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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 20 "Homework 1 Solutions" }}{SECT 1 
{PARA 3 "" 0 "" {TEXT 269 26 "1. Write our name and such" }}{EXCHG 
{PARA 261 "" 0 "" {TEXT 267 10 "Homework 1" }{TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT -1 5 "CS300" }}{PARA 260 "" 0 "" {TEXT -1 11 "Spring
 2003" }}{PARA 0 "" 0 "" {TEXT -1 32 "Jeff Abrahamson & Jeremy Johnson
" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 256 43 "2. Verif
ying the correctness of Winograd's " }{TEXT 257 9 "algorithm" }}{PARA 
258 "" 0 "" {TEXT 258 85 "We define two squares matrices A and B with \+
coefficients a, b, c, d, and e, f, g,  h." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT 266 83 "Let's now plug in values initials variables A11, A12, A2
1,A22,B11,B12, B21,and B22." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
217 "A11:= a: A12:= b: A21:= c: A22:= d: \nB11:= e: B12:= f: B21:= g: \+
B22:= h:                                                              \+
             A:= Matrix([[A11,A12],[A21,A22]]): \nB:= Matrix([[B11,B12
],[B21,B22]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 259 85 "Let SC (Standard C matrix) be the
 product of A and B . Using the standard algorithm ." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "SC:= A.B; " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#SCG-%'RTABLEG6%\")C!RE\"-%'MATRIXG6#7$7$,&*&%\"aG\"\"\"%\"eGF
1F1*&%\"bGF1%\"gGF1F1,&*&F0F1%\"fGF1F1*&F4F1%\"hGF1F17$,&*&%\"cGF1F2F1
F1*&%\"dGF1F5F1F1,&*&F>F1F8F1F1*&F@F1F:F1F1%'MatrixG" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 167 "We now follow
 Winograd's algorithm to evaluate WC, the product according to Winogra
d's algorithm. Assign S1, S2, S3,  S4, T1, T2, T3 and T4 according to \+
the algorithm." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "S1:= A21
+A22: S2:= S1 - A11: S3:= A11 - A21: S4:= A12 - S2:                   \+
              T1:= B12 - B11: T2:= B22 - T1: T3:= B22 - B12: T4:= B21 \+
- T2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 262 72 "Assign P1, P2, P3, P4, P5, P6, and P7 without display
ing their results.." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "P1:
= A11*B11: P2:= A12*B21: P3:= S1*T1: P4:= S2*T2:                      \+
                   P5:= S3*T3: P6:= S4*B22: P7:= A22*T4:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 66 "Calculat
e U1,U2,U3,U4,U5,U6, U7 and assign them to WC11,WC12,WC21," }}{PARA 0 
"" 0 "" {TEXT 268 54 "and WC22 accordingly; WC stands for Winograd C m
atrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "U1:= expand(P1+P2
); U2:= expand(P1+P4): \nU3:= expand(U2+P5): U4:= expand(U3+P7);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U1G,&*&%\"aG\"\"\"%\"eGF(F(*&%\"bGF
(%\"gGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U4G,&*&%\"cG\"\"\"%\"
eGF(F(*&%\"dGF(%\"gGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
132 "U5:= expand(U3+P3); U6:= expand(U2+P3): \nU7:= expand(U6+P6); WC1
1:= U1: WC12:= U7: \nWC21:= U4: WC22:= U5:                           \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U5G,&*&%\"cG\"\"\"%\"fGF(F(*&%
\"dGF(%\"hGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U7G,&*&%\"aG\"\"
\"%\"fGF(F(*&%\"bGF(%\"hGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "
               " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "WC:= Mat
rix([[WC11, WC12],[WC21,WC22]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#WCG-%'RTABLEG6%\")cPz7-%'MATRIXG6#7$7$,&*&%\"aG\"\"\"%\"eGF1F1*&%\"bG
F1%\"gGF1F1,&*&F0F1%\"fGF1F1*&F4F1%\"hGF1F17$,&*&%\"cGF1F2F1F1*&%\"dGF
1F5F1F1,&*&F>F1F8F1F1*&F@F1F:F1F1%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 24 "Let's compare SC and WC.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "difference:= SC - WC;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+differenceG-%'RTABLEG6%\")C%eG\"-
%'MATRIXG6#7$7$\"\"!F.F-%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
264 64 "Winograd's algorithm thus correctly computes the matrix produc
t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
265 43 " 3. Using rsolve command to plot the ratio." }}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 8 "Let n = " }{XPPEDIT 18 0 "2^k" "6#)\"\"#%\"kG
" }{TEXT -1 42 ". Winograd's algoritm's computing time is " }{XPPEDIT 
18 0 "WT(n) = 7 * ( WT(n/2) ) + 15 (n/2)^2)" "6#/-%#WTG6#%\"nG,&*&\"\"
(\"\"\"-F%6#*&F'F+\"\"#!\"\"F+F+*$-\"#:6#*&F'F+F/F0F/F+" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 51 "(7 multiplications and 15 additions/
substractions)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 35 "How many arithmetic operations for " }{XPPEDIT 18 0 
"n =2^k" "6#/%\"nG)\"\"#%\"kG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 63 "wt:=rsolve(\{WT(n)= 7* WT(n/2) + 15*((n/2)^2),WT(1)=1\}, WT(n));
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#wtG,&*&\"\"'\"\"\")%\"nG*&-%#l
nG6#\"\"(F(-F-6#\"\"#!\"\"F(F(*&\"\"&F()F*F2F(F3" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 33 "which is O(n^lg 7) = O(n^2.81...)" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The standard m
ultiplication algorithm, which requires 8 multiplications and 4 additi
ons, is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "st:=rsolve(\{ST(
n)= 8*ST(n/2) + 4*((n/2)^2),ST(1)=1\}, ST(n)); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#stG*&)%\"nG\"\"#\"\"\",&*&F(F)F'F)F)F)!\"\"F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "which is O(n^3). Here \"st\" means
 the standard algorithm, not Strassen's, sorry for the bad naming." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "wt2:=unapply(wt,n); st2:=unapply(st,n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$wt2Gf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&*&\"\"'\"\"
\")9$*&-%#lnG6#\"\"(F/-F46#\"\"#!\"\"F/F/*&\"\"&F/)F1F9F/F:F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$st2Gf*6#%\"nG6\"6$%)operatorG%&arro
wGF(*&)9$\"\"#\"\"\",&*&F/F0F.F0F0F0!\"\"F0F(F(F(" }}}{PARA 11 "" 1 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "wt_st_rati
o:=wt2(n)/st2(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,wt_st_ratioG*(
,&*&\"\"'\"\"\")%\"nG*&-%#lnG6#\"\"(F)-F.6#\"\"#!\"\"F)F)*&\"\"&F))F+F
3F)F4F)F+!\"#,&*&F3F)F+F)F)F)F4F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
68 "Let's check that this seems reasonable by evaluating the base case
s:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "wt2(2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&\"\"'\"\"\")\"\"#*&-%#lnG6#\"\"(F&-F+6#F(!\"\"F&F&
\"#?F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "st2(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We can see that the cross-over hap
pens between k=8 and k=9:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(ev
alf(wt2(2^k)/st2(2^k)),k=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$
\"+LLLL=!\"*$\"+'G92\">F%$\"+nmT5=F%$\"+a=)Rl\"F%$\"+jfy$[\"F%$\"+Uth<
8F%$\"+?)*4j6F%$\"+%GfI-\"F%$\"+F^nz*)!#5$\"+@opryF6" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 27 "Let's see this graphically:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warnin
g, the name changecoords has been redefined\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7Z%(animateG%*animate3dG%-animatecurveG%&arrowG%-change
coordsG%,complexplotG%.complexplot3dG%*conformalG%,conformal3dG%,conto
urplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,dens
ityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+g
radplot3dG%,graphplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%,in
teractiveG%-listcontplotG%/listcontplot3dG%0listdensityplotG%)listplot
G%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%(odeplotG%'paretoG%
,plotcompareG%*pointplotG%,pointplot3dG%*polarplotG%,polygonplotG%.pol
ygonplot3dG%4polyhedra_supportedG%.polyhedraplotG%'replotG%*rootlocusG
%,semilogplotG%+setoptionsG%-setoptions3dG%+spacecurveG%1sparsematrixp
lotG%+sphereplotG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "p1 := plot(evalf(wt2(2^k)/st
2(2^k)),k=1..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p2 := \+
plot(1,k=1..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display
(\{p1,p2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6&
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "For what n value does Winograd bec
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 to 300 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "  if evalf(wt2(i)/st2
(i)) < 1 then print(i); break; fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "od:" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$!H" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Now plot the ratio..." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "smartplot(wt_st_ratio);" }}{PARA 13 "" 1 "" 
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3?+++I!o-*\\F-$\"3riWR'Q)Q*)=F-7$$\"3N+++5k.6aF-$\"3-^4J7^cy=F-7$$\"3J
+++?WTAeF-$\"3_\")f*[tIw'=F-7$$\"3w******f!*3`iF-$\"3fFGvq.1c=F-7$$\"3
y******H*zym'F-$\"3$[(RTc5&\\%=F-7$$\"3-+++5N1#4(F-$\"3Y'yl]3DP$=F-7$$
\"3v******HYt7vF-$\"3Zx?U5\"*yA=F-7$$\"3O++++xG**yF-$\"3&3Aa!=i%H\"=F-
7$$\"3C******R6KU$)F-$\"3L5t7^5$>!=F-7$$\"3K+++IbdQ()F-$\"3G_y,@.L#z\"
F-7$$\"3F+++g`1h\"*F-$\"3Q)*>mO)fBy\"F-7$$\"3l+++S?Wl&*F-$\"3$optnWtIx
\"F-7$$\"#5\"\"!$\"3U0Q&=$*pLw\"F--%'LEGENDG6#Q(Curve~16\"-%'COLOURG6&
%$RGBG$\"*++++\"!\")$Fe_lFe_lFd`l-%+AXESLABELSG6$Q\"nF\\`lQ!F\\`l-%%VI
EWG6$;$!#5Fe_lFc_l%(DEFAULTG" 1 2 0 1 10 0 2 9 0 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "plot(wt2(n)-st2(n),n=0..400);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"\"!F)F(7$$\"3=
nmmm;')=()!#<$\"3Yv-jfTj1**!#:7$$\"3VLLLe'40j\"!#;$\"3?&oe-F&\\da!#97$
$\"3wmmm;6m$[#F4$\"3;;p9YP()R;!#87$$\"3Nnmm;yYULF4$\"3Ag\"4S^_0[$F=7$$
\"3GLLLeF>(>%F4$\"3SyGV)[;M5'F=7$$\"3Zmmm\">K'*)\\F4$\"3P$obz&=H`#*F=7
$$\"38+++Dt:5eF4$\"3lUej)pkIK\"!#77$$\"3+nmm\"fX(emF4$\"3,je1&\\Cv!=FR
7$$\"3a+++DCh/vF4$\"3^TT!G-e*eBFR7$$\"3uMLLL/pu$)F4$\"3?\"4^hQK+*HFR7$
$\"3-omm;c0T\"*F4$\"31dFw\"[4Af$FR7$$\"3@+++I,Q+5F0$\"3/1'epM%f7VFR7$$
\"3.+++]*3q3\"F0$\"34Uu4B'>+2&FR7$$\"3=+++q=\\q6F0$\"3&)3.okGP?eFR7$$
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3LLLL`Q\"GT\"F0$\"3[YVXd[6%*zFR7$$\"39++]s]k,:F0$\"3.+_!p3V8u)FR7$$\"3
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++]<ep[<F0$\"3Mwml[/HZ5!#67$$\"3iLLLe/TM=F0$\"3z>J-)G&))*3\"Fcr7$$\"3Z
LL$eDBJ\">F0$\"3<C\\#[)[[=6Fcr7$$\"3%omm;kD!)*>F0$\"38c%\\$RtHO6Fcr7$$
\"3(omm\"f`@'3#F0$\"3[^b&*fU[Q6Fcr7$$\"3?++]nZ)H;#F0$\"3%e>x$>w<D6Fcr7
$$\"3ymmmJy*eC#F0$\"3=@IZC%HJ4\"Fcr7$$\"36+++S^bJBF0$\"3R<58t&)yQ5Fcr7
$$\"37+++0TN:CF0$\"3?(fagCKti*FR7$$\"33++]7RV'\\#F0$\"3uZ&Q#)y\"zc')FR
7$$\"3')*****\\@fke#F0$\"3G$3\\()R`fG(FR7$$\"3MLLL`4NnEF0$\"3!=mv\"*fV
/x&FR7$$\"3L+++],s`FF0$\"3eR>$32cN$QFR7$$\"3[mm;zM)>$GF0$\"3U!\\P(H(>
\\x\"FR7$$\"3Z+++qfa<HF0$!3=+P3fD?v#)F=7$$\"37LL$eg`!)*HF0$!3wT)HCZS@j
$FR7$$\"3O++]#G2A3$F0$!3\\Eo;,aaapFR7$$\"3BLLL$)G[kJF0$!32,Fs$f`51\"Fc
r7$$\"3C++]7yh]KF0$!3!G!oPy-O*[\"Fcr7$$\"3=nmm')fdLLF0$!3z5'QCSE#[>Fcr
7$$\"3vmmm,FT=MF0$!35mT:no!pY#Fcr7$$\"3%RLLe#pa-NF0$!3lk+@+P.LIFcr7$$
\"3V+++Sv&)zNF0$!3eo&\\nLP2g$Fcr7$$\"3\\LLLGUYoOF0$!3\"*HQb'G?)4VFcr7$
$\"3+nmm1^rZPF0$!3wjb*z?M\"**\\Fcr7$$\"3_++]sI@KQF0$!3#*[&4TJ0Qz&Fcr7$
$\"3F+++S2lsQF0$!3yKg![SFm>'Fcr7$$\"3f++]2%)38RF0$!3psQpyfS9mFcr7$$\"3
H++v.UacRF0$!3z?\\!euc.3(Fcr7$$\"$+%F)$!3*Rv0l;[Uc(Fcr-%'COLOURG6&%$RG
BG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"n6\"Q!F`\\l-%%VIEWG6$;F(Fa[l%(DEFAU
LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Let's take a closer look, we
 can zoom in as much as we want:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"plot(wt2(n)-st2(n),n=280..300);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"$!G\"\"!$\"3]zb%33&G_E!#77$$
\"39LL$3VfV!G!#:$\"3164J3.nNDF-7$$\"3!pm\"H[D:3GF1$\"3y]o\"))pXMV#F-7$
$\"3MLLe0$=C\"GF1$\"3IP2EWtj<BF-7$$\"31LL3RBr;GF1$\"30rWk&*Q:+AF-7$$\"
3zm;zjf)4#GF1$\"3eJ=dj>K#3#F-7$$\"3dL$e4;[\\#GF1$\"3gm<MQfEs>F-7$$\"3*
)**\\i'y]!HGF1$\"3$e2O!GZ[d=F-7$$\"33L$ezs$HLGF1$\"3W#*o))zE*yt\"F-7$$
\"3-+]7iI_PGF1$\"3Q$Hm,m&y<;F-7$$\"3Vmm;_M(=%GF1$\"3Ep^k;>I$\\\"F-7$$
\"3QLL3y_qXGF1$\"3bDgQwa'GQ\"F-7$$\"3#)****\\1!>+&GF1$\"3Os#f`)3ld7F-7
$$\"3v****\\Z/NaGF1$\"3\"4iz3^r48\"F-7$$\"3/++]$fC&eGF1$\"3&[$>T0$))z+
\"F-7$$\"3UL$ez6:B'GF1$\"3n%eKv?m`&*)!#87$$\"3Zmm;=C#o'GF1$\"3dTTD#eC'
3wFdp7$$\"3]mmm#pS1(GF1$\"3n-B)p:*efkFdp7$$\"38+]i`A3vGF1$\"3-O]g0;`8^
Fdp7$$\"3`mmm(y8!zGF1$\"3WA@YnC`8RFdp7$$\"3,+]i.tK$)GF1$\"3c=<r44s(e#F
dp7$$\"3?+](3zMu)GF1$\"3)R+Z:))QiJ\"Fdp7$$\"3gmm\"H_?<*GF1$!31l:UKzI!)
>F17$$\"3[m;zihl&*GF1$!3@[&eSYs^D\"Fdp7$$\"3;LL3#G,***GF1$!3Q/j)eq)z'f
#Fdp7$$\"3\\L$ezw5V!HF1$!3nzB\\:gT+SFdp7$$\"3v**\\PQ#\\\"3HF1$!3](G!Gm
oiI_Fdp7$$\"33LLe\"*[H7HF1$!3u8z4b>0olFdp7$$\"3-+++dxd;HF1$!3)**Qs'[$G
$fzFdp7$$\"3H++D0xw?HF1$!3HF8C+d()H$*Fdp7$$\"3>+]i&p@[#HF1$!3smrl#z([m
5F-7$$\"36++vgHKHHF1$!3ua[Ja([d@\"F-7$$\"3smmmZvOLHF1$!37p!oc8&z]8F-7$
$\"3>++]2goPHF1$!3AY+zR$ff\\\"F-7$$\"3/L$eR<*fTHF1$!3Z@(zmKq$G;F-7$$\"
3<++])Hxe%HF1$!3A(>]pW!3u<F-7$$\"3dm;H!o-*\\HF1$!333f0[447>F-7$$\"3#)*
*\\7k.6aHF1$!3=WmKXhHd?F-7$$\"3vmm;WTAeHF1$!3Iev<En>+AF-7$$\"35+]i!*3`
iHF1$!3TB)oy![z]BF-7$$\"3DLLL*zym'HF1$!3?^OVlI!o\\#F-7$$\"3aLL3N1#4(HF
1$!3_OnvB\"*4ZEF-7$$\"3(om\"HYt7vHF1$!3_3Ex0P8(z#F-7$$\"3-+++xG**yHF1$
!3C8ul@a'e$HF-7$$\"3tmmT6KU$)HF1$!3!omBB***)e4$F-7$$\"3YLLLbdQ()HF1$!3
^7]+pb%*RKF-7$$\"3%)**\\i`1h\"*HF1$!3'[H\"*z_1XR$F-7$$\"3?+]P?Wl&*HF1$
!3>9>q\\!zLa$F-7$$\"$+$F*$!3K3_J-&)Q/PF--%'COLOURG6&%$RGBG$\"#5!\"\"$F
*F*Fa[l-%+AXESLABELSG6$Q\"n6\"Q!Ff[l-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 
13 "" 1 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 270 27 "4. O
ptimal hybrid algorithm" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Recall \+
that st2 and wt2 are the standard and Winograd methods, respectively:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "st2(n);wt2(n);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&)%\"nG\"\"#\"\"\",&*&F&F'F%F'F'F'!\"\"F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"'\"\"\")%\"nG*&-%#lnG6#\"\"(F&
-F+6#\"\"#!\"\"F&F&*&\"\"&F&)F(F0F&F1" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 156 "We want a procedure something like this (you can't just \+
plug in wt2 since different values of t will lead to different solutio
ns of the recurrence relation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "hybrid:=proc(n,t)\nif(n<=t) then re
turn st2(n); fi;\nreturn 7*hybrid(n/2,t) + 15*(n/2)^2;\nend;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%'hybridGf*6$%\"nG%\"tG6\"F)F)C$@$19$9%O-%$
st2G6#F-O,&*&\"\"(\"\"\"-F$6$,$*&#F7\"\"#F7F-F7F7F.F7F7*&#\"#:\"\"%F7*
$)F-F=F7F7F7F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 279 "This leads
 to a family of algorithms parameterized by t.  Different algorithms w
ill have different number of operations and the cross over point when \+
comparing to the standard algorithm will also differ from algorithm to
 algorithm.  The goal is to find the optimal setting for t." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "First we empiri
cally explore different values of t." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "rsolve(\{w(n)=7*w(n/2)+15*(n/2)^2,w(2)=st2(2)\},w(n))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"#K\"\"(\"\"\")%\"nG*&-%#ln
G6#F'F(-F-6#\"\"#!\"\"F(F(*&\"\"&F()F*F1F(F2" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&
$\"+r&G9d%!\"*\"\"\")%\"nG$\"+A\\N2GF'F(F(*&$\"\"&\"\"!F()F*\"\"#F(!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rsolve(\{w(n)=7*w(n/2
)+15*(n/2)^2,w(4)=st2(4)\},w(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
&*&#\"$#>\"#\\\"\"\")%\"nG*&-%#lnG6#\"\"(F(-F-6#\"\"#!\"\"F(F(*&\"\"&F
()F*F2F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+ZtO=R!\"*\"\"\")%\"nG$\"+A\\N2
GF'F(F(*&$\"\"&\"\"!F()F*\"\"#F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "rsolve(\{w(n)=7*w(n/2)+15*(n/2)^2,w(8)=st2(8)\},w(n))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"%!G\"\"$V$\"\"\")%\"nG*&-%
#lnG6#\"\"(F(-F-6#\"\"#!\"\"F(F(*&\"\"&F()F*F2F(F3" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&*&$\"+E%y<t$!\"*\"\"\")%\"nG$\"+A\\N2GF'F(F(*&$\"\"&\"\"!F()F*\"\"#
F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "rsolve(\{w(n)=7*
w(n/2)+15*(n/2)^2,w(16)=st2(16)\},w(n));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&#\"%;#*\"%,C\"\"\")%\"nG*&-%#lnG6#\"\"(F(-F-6#\"\"#!\"\"F(F(
*&\"\"&F()F*F2F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+m1SQQ!\"*\"\"\")%\"nG$
\"+A\\N2GF'F(F(*&$\"\"&\"\"!F()F*\"\"#F(!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "s := seq(st2(2^k),k=1..10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sG6,\"#7\"$7\"\"$g*\"%Oz\"&7X'\"'#>?&\"(?z<%\")'*))
[L\"*7L<o#\"+s]VY@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "w1 :=
 seq(hybrid(2^k,1),k=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w1G
6,\"#A\"$9#\"%Q<\"&EJ\"\"&Ad*\"'9ao\"(Q$f[\")E6EM\"*A4\"3C\"+9'3'*o\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "zip((x,y)->evalf(x/y),[
w1],[s]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,$\"+LLLL=!\"*$\"+'G92\"
>F&$\"+nmT5=F&$\"+b=)Rl\"F&$\"+ify$[\"F&$\"+Tth<8F&$\"+?)*4j6F&$\"+&Gf
I-\"F&$\"+H^nz*)!#5$\"+CopryF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "w2 := seq(hybrid(2^k,2),k=1..10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#w2G6,\"#7\"$W\"\"%[7\"%'p*\"&7<(\"'Wt^\"([Go$\")'pDg
#\"*7H;$=\"+WD2'G\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "zip(
(x,y)->evalf(x/y),[w2],[s]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,$\"
\"\"\"\"!$\"+'G9dG\"!\"*$\"+++++8F)$\"+%>u<A\"F)$\"+Vrg66F)$\"+%)4DX**
!#5$\"+uv-:))F2$\"+?/WrxF2$\"+L,-IoF2$\"+$en;*fF2" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "w4 := seq(hybrid(2^k,4),k=1..10);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#w4G6,\"#7\"$7\"\"%C5\"%G\")\"&O2'\"'70W\"
(C]9$\")G4EA\"*O&4o:\"+7*)f,6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "zip((x,y)->evalf(x/y),[w4],[s]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7,$\"\"\"\"\"!F$$\"+nmmm5!\"*$\"+[N>C5F)$\"+SDo9%*!#5$\"+uyDo%)F
.$\"+;nsFvF.$\"+_kDZmF.$\"+12KZeF.$\"+WcAK^F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "w8 := seq(hybrid(2^k,8),k=1..10);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#w8G6,\"#7\"$7\"\"$g*\"%!o(\"&+w&\"'g&=%\"(g8*
H\")!G&=@\"*++G\\\"\"+g@*)[5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "zip((x,y)->evalf(x/y),[w8],[s]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7,$\"\"\"\"\"!F$F$$\"+b$>un*!#5$\"+H9dG*)F)$\"+V)fi/)F)$\"+rk#*f
rF)$\"+K51EjF)$\"+T%\\lc&F)$\"+/3n')[F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "w16 := seq(hybrid(2^k,16),k=1..10);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$w16G6,\"#7\"$7\"\"$g*\"%Oz\"&#Rf\"'/6V\"(o\"zI\")
O**z@\"*#f#e`\"\"+/.,z5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
zip((x,y)->evalf(x/y),[w16],[s]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7
,$\"\"\"\"\"!F$F$F$$\"+1#\\j?*!#5$\"+v:S(G)F)$\"+R!)4qtF)$\"+s7g4lF)$
\"+k,*ps&F)$\"+cy)p-&F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 292 "This suggests the optimal value of t is \+
between 8 and 16.  If restricted to a power of two it appears to be 8.
  Altneratively we can find the optimal value of t by determining the \+
smallest value of n such that one application of Winograd requires few
er operations than the standard algorithm." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "To find the threshold v
alue t, we want to know when to switch from Winograd to standard:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solve(7
 * st2(n/2) + 15 * (n/2)^2 > st2(n),n);" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 208 "Since we don't care about what happens for matrices
 with negative dimension (whatever they might be), this tells us that \+
it's worth using the standard algorithm up to 12x12, then we should sw
itch to Winograd." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$,$
%)infinityG!\"\"-%%OpenG6#\"\"!-F$6$F)-F*6#\"#7" }}}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We could also find
 a formula for this, by the way (courtesy of art25):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 48 "isolate(7 * st2(n/2) + 15 * (n/2)^2 > st2(n),n);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2,&*&#\"\"\"\"\"%F'*$)%\"nG\"\"$F'
F'F'*&F,F')F+\"\"#F'!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 65 "The en
d result, though, is that we want t=12 in procedure hybrid." }}}{SECT 
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;F(F\\ezFaagl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
32 "Let's compare these with ratios:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "plot([seq([n,pad(n)/even_up(n)],n=1..1000)]);" }}{PARA 0 "" 0 "
" {TEXT -1 59 "The ratio stays positive, so pad takes longer than even
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