(%i41)
(%o42) "Show solutions for the polynomial system"
(%i43)
"x2eq follows"
-1.5116048179098621E+89*x2^4-2.4669399315392145E+91*x2^3
+1.8433697584572433E+94*x2^2
-1.4139077544254316E+96*x2+2.5287228946699688E+97
i = 1
nmbr_x2roots = 4
"x2eq follows"
-1.5116048179098621E+89*x2^4-5.0946591076946017E+90*x2^3
+1.7474971223306917E+93*x2^2
-4.5298107876566169E+94*x2+2.4360500183212742E+95
i = 2
nmbr_x2roots = 4
i = 2
j = 3
k = 1
x1pflt = -18.00183546310291
x2pflt = 21.81927785230801
x3pflt = 5.672693971078843
solution_error = 4.9511023221242264E-6
basic_set_error = 82.48486975990423
"x2eq follows"
-1.5116048179098621E+89*x2^4+4.0660365335241104E+90*x2^3
-7.4669028173405442E+91*x2^2
+4.5953614875020498E+92*x2-4.1470405336221688E+92
i = 3
nmbr_x2roots = 2
i = 3
j = 2
k = 2
x1pflt = -5.559397575911135
x2pflt = 7.78733742935583
x3pflt = 3.803053039591759
solution_error = 2.0099677562970053E-7
basic_set_error = 34.63232557644681
"x2eq follows"
-1.5116048179098621E+89*x2^4+4.0904011757632389E+90*x2^3
-7.442580621995423E+91*x2^2
+4.4656525009345349E+92*x2-3.7072093595402995E+92
i = 4
nmbr_x2roots = 2
"x2eq follows"
-1.5116048179098621E+89*x2^4+5.1409560688342387E+90*x2^3
-3.8255990187676112E+91*x2^2
-1.2611058824083709E+92*x2+6.7288623013850759E+92
i = 5
nmbr_x2roots = 4
i = 5
j = 3
k = 1
x1pflt = -4.099397148471326
x2pflt = 15.89904746459797
x3pflt = 0.54241102328524
solution_error = 1.8403425583493835E-6
basic_set_error = 42.17703830759238
"x2eq follows"
-1.5116048179098621E+89*x2^4+7.6554097707741201E+90*x2^3
+2.5218225849857932E+92*x2^2
+2.5125216800037992E+92*x2-1.5073731397125987E+93
i = 6
nmbr_x2roots = 4
i = 6
j = 2
k = 2
x1pflt = -0.68416153406724
x2pflt = -3.196148897986859
x3pflt = -0.16304807597771
solution_error = 7.1063968203207986E-7
basic_set_error = 39.42394086929675
"x2eq follows"
-1.5116048179098621E+89*x2^4+9.087086028306474E+90*x2^3
+5.460388281056032E+92*x2^2
+3.1947578902781098E+93*x2+1.0444426345206805E+93
i = 7
nmbr_x2roots = 4
"x2eq follows"
-1.5116048179098621E+89*x2^4+9.5896803415416726E+90*x2^3
+6.713075539309518E+92*x2^2
+4.9468725753689177E+93*x2+4.4490751192180965E+93
i = 8
nmbr_x2roots = 4
"x2eq follows"
-1.5116048179098621E+89*x2^4+1.0314153927999219E+91*x2^3
+8.7209625281734618E+92*x2^2
+8.2660005554499686E+93*x2+1.3088514201687517E+94
i = 9
nmbr_x2roots = 4
i = 9
j = 1
k = 1
x1pflt = 2.927055378910154
x2pflt = -41.07100304448977
x3pflt = -19.40053169103339
solution_error = 7.9548948135751705E-6
basic_set_error = 170.3844219969379
"x2eq follows"
-1.5116048179098621E+89*x2^4+1.3010044619697781E+91*x2^3
+1.8289751452980222E+93*x2^2
+3.093253347372138E+94*x2+1.1265173828201661E+95
i = 10
nmbr_x2roots = 4
i = 10
j = 3
k = 1
x1pflt = 6.588726260233671
x2pflt = -5.148092913907021
x3pflt = -8.703912747558206
solution_error = 2.8609446729543209E-6
basic_set_error = 55.47286469545191
(%i44)
case_cntr = 72
Sbound = 1.0000000000000001E-9
(%o45) "end of run"
(%i46)
SECTION TWO
Maxima 5.18.0 http://maxima.sourceforge.net
Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
"Use fn1:simplode([maxpath, file-name-in-dbl-quotes])\; etc., to create in
fn1 a file name"
(%i1) /* Three surface program Test Case 2.wpd November 10, 2011 (3:03AM) */
/* begin one time code */
a[1]:-1.03$ a[2]:2$ a[3]:-0.01$
b[1]:-1$ b[2]:0$ b[3]:0$
c[1]:1$ c[2]:0.01$ c[3]:0.04$
d[1]:1.01$ d[2]:0.02$ d[3]:2$
e1 :3.9$ e2 :4.1$ e3 :8.4$
--Balance of program listing (and associated statement numbers) deleted--
(%i32)
(%o33) "Equations eq1, eq2, eq3 follow:"
(%i34)
(%o34) -(6083360000*x3^2-243332800*x3+x2*(-320000*x3-60833200)
+x1*(-64000000*x3-16000000*x2+1360000)+6083960000*x2^2
+4484000000*x1^2-17045238889)
/100000000
(%i35)
(%o35) -(67239600*x3^2+x2*(160000*x3-101992000)+509960*x3
+x1*(-2400*x3+480000*x2+136009880)+51240000*x2^2
+67236400*x1^2-95297001)
/1000000
(%i36)
(%o36) -(366210000*x3^2-746068800*x3+x1*(-980000*x3-5000*x2-928117800)
+x2*(-980000*x3-12862800)+462247500*x2^2+462247500*x1^2
-845811859)
/6250000
(%i37)
(%i38)
(%i39)
"x1eq follows"
9.539870758870685E+281*x1^16-3.7765086630154793E+282*x1^15
+9.6603201487019756E+283*x1^14
-2.5557056703846289E+284*x1^13
+3.7986141330916184E+285*x1^12
-6.6573583583257668E+285*x1^11
+9.1306619965221758E+286*x1^10
-1.0999751539948712E+287*x1^9
+1.536520659347158E+288*x1^8
-8.7152655362035653E+287*x1^7
+1.4763102552535207E+289*x1^6
-1.0684586470257733E+288*x1^5
+1.0723466647490371E+290*x1^4
+4.6548729522209408E+289*x1^3
-1.4233282746449918E+290*x1^2
-2.7228272840659548E+289*x1
+4.5523732387273018E+289
(%i40)
(%i41)
(%o42) "Show solutions for the polynomial system"
(%i43)
"x2eq follows"
1.6753846674658863E+103*x2^4+1.0087515618371521E+104*x2^3
+5.7411516886132255E+104*x2^2
+2.1069927573669284E+104*x2+1.262983194020069E+103
i = 1
nmbr_x2roots = 2
"x2eq follows"
1.6753846674658863E+103*x2^4+1.00874735111057E+104*x2^3
+5.7381873973756773E+104*x2^2
+2.0981685226902E+104*x2+1.0812746361487799E+103
i = 2
nmbr_x2roots = 2
i = 2
j = 2
k = 2
x1pflt = -0.85428428137675
x2pflt = -0.061900383327156
x3pflt = 1.518878292758018
solution_error = 7.485131814593818E-8
basic_set_error = 6.0992575567163715E-10
"x2eq follows"
1.6753846674658863E+103*x2^4+1.0013533796888142E+104*x2^3
+9.5457384851200266E+103*x2^2
-1.2002239743384144E+105*x2
+1.1612294155314822E+105
i = 3
nmbr_x2roots = 2
i = 3
j = 2
k = 1
x1pflt = 0.6941764797084
x2pflt = 1.552698559593409
x3pflt = -0.2095697154291
solution_error = 1.2095959998930184E-7
basic_set_error = 1.0138076272370094E-9
"x2eq follows"
1.6753846674658863E+103*x2^4+1.0013417282743036E+104*x2^3
+9.4770091662776258E+103*x2^2
-1.2022273641734427E+105*x2
+1.1689456122146154E+105
i = 4
nmbr_x2roots = 0
(%i44)
case_cntr = 12
Sbound = 1.0000000000000001E-9
(%o45) "end of run"
(%i46)
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Rev. 11/11/11