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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 256 53 "The following examples is Jan Verschelde
's  examples:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 257 183 "
The purpose of this worksheet is to generate instances of multivariate
 polynomials with complex coefficients.  Executing this worksheet gene
rates the files exf1, exf2, exf3, and exf4." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "The Jacobian for a Ste
wart-Gough Platform" }}{PARA 0 "" 0 "" {TEXT -1 58 "We derive the Jaco
bian for a platform, first symbolically." }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 32 "1. The Symbolic Jacobian Matrix." }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 81 "A rotation is represented by a quaternion q, as an elem
ent in projective 3-space:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "q := array(0..3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Q := sum(q
['i']^2,'i'=0..3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Rhat := Matri
x(3,3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "R := Rhat/Q:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 47 "Rhat[1,1] := q[0]^2 + q[1]^2 - q[2]^2 - q[3
]^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Rhat[2,2] := q[0]^2 - q[1]^
2 + q[2]^2 - q[3]^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Rhat[3,3] :
= q[0]^2 - q[1]^2 - q[2]^2 + q[3]^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "Rhat[1,2] := 2*(q[1]*q[2] + q[0]*q[3]):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "Rhat[2,1] := 2*(q[1]*q[2] - q[0]*q[3]):" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 39 "Rhat[1,3] := 2*(q[1]*q[3] - q[0]*q[2]):" }
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Rhat[3,1] := 2*(q[1
]*q[3] + q[0]*q[2]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Rhat[2,3] :
= 2*(q[2]*q[3] + q[0]*q[1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Rha
t[3,2] := 2*(q[2]*q[3] - q[0]*q[1]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**$)&%\"qG6#
\"\"!\"\"#\"\"\"F,*$)&F(6#F,F+F,F,*$)&F(6#F+F+F,F,*$)&F(6#\"\"$F+F,F,!
\"\"-%'RTABLEG6%\")#\\>u#-%'MATRIXG6#7%7%,*F%F,F-F,F1F:F5F:,&*(F+F,F/F
,F3F,F,*(F+F,F'F,F7F,F,,&*(F+F,F/F,F7F,F,*(F+F,F'F,F3F,F:7%,&*(F+F,F/F
,F3F,F,*(F+F,F'F,F7F,F:,*F%F,F-F:F1F,F5F:,&*(F+F,F3F,F7F,F,*(F+F,F'F,F
/F,F,7%,&*(F+F,F/F,F7F,F,*(F+F,F'F,F3F,F,,&*(F+F,F3F,F7F,F,*(F+F,F'F,F
/F,F:,*F%F,F-F:F1F:F5F,%'MatrixGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "Next we define the position vector pt:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "pt := <p[1],p[2],p[3]>:" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 65 "and we define the locations at endplate and the basepl
ate joints:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "for i from 1 to 6 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  a
a||i := <a||i[1],a||i[2],a||i[3]>:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "  bb||i := <b||i[1],b||i[2],b||i[3]>:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Now we
 are ready to define the Jacobian matrix:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "jp := Matrix(6,6):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to 6 do" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 90 "  jpA||i := LinearAlgebra:-ScalarMultiply(pt-bb||i,
Q)+LinearAlgebra:-Multiply(Rhat,aa||i);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "  for j from 1 to 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " \+
   jp[i,j] := jpA||i[j];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "  jpB||i := LinearAlgebra:-Cro
ssProduct(LinearAlgebra:-Multiply(Rhat,aa||i),pt-bb||i);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "  for j from 1 to 3 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "    jp[i,j+3] := jpB||i[j];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We better do not ask here to \+
compute the determinant..." }{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 
0 "" {TEXT 258 22 "2. Numerical Instances" }{TEXT 259 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "We have three vectors to specify: the po
sition vector and the locations at endplate and the baseplate joints.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 " For the purposes of applyin
g homotopy continuation to factor the polynomials, we generate random \+
complex number of modulus one, using 20 digit" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "precision" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "Digits:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Dig
itsG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "randcomp := pro
c()" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "  description `returns a ran
dom complex number of modulus one`;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "  local angle;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "angle := stat
s[random,uniform[0,2*Pi]](1):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  \+
evalc(cos(angle)+sin(angle)*I):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e
nd proc:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "randcomp();" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#^$$!5M'[R>54a!y*)!#?$\"5[a!*H$*)yBRS%F&" }}}
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "In this first instance, we pick the locations and positio
n vector at random:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rando
mize:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to 3 \+
do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  p[i] := randcomp(): " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "for i from 1 to 6 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 22 "  for j from 1 to 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " \+
   a||i[j] := randcomp();" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "    b|
|i[j] := randcomp();" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "f1 := LinearAlgebra:-Determinant(jp):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "save f1, \"verschelde/svw_exf1\";" 
}}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "Here we set the third element of all locations equal to z
ero:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=20;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "for i from 1 to 6 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "  a||i[3] := 0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"  b||i[3] := 0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f2 := LinearAlgebra:-Determi
nant(jp):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "save f2,\"vers
chelde/svw_exf2\";" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Example 3
" }}{PARA 0 "" 0 "" {TEXT -1 47 "In addition we set q[1] and q[2] equa
l to zero:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=20;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "q[1] := 0: q[2] := 0:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "f3 := LinearAlgebra:-Determinant(jp):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "f3:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "ff:=f3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "save ff, \"verschelde/svw_exf3\";" }}}}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Finally, \+
we let the point pt have arbitrary coordinates:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "Digits:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%'DigitsG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i fr
om 1 to 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  p[i] := evaln(p[i
]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "pt := <p[1],p[2],p[3]>;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#ptG-%'RTABLEG6%\")W()HC-%'MATRIXG6#7%7#&%\"pG6#\"\"
\"7#&F/6#\"\"#7#&F/6#\"\"$&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "jp := Matrix(6,6);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "for i from 1 to 6 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 90 "  jpA||i := LinearAlgebra:-ScalarMultiply(pt-bb||i,Q)+LinearAlge
bra:-Multiply(Rhat,aa||i);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for
 j from 1 to 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "    jp[i,j] :=
 jpA||i[j];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 86 "  jpB||i := LinearAlgebra:-CrossProduct(Lin
earAlgebra:-Multiply(Rhat,aa||i),pt-bb||i);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "  for j from 1 to 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "    jp[i,j+3] := jpB||i[j];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#jpG-%'RTABLEG6%\")wGwC-%'MATRIXG6#7(7(\"\"!F
.F.F.F.F.F-F-F-F-F-%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "f4 :=LinearAlgebra:- Determinant(jp):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "save f4, \"verschelde/svw_exf4\";" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}
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