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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(MS):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 90 "This example looks at the Lucas numbers t
ype I.  Consider the linear recurrence relation  " }{XPPEDIT 18 0 "b[i
] = b[i-1]+b[i-2];" "6#/&%\"bG6#%\"iG,&&F%6#,&F'\"\"\"\"\"\"!\"\"F,&F%
6#,&F'F,\"\"#F.F," }{TEXT -1 7 " where " }{XPPEDIT 18 0 "b[0] = 2;" "6
#/&%\"bG6#\"\"!\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[1] = 1;" "
6#/&%\"bG6#\"\"\"\"\"\"" }{TEXT -1 49 ".  Multisection this by 8 at 2.
  Notice that 8 = " }{XPPEDIT 18 0 "2^3;" "6#*$\"\"#\"\"$" }{TEXT -1 
42 " and further that 2 =  0 (4) + 1 (2) + 0. " }{TEXT -1 112 " Any me
thod can be used to compute the intermediate multisectioning.  For thi
s example the naive method is used." }}{PARA 0 "" 0 "" {TEXT -1 51 "So
 the first step is to calculate s^0_2 (x), where " }{XPPEDIT 18 0 "s(x
) = sum(b[i]*x^i/i!,i = 0 .. infinity);" "6#/-%\"sG6#%\"xG-%$sumG6$*(&
%\"bG6#%\"iG\"\"\")F'F/F0-%*factorialG6#F/!\"\"/F/;\"\"!%)infinityG" }
{TEXT -1 10 " with the " }{XPPEDIT 18 0 "b[i];" "6#&%\"bG6#%\"iG" }
{TEXT -1 21 "s defined as above.  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "s := b(i) = b(i-1) + b(i-2) , b, i, [b(0) = 2, b(1) =
 1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6&/-%\"bG6#%\"iG,&-F(6#,
&F*\"\"\"!\"\"F/F/-F(6#,&F*F/!\"#F/F/F(F*7$/-F(6#\"\"!\"\"#/-F(6#F/F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t := `egf/ms/naive`(s,2
,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG6&/-%\"bG6#%\"iG,&-F(6#,
&F*\"\"\"!\"#F/\"\"$-F(6#,&F*F/!\"%F/!\"\"F(F*7&/-F(6#\"\"!\"\"#/-F(6#
F/F;/-F(6#F<F1/-F(6#F1F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Now c
ompress this result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "s2 \+
:= readlib(`egf/compress`)(t, 2, 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#s2G6&/-%\"bG6#%\"iG,&-F(6#,&F*\"\"\"!\"\"F/\"\"$-F(6#,&F*F/!\"#F/
F0F(F*7$/-F(6#\"\"!\"\"#/-F(6#F/F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 74 "The second step is to calculate the multisectioning of the abov
e function " }{XPPEDIT 18 0 "s2;" "6#%#s2G" }{TEXT -1 11 " by 2 at 1.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "t2 := `egf/ms/naive`(s2
, 2, 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G6&/-%\"bG6#%\"iG,&-F
(6#,&F*\"\"\"!\"#F/\"\"(-F(6#,&F*F/!\"%F/!\"\"F(F*7&/-F(6#\"\"!F;/-F(6
#F/\"\"$/-F(6#\"\"#F;/-F(6#F?\"#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
24 "Now compress the result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "s3 := `egf/compress`(t2, 2, 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#s3G6&/-%\"bG6#%\"iG,&-F(6#,&F*\"\"\"!\"\"F/\"\"(-F(6#,&F*F/!\"#F
/F0F(F*7$/-F(6#\"\"!\"\"$/-F(6#F/\"#=" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "Now the last step is to  multisection the above function \+
" }{XPPEDIT 18 0 "s3;" "6#%#s3G" }{TEXT -1 11 " by 2 at 0." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "t3 := `egf/ms/naive`(s3, 2, 0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G6&/-%\"bG6#%\"iG,&-F(6#,&F*\"\"
\"!\"#F/\"#Z-F(6#,&F*F/!\"%F/!\"\"F(F*7&/-F(6#\"\"!\"\"$/-F(6#F/F;/-F(
6#\"\"#\"$B\"/-F(6#F<F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "By co
mpressing this result, a linear recurrence relation for the Lucas type
 I numbers is found using only every 8-th term." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "s4 := `egf/compress`(t3, 2, 0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#s4G6&/-%\"bG6#%\"iG,&-F(6#,&F*\"\"\"!\"\"F/\"
#Z-F(6#,&F*F/!\"#F/F0F(F*7$/-F(6#\"\"!\"\"$/-F(6#F/\"$B\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 83 "Uncompressing this result to gives the an
swer, as expected from the other commands." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "readlib(`egf/uncompress`)(s4, 8, 2);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6&/-%\"bG6#%\"iG,&-F%6#,&F'\"\"\"!\")F,\"#Z-F%6#,&
F'F,!#;F,!\"\"F%F'72/-F%6#\"\"!F8/-F%6#F,F8/-F%6#\"\"#\"\"$/-F%6#F@F8/
-F%6#\"\"%F8/-F%6#\"\"&F8/-F%6#\"\"'F8/-F%6#\"\"(F8/-F%6#\"\")F8/-F%6#
\"\"*F8/-F%6#\"#5\"$B\"/-F%6#\"#6F8/-F%6#\"#7F8/-F%6#\"#8F8/-F%6#\"#9F
8/-F%6#\"#:F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Notice that usi
ng the naive method directly to multisection by 8 at 2 gives the same \+
result, but the method takes much longer to work." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "`egf/ms/naive`(s,8,2);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6&/-%\"bG6#%\"iG,&-F%6#,&F'\"\"\"!\")F,\"#Z-F%6#,&F'F,!#;
F,!\"\"F%F'72/-F%6#\"\"!F8/-F%6#F,F8/-F%6#\"\"#\"\"$/-F%6#F@F8/-F%6#\"
\"%F8/-F%6#\"\"&F8/-F%6#\"\"'F8/-F%6#\"\"(F8/-F%6#\"\")F8/-F%6#\"\"*F8
/-F%6#\"#5\"$B\"/-F%6#\"#6F8/-F%6#\"#7F8/-F%6#\"#8F8/-F%6#\"#9F8/-F%6#
\"#:F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "This process has been \+
automated with the Maple command `egf/ma/compress`.  That last option \+
of the command specifies to use the naive method to do the underlying \+
computation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "`egf/ms/com
press`(s, 8, 2, naive);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&/-%\"bG6#%
\"iG,&-F%6#,&F'\"\"\"!\")F,\"#Z-F%6#,&F'F,!#;F,!\"\"F%F'72/-F%6#\"\"!F
8/-F%6#F,F8/-F%6#\"\"#\"\"$/-F%6#F@F8/-F%6#\"\"%F8/-F%6#\"\"&F8/-F%6#
\"\"'F8/-F%6#\"\"(F8/-F%6#\"\")F8/-F%6#\"\"*F8/-F%6#\"#5\"$B\"/-F%6#\"
#6F8/-F%6#\"#7F8/-F%6#\"#8F8/-F%6#\"#9F8/-F%6#\"#:F8" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 29 "Which gives the same results." }}}}{MARK "16 0 \+
0" 107 }{VIEWOPTS 1 1 0 1 1 1803 }