For Formula (19.125)  on page 217 of ChalkleyJan05.dvi
Verification when n = 1, 2, 3, 4, and 5.

For this particular computation, Version 5.1 of MATHEMATICA performs much better than previous versions.  (The particular computer on which this was run with Version 4.1 lacked the ability to do the case n = 4.)

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

\(difference[n_]\  := \ 
      Module[\ \ {vect, \ m, \ M, \ F, \ Fprime, \ G, Gpri, 
          difference}, 
		vect[z_]\  = \ 
          Table[\ \(f[i]\)[z], {i, 0, n}]; \ 
		\(m[i_, j_]\)[z_]\  := \ 
          Sum[\ Binomial[n - i, j - nu]*Binomial[i, nu]*
              a^\((n - i - j + nu)\)*b^\((j - nu)\)*c^\((i - nu)\)*d^nu, {nu, 
              0, j}]; 
		M[z_]\  = \ 
          Table[\ \(m[i, j]\)[z], {i, 0, n}, {j, 0, 
              n}]; 
		\ Do[\ \ \(g[i]\)[
              z_]\  = \ \(Flatten[\((vect[z] . M[z])\)]\)[\([i + 1]\)], \ {i, 
            0, n}]; 
		F[Y_]\  = \ 
          Sum[\ \(f[i]\)[z]*Y^i, {i, 0, n}\ ]; 
		Fprime[
            Y_]\  = \ \ Sum[\ \(f[i]'\)[z]*Y^i, {i, 0, n}\ ]; 
		G[
            Y_]\  = \ Sum[\ \(g[i]\)[z]*Y^i, {i, 0, n}]; 
		Gpri[Y_]\  = \ 
          Sum[\ \(g[i]'\)[z]*Y^i, {i, 0, n}]; 
		difference\  = \ 
          Expand[\ \ \ \((\ 
              Resultant[G[Y], Gpri[Y], Y] - \((a*d - b*c)\)^\((n^2)\)*
                  Resultant[F[Y], Fprime[Y], 
                    Y])\)\ \ ]; 
		Print[\ "difference[", 
          n, "] = ", \ difference]\ \ ]\)

\(ratio[n_]\  := \ 
      Module[\ \ {vect, \ m, \ M, \ F, \ Fprime, \ G, Gpri, 
          ratio}, 
		vect[z_]\  = \ 
          Table[\ \(f[i]\)[z], {i, 0, n}]; \ 
		\(m[i_, j_]\)[z_]\  := \ 
          Sum[\ Binomial[n - i, j - nu]*Binomial[i, nu]*
              a^\((n - i - j + nu)\)*b^\((j - nu)\)*c^\((i - nu)\)*d^nu, {nu, 
              0, j}]; 
		M[z_]\  = \ 
          Table[\ \(m[i, j]\)[z], {i, 0, n}, {j, 0, 
              n}]; 
		\ Do[\ \ \(g[i]\)[
              z_]\  = \ \(Flatten[\((vect[z] . M[z])\)]\)[\([i + 1]\)], \ {i, 
            0, n}]; 
		F[Y_]\  = \ 
          Sum[\ \(f[i]\)[z]*Y^i, {i, 0, n}\ ]; 
		Fprime[
            Y_]\  = \ \ Sum[\ \(f[i]'\)[z]*Y^i, {i, 0, n}\ ]; 
		G[
            Y_]\  = \ Sum[\ \(g[i]\)[z]*Y^i, {i, 0, n}]; 
		Gpri[Y_]\  = \ 
          Sum[\ \(g[i]'\)[z]*Y^i, {i, 0, n}]; 
		ratio\  = \ 
          Factor[\((\ 
              Resultant[G[Y], Gpri[Y], Y]/
                Resultant[F[Y], Fprime[Y], 
                  Y])\)\ ]; 
		Print[\ "ratio[", n, "] = ", \ 
          ratio]\ \ ]\)

Date[]

difference[1]

Date[]

ratio[1]

Date[]

difference[2]

Date[]

ratio[2]

Date[]

difference[3]

Date[]

ratio[3]

Date[]

difference[4]

Date[]

ratio[4]

Date[]

difference[5]

Date[]

ratio[5]

Date[]

MemoryInUse[]

MaxMemoryUsed[]

Date[]

difference[6]

We terminated this computation after about 15 minutes because by that time it was using virtual memory (writing to the hard disc) almost continuously.

The case n = 5 was run fairly rapidly here with Version 5.1 whereas previous versions had difficulty beyond n = 3.

System Information report written at: 02/06/2005 11:34:52 AM
[System Summary]

Item	Value
OS Name	Microsoft Windows 2000 Professional
Version	5.0.2195 Service Pack 4 Build 2195
OS Manufacturer	Microsoft Corporation
System Name	CHALKLEY
System Manufacturer	Dell Computer Corporation      
System Model	XPS-Z                          
System Type	X86-based PC
Processor	x86 Family 6 Model 8 Stepping 3 GenuineIntel ~930 Mhz
BIOS Version	08/29/00
Windows Directory	C:\WINNT
System Directory	C:\WINNT\system32
Boot Device	\Device\Harddisk0\Partition1
Locale	United States
User Name	CHALKLEY\chalklr
Time Zone	Eastern Standard Time
Total Physical Memory	523,568 KB
Available Physical Memory	110,160 KB
Total Virtual Memory	1,801,932 KB
Available Virtual Memory	916,352 KB
Page File Space	1,278,364 KB
Page File	C:\pagefile.sys