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T" }{TEXT 205 18 "\303\266rt\303\251" }{TEXT 205 3 "ne t" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 18 "2. Algebrai alapok" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 7 "3. Norm" }{TEXT 205 8 "\303\241" }{TEXT 205 6 "l form" }{TEXT 205 8 "\303\241" }{TEXT 205 12 "k, reprez ent" }{TEXT 205 18 "\303\241ci\303\263" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 13 "4. Aritmetika" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 4 "5 . K" }{TEXT 205 8 "\303\255" }{TEXT 205 9 "nai marad" }{TEXT 205 8 "\3 03\251" }{TEXT 205 3 "kol" }{TEXT 205 8 "\303\241" }{TEXT 205 1 "s" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT 205 14 "6. Newton-iter" }{TEXT 205 8 " \303\241" }{TEXT 205 2 "ci" }{TEXT 205 8 "\303\263" }{TEXT 205 16 ", H ensel-felemel" }{TEXT 205 8 "\303\251" }{TEXT 205 1 "s" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 15 "7. Legnagyobb k" }{TEXT 205 8 "\303\266" }{TEXT 205 1 "z" }{TEXT 205 8 "\303\266" }{TEXT 205 6 "s oszt" }{TEXT 205 8 "\303\263" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 11 "8. Faktoriz " }{TEXT 205 8 "\303\241" }{TEXT 205 1 "l" }{TEXT 205 8 "\303\241" } {TEXT 205 1 "s" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 21 "9. Egyenletre ndszerek" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 6 "10. Gr" }{TEXT 205 27 "\303\266bner-b\303\241zisok" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 10 "11. Racion" }{TEXT 205 8 "\303\241" }{TEXT 205 5 "lis t" }{TEXT 205 8 "\303\266" }{TEXT 205 3 "rtf" }{TEXT 205 8 "\303\274" }{TEXT 205 3 "ggv" }{TEXT 205 8 "\303\251" }{TEXT 205 11 "nyek integr" }{TEXT 205 8 "\303\241" }{TEXT 205 1 "l" }{TEXT 205 8 "\303\241" }{TEXT 205 2 "sa" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11 .1. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(1/(x+1),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*$),&I\"xG6\"\"\"\"F(F(\"\"#!\"\"F*" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.2. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(1/(x^3+x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",& -I#lnG6$%*protectedGI(_syslibG6\"6#I\"xGF(\"\"\"*&#F+\"\"#F+-F$6#,&*$) F*F.F+F+F+F+F+!\"\"" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.3. \+ P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int(1/(x^2-2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"\"\"\"#F%)F&F$F%-I(arcta nhG6$%*protectedGI(_syslibG6\"6#,$*(F$F%I\"xGF-F%F'F%F%F%!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "diff(2^(1/2)/4*ln(x-2^(1/2)) -2^(1/2)/4*ln(x+2^(1/2)),x); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"%F%)\"\"##F%F(F%,&I\"xG6\"F%*$F'F%!\"\"F.F %*(F$F%F'F%,&F+F%F-F%F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "*$,&*$)I\"x G6\"\"\"#\"\"\"F)F(!\"\"F*" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 20 " A 11.1. Algoritmus. " }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "SquareFree:=proc(a,x) lo cal i,out,b,c,y,z,w;\n" }{MPLTEXT 1 0 30 "i:=1; out:=[]; b:=diff(a,x); \n" }{MPLTEXT 1 0 29 "c:=gcd(a,b); w:=quo(a,c,x);\n" }{MPLTEXT 1 0 15 "while c<>1 do\n" }{MPLTEXT 1 0 16 " y:=gcd(w,c);\n" }{MPLTEXT 1 0 18 " z:=quo(w,y,x);\n" }{MPLTEXT 1 0 21 " out:=[op(out),z];\n" } {MPLTEXT 1 0 11 " i:=i+1;\n" }{MPLTEXT 1 0 24 " w:=y; c:=quo(c,y,x); \n" }{MPLTEXT 1 0 26 "od; out:=[op(out),w]; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"aG6\"I\"xGF%6)I\"iGF%I$outGF%I\"bGF%I\"cGF%I\"yGF% I\"zGF%I\"wGF%F%F%C)>F(\"\"\">F)7\">F*-I%diffG%*protectedGF#>F+-I$gcdG 6$F7I(_syslibGF%6$F$F*>F.-I$quoGF;6%F$F+F&?(F%F1F1F%0F+F1C(>F,-F:6$F.F +>F--F@6%F.F,F&>F)7$-I#opGF76#F)F->F(,&F(F1F1F1>F.F,>F+-F@6%F+F,F&>F)7 $FMF.F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "SquareFree(( -12*x^3+9*x+3)/(-12),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"xG6\" \"\"\"F&!\"\",&#F&\"\"#F&F$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sqrfree(-12*x^3+9*x+3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\"$7 $7$,&I\"xG6\"\"\"\"F)!\"\"F)7$,&F)F)*&\"\"#F)F'F)F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "PolynomialDiophant:=proc(a,b,r,x) l ocal y,z,q;\n" }{MPLTEXT 1 0 40 "gcdex(a,b,x,'y','z'); q:=quo(y*r,b,x) ;\n" }{MPLTEXT 1 0 38 "[expand(y*r-q*b),expand(z*r+q*a)] end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"aG6\"I\"bGF%I\"rGF%I\"xGF%6%I\"yG F%I\"zGF%I\"qGF%F%F%C%-I&gcdexG6$%*protectedGI(_syslibGF%6'F$F&F(.F*.F +>F,-I$quoGF06%*&F*\"\"\"F'F;F&F(7$-I'expandGF16#,&F:F;*&F,F;F&F;!\"\" -F>6#,&*&F+F;F'F;F;*&F,F;F$F;F;F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "PartialFractions1:=proc(r,L,x) local a,b,l,i,c;\n" } {MPLTEXT 1 0 48 "l:=nops(L); if l<2 then return([[r,L[1]]]) fi;\n" } {MPLTEXT 1 0 50 "a:=1; for i to l-1 do a:=a*L[i]^i od; b:=L[l]^l;\n" } {MPLTEXT 1 0 33 "c:=PolynomialDiophant(a,b,r,x);\n" }{MPLTEXT 1 0 56 " [op(PartialFractions1(c[2],L[1..l-1],x)),[c[1],L[l]]];\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"rG6\"I\"LGF%I\"xGF%6 'I\"aGF%I\"bGF%I\"lGF%I\"iGF%I\"cGF%F%F%C)>F+-I%nopsG%*protectedG6#F&@ $2F+\"\"#O7#7$F$&F&6#\"\"\">F)FF)*&F)F <)&F&6#F,F,F<>F*)&F&6#F+F+>F--I3PolynomialDiophantGF%6&F)F*F$F'7$-I#op GF26#-I2PartialFractions1GF%6%&F-6#F6&F&6#;F " 0 "" {MPLTEXT 1 0 53 "PartialFractions2:=proc(r,q, e,x) local a,b,l,i,u,v;\n" }{MPLTEXT 1 0 29 "if e<2 then return([r]) f i;\n" }{MPLTEXT 1 0 18 "u:=quo(r,q,x,v);\n" }{MPLTEXT 1 0 39 "[op(Part ialFractions2(u,q,e-1,x)),v];\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"rG6\"I\"qGF%I\"eGF%I\"xGF%6(I\"aGF%I\"bGF%I\" lGF%I\"iGF%I\"uGF%I\"vGF%F%F%C%@$2F'\"\"#O7#F$>F.-I$quoG6$%*protectedG I(_syslibGF%6&F$F&F(F/7$-I#opGF:6#-I2PartialFractions2GF%6&F.F&,&F'\" \"\"FE!\"\"F(F/F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Pa rtialFractions:=proc(r,L,x) local i,LL,LLL;\n" }{MPLTEXT 1 0 40 "LL:=P artialFractions1(r,L,x); LLL:=[];\n" }{MPLTEXT 1 0 22 "for i to nops(L L) do\n" }{MPLTEXT 1 0 71 " LLL:=[op(LLL),[LL[i][2],PartialFractions2 (LL[i][1],LL[i][2],i,x)]];\n" }{MPLTEXT 1 0 8 "od; end;" }}{PARA 205 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"rG6 \"I\"LGF%I\"xGF%6%I\"iGF%I#LLGF%I$LLLGF%F%F%C%>F*-I2PartialFractions1G F%F#>F+7\"?(F)\"\"\"F3-I%nopsG%*protectedG6#F*I%trueGF6>F+7$-I#opGF66# F+7$&&F*6#F)6#\"\"#-I2PartialFractions2GF%6&&F@6#F3F?F)F'F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "HermiteReduction:=proc(p,q,x ) local pp,rp,r,qq,ip,i,qi,ri,n,c;\n" }{MPLTEXT 1 0 32 "pp:=quo(p,q,x) ; r:=rem(p,q,x);\n" }{MPLTEXT 1 0 22 "qq:=SquareFree(q,x);\n" } {MPLTEXT 1 0 30 "r:=PartialFractions(r,qq,x);\n" }{MPLTEXT 1 0 15 "rp: =0; ip:=0;\n" }{MPLTEXT 1 0 21 "for i to nops(r) do\n" }{MPLTEXT 1 0 35 " qi:=r[i][1]; ri:=r[i][2]; n:=i;\n" }{MPLTEXT 1 0 16 " while n>1 do\n" }{MPLTEXT 1 0 22 " if ri[n]<>0 then\n" }{MPLTEXT 1 0 53 " \+ c:=PolynomialDiophant(qi,diff(qi,x),ri[n],x);\n" }{MPLTEXT 1 0 35 " rp:=rp-c[2]/(n-1)/qi^(n-1);\n" }{MPLTEXT 1 0 49 " ri[n-1]:= ri[n-1]+c[1]+diff(c[2],x)/(n-1);\n" }{MPLTEXT 1 0 17 " fi; n:=n-1; \n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 20 " ip:=ip+ri[1]/qi;\n" }{MPLTEXT 1 0 32 "od; rp+int(pp,x)+Int(ip,x); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"pG6\"I\"qGF%I\"xGF%6,I#ppGF%I#rpGF%I\"rGF%I#qqGF% I#ipGF%I\"iGF%I#qiGF%I#riGF%I\"nGF%I\"cGF%F%F%C*>F)-I$quoG6$%*protecte dGI(_syslibGF%F#>F+-I$remGF7F#>F,-I+SquareFreeGF%6$F&F'>F+-I1PartialFr actionsGF%6%F+F,F'>F*\"\"!>F-FF?(F.\"\"\"FI-I%nopsGF86#F+I%trueGF8C'>F /&&F+6#F.6#FI>F0&FQ6#\"\"#>F1F.?(F%FIFIF%2FIF1C$@$0&F06#F1FFC%>F2-I3Po lynomialDiophantGF%6&F/-I%diffGF86$F/F'FhnF'>F*,&F*FI*(&F2FVFI,&F1FIFI !\"\"Fgo)F/FfoFgoFgo>&F06#Ffo,(FjoFI&F2FSFI*&-F`o6$FeoF'FIFfoFgoFI>F1F fo>F-,&F-FI*&&F0FSFIF/FgoFI,(F*FI-I$intGF76$F)F'FI-I$IntGF76$F-F'FIF%F %F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.4. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "fp:=441*x^7+780*x^6-2861*x^5 +4085*x^4+7695*x^3+3713*x^2-43253*x+24500;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&\"$T%\"\"\")I\"xG6\"\"\"(F%F%*&\"$!yF%)F'\"\"'F%F%*& \"%hGF%)F'\"\"&F%!\"\"*&\"%&3%F%)F'\"\"%F%F%*&\"%&p(F%)F'\"\"$F%F%*&\" %8PF%)F'\"\"#F%F%*&\"&`K%F%F'F%F2\"&+X#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fq:=9*x^6+6*x^5-65*x^4+20*x^3+135*x^2-154*x+49;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",0*&\"\"*\"\"\")I\"xG6\"\"\"'F%F%*&F)F%) F'\"\"&F%F%*&\"#lF%)F'\"\"%F%!\"\"*&\"#?F%)F'\"\"$F%F%*&\"$N\"F%)F'\" \"#F%F%*&\"$a\"F%F'F%F1\"#\\F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fp:=fp/9; fq:=fq/9; fr:=rem(fp,fq,x); f:=fp/fq;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&\"#\\\"\"\")I\"xG6\"\"\"(F%F%*&#\"$g#\"\"$F%)F' \"\"'F%F%*&#\"%hG\"\"*F%)F'\"\"&F%!\"\"*&#\"%&3%F3F%)F'\"\"%F%F%*&\"$b )F%)F'F-F%F%*&#\"%8PF3F%)F'\"\"#F%F%*&#\"&`K%F3F%F'F%F6#\"&+X#F3F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",0*$)I\"xG6\"\"\"'\"\"\"F(*&#\"\"#\"\"$F ()F%\"\"&F(F(*&#\"#l\"\"*F()F%\"\"%F(!\"\"*&#\"#?F2F()F%F,F(F(*&\"#:F( )F%F+F(F(*&#\"$a\"F2F(F%F(F5#\"#\\F2F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*#\"&a=#\"\"*\"\"\"*&\"$N(F&)I\"xG6\"\"\"%F&F&*&\"$T%F&)F*\"\"#F&F& *&#\"&YC\"\"\"$F&F*F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,2*&\"# \\\"\"\")I\"xG6\"\"\"(F&F&*&#\"$g#\"\"$F&)F(\"\"'F&F&*&#\"%hG\"\"*F&)F (\"\"&F&!\"\"*&#\"%&3%F4F&)F(\"\"%F&F&*&\"$b)F&)F(F.F&F&*&#\"%8PF4F&)F (\"\"#F&F&*&#\"&`K%F4F&F(F&F7#\"&+X#F4F&F&,0*$F/F&F&*&#FDF.F&F5F&F&*&# \"#lF4F&F;F&F7*&#\"#?F4F&F?F&F&*&\"#:F&FCF&F&*&#\"$a\"F4F&F(F&F7#F%F4F &F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "qq:=SquareFree(fq,x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\",&I\"xG6\"F##\"\"(\"\"$F#F# ,&F%F#F#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "PartialFra ctions1(fr,qq,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&7$\"\"!\"\"\"7$\" $%H,&I\"xG6\"F%#\"\"(\"\"$F%F#7$,(\"$#RF%*&\"$T%F%)F)\"\"#F%F%*&\"$#)) F%F)F%!\"\",&F)F%F%F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Pa rtialFractions2(392+441*x^2-882*x,x-1,4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"!\"$T%F#!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "PartialFractions(fr,qq,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7& 7$\"\"\"7#\"\"!7$,&I\"xG6\"F$#\"\"(\"\"$F$7$F&\"$%H7$F$7%F&F&F&7$,&F)F $F$!\"\"7&F&\"$T%F&!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " convert(f,parfrac,x,sqrfree);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&\"# \\\"\"\"I\"xG6\"F%F%\"#aF%*&\"$T%F%),&F&F%F%!\"\"\"\"#F-F%*&F$F%)F,\" \"%F-F-*&\"%YEF%),&*&\"\"$F%F&F%F%\"\"(F%F.F-F%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "PolynomialDiophant(x+7/3,1,294,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!\"$%H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(294/(x+7/3)^2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"$%H\"\"\",&I\"xG6\"F%#\"\"(\"\"$F%!\"\"F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "int(441/(x-1)^2-49/(x-1)^4,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 ",&*&\"$T%\"\"\",&I\"xG6\"F%F%!\"\"F)F)*&#\"#\\\"\" $F%)F&F-F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "HermiteRedu ction(fp,fq,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&\"$%H\"\"\",&I\"x G6\"F%#\"\"(\"\"$F%!\"\"F,*&#\"#\\F+F%),&F'F%F%F,F+F,F%*&\"$T%F%F1F,F, *&#F/\"\"#F%)F'F6F%F%*&\"#aF%F'F%F%-I$IntG6$%*protectedGI(_syslibGF(6$ \"\"!F'F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.5. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "gp:=36*x^6+126*x^5+183*x ^4+13807/6*x^3-407*x^2-3242/5*x+3044/15;;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&\"#O\"\"\")I\"xG6\"\"\"'F%F%*&\"$E\"F%)F'\"\"&F%F%*& \"$$=F%)F'\"\"%F%F%*&#\"&2Q\"F)F%)F'\"\"$F%F%*&\"$2%F%)F'\"\"#F%!\"\"* &#\"%UKF-F%F'F%F;#\"%WI\"#:F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gq:=(x^2+7/6*x+1/3)^2*(x-2/5)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&),(*$)I\"xG6\"\"\"#\"\"\"F**&#\"\"(\"\"'F*F'F*F*#F*\"\"$F*F)F*),&F 'F*#F)\"\"&!\"\"F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g:=g p/gq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,0*&\"#O\"\"\")I\"xG6\"\"\"'F &F&*&\"$E\"F&)F(\"\"&F&F&*&\"$$=F&)F(\"\"%F&F&*&#\"&2Q\"F*F&)F(\"\"$F& F&*&\"$2%F&)F(\"\"#F&!\"\"*&#\"%UKF.F&F(F&F<#\"%WI\"#:F&F&),(*$F:F&F&* &#\"\"(F*F&F(F&F&#F&F7F&F;F<),&F(F&#F;F.F " 0 "" {MPLTEXT 1 0 29 "convert(g,parfrac,x,sqrfree);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&\"%q<\"\"\"),&*&\"\"&F%I\"xG6\"F%F%\"\"#!\"\"F,F-F %*&\"%?VF%)F'\"\"$F-F%*&#\"'bs=\"#;F%F'F-F%*(#F%F5F%,&\"'DmMF-*&\"']7A F%F*F%F-F%,(*&\"\"'F%)F*F,F%F%*&\"\"(F%F*F%F%F,F%F-F%*&,&\"&Dq%F-*&\"& ]'zF%F*F%F-F%)F " 0 "" {MPLTEXT 1 0 74 "HorowitzReduction:=proc(p,q, x) local pop,pp,d,b,m,n,a,aa,c,cc,r,i,j,e,s;\n" }{MPLTEXT 1 0 34 "pop: =quo(p,q,x); pp:=rem(p,q,x);\n" }{MPLTEXT 1 0 37 "d:=gcd(q,diff(q,x)); b:=quo(q,d,x);\n" }{MPLTEXT 1 0 29 "m:=degree(b); n:=degree(d);\n" } {MPLTEXT 1 0 29 "aa:=sum(a[i]*x^i,i=0..m-1);\n" }{MPLTEXT 1 0 29 "cc:= sum(c[i]*x^i,i=0..n-1);\n" }{MPLTEXT 1 0 55 "r:=expand(b*diff(cc,x)-cc *quo(b*diff(d,x),d,x)+d*aa);\n" }{MPLTEXT 1 0 64 "for i from 0 to m+n- 1 do e[i]:=coeff(pp,x,i)=coeff(r,x,i); od;\n" }{MPLTEXT 1 0 60 "s:=sol ve([e[j]$j=0..m+n-1],[a[j]$j=0..m-1,c[j]$j=0..n-1]);\n" }{MPLTEXT 1 0 49 "aa:=sum(a[j]*x^j,j=0..m-1); aa:=subs(op(s),aa);\n" }{MPLTEXT 1 0 49 "cc:=sum(c[j]*x^j,j=0..n-1); cc:=subs(op(s),cc);\n" }{MPLTEXT 1 0 30 "cc/d+Int(pop,x)+Int(aa/b,x);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"pG6\"I\"qGF%I\"xGF%61I$popGF%I#ppGF%I\"dGF %I\"bGF%I\"mGF%I\"nGF%I\"aGF%I#aaGF%I\"cGF%I#ccGF%I\"rGF%I\"iGF%I\"jGF %I\"eGF%I\"sGF%F%F%C2>F)-I$quoG6$%*protectedGI(_syslibGF%F#>F*-I$remGF F+-I$gcdGF<6$F&-I%diffGF=6$F&F'>F,-F;6%F&F+F'>F--I'degreeGF=6#F,>F .-FN6#F+>F0-I$sumGF<6$*&&F/6#F4\"\"\")F'F4FZ/F4;\"\"!,&F-FZFZ!\"\">F2- FU6$*&&F1FYFZFenFZ/F4;Fhn,&F.FZFZFjn>F3-I'expandGF=6#,(*&F,FZ-FG6$F2F' FZFZ*&F2FZ-F;6%*&F,FZ-FG6$F+F'FZF+F'FZFjn*&F+FZF0FZFZ?(F4FhnFZ,(F-FZF. FZFZFjnI%trueGF=>&F6FY/-I&coeffGF=6%F*F'F4-Fip6%F3F'F4>F7-I&solveGF<6$ 7#-I\"$GF=6$&F66#F5/F5;FhnFcp7$-Fcq6$&F/Ffq/F5Fgn-Fcq6$&F1Ffq/F5Fao>F0 -FU6$*&F\\rFZ)F'F5FZF]r>F0-I%subsGF=6$-I#opGF=6#F7F0>F2-FU6$*&F`rFZFfr FZFar>F2-Fir6$F[sF2,(*&F2FZF+FjnFZ-I$IntGF<6$F)F'FZ-Fhs6$*&F0FZF,FjnF' FZF%F%F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.6. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "debug(HorowitzReduction) ; HorowitzReduction(fp,fq,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I2Horow itzReductionG6\"" }}{PARA 9 "" 1 "" {TEXT 207 167 "\{--> enter Horowit zReduction, args = 49*x^7+260/3*x^6-2861/9*x^5+4085/9*x^4+855*x^3+3713 /9*x^2-43253/9*x+24500/9, x^6+2/3*x^5-65/9*x^4+20/9*x^3+15*x^2-154/9*x +49/9, x" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#\\\"\"\"I\"xG6\"F%F% \"#aF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*#\"&a=#\"\"*\"\"\"*&\"$N(F&) I\"xG6\"\"\"%F&F&*&\"$T%F&)F*\"\"#F&F&*&#\"&YC\"\"\"$F&F*F&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",,#\"\"(\"\"$!\"\"*&\"\"'\"\"\"I\"xG6\"F )F)*&\"\"%F))F*\"\"#F)F&*&#F/F%F))F*F%F)F&*$)F*F-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(*&#\"\"%\"\"$F(F%F(F(#\"\"( F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&&I\"aG6\"6#\"\"! \"\"\"*&&F$6#F(F(I\"xGF%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*&I\"cG 6\"6#\"\"!\"\"\"*&&F$6#F(F(I\"xGF%F(F(*&&F$6#\"\"#F()F,F0F(F(*&&F$6#\" \"$F()F,F5F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",L*(#\"#5\"\"$\"\"\"&I \"cG6\"6#\"\"#F')I\"xGF*F,F'!\"\"*&#\"\"(F&F'&I\"aGF*6#\"\"!F'F/*(F1F' &F46#F'F'F.F'F/*(#\"#9F&F'&F)F9F'F.F'F/*(F,F'&F)6#F&F')F.F&F'F/*(F;F'F (F'F.F'F/*(F2F'F?F'F-F'F/*(F&F'F-F'F=F'F/*(F,F'FAF'F(F'F/*&)F.\"\"%F'F ?F'F/*(FHF'&F)F5F'F.F'F/*(\"\"'F'F.F'F3F'F'*(FLF'F8F'F-F'F'*(FHF'F-F'F 3F'F/*(FHF'FAF'F8F'F/*(#F,F&F'FAF'F3F'F/*(FQF'FGF'F8F'F/*&FGF'F3F'F'*& )F.\"\"&F'F8F'F'*&FLF'FJF'F/*&F1F'F=F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "/#\"&a=#\"\"*,(*&#\"\"(\"\"$\"\"\"&I\"aG6\"6#\"\"!F+!\"\"*&\"\"'F+ &I\"cGF.F/F+F1*&F(F+&F56#F+F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "/#!&Y C\"\"\"$,,*&#\"\"(F%\"\"\"&I\"aG6\"6#F*F*!\"\"*&#\"#9F%F*&I\"cGF-F.F*F /*&F1F*&F46#\"\"#F*F/*&\"\"%F*&F46#\"\"!F*F/*&\"\"'F*&F,F " 0 "" {MPLTEXT 1 0 53 "debug(HorowitzRe duction); HorowitzReduction(gp,gq,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I2HorowitzReductionG6\"" }}{PARA 9 "" 1 "" {TEXT 207 129 "\{--> enter HorowitzReduction, args = 36*x^6+126*x^5+183*x^4+13807/6*x^3-407*x^2- 3242/5*x+3044/15, (x^2+7/6*x+1/3)^2*(x-2/5)^3, x" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&\"#O\"\"\")I\" xG6\"\"\"'F%F%*&\"$E\"F%)F'\"\"&F%F%*&\"$$=F%)F'\"\"%F%F%*&#\"&2Q\"F)F %)F'\"\"$F%F%*&\"$2%F%)F'\"\"#F%!\"\"*&#\"%UKF-F%F'F%F;#\"%WI\"#:F%" } }{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*$)I\"xG6\"\"\"#\"\"\"F)*&#\"\"(\" \"'F)F&F)F)#F)\"\"$F)F)),&F&F)#F(\"\"&!\"\"F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\"xG6\"\"\"$\"\"\"F(*&#\"#B\"#IF()F%\"\"#F(F(*&#F. \"#:F(F%F(!\"\"F0F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(&I\"aG6 \"6#\"\"!\"\"\"*&&F$6#F(F(I\"xGF%F(F(*&&F$6#\"\"#F()F,F0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*&I\"cG6\"6#\"\"!\"\"\"*&&F$6#F(F(I\"xGF%F(F( *&&F$6#\"\"#F()F,F0F(F(*&&F$6#\"\"$F()F,F5F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",hn*(\"\"$\"\"\")I\"xG6\"F$F%&I\"cGF(6#F%F%!\"\"*(\"\"#F% )F'\"\"%F%&F*6#F.F%F,*&)F'\"\"&F%&F*6#F$F%F,*(#\"#H\"#:F%)F'F.F%F)F%F, *(#\"\"(\"\"'F%F&F%F1F%F,*(#F.F5F%F/F%F6F%F,*(F0F%&F*6#\"\"!F%F " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.8. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a:=1; b:=x^3+x; resultant(a-z*diff(b,x),b,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,&*$)I\"xG6\"\"\"$\"\"\"F(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&\" \"\"F$I\"zG6\"!\"\"F$),&*&\"\"#F$F%F$F$F$F$F+F$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "gcd(a-1*diff(b,x),b); gcd(a+1/2*diff(b,x),b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F(F(" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 11.9. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a:=1; b:=x^2-2; resultant(a-z*diff(b,x),b,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,&*$)I\"xG6\"\"\"#\"\"\"F(F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&* &\"\")\"\"\")I\"zG6\"\"\"#F%!\"\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "alpha:=2^(1/2)/4; gcd(a+alpha*diff(b,x),b); gcd(a-alp ha*diff(b,x),b);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"%F%) \"\"##F%F(F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)\"\"##\"\"\"F%F'F 'I\"xG6\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)\"\"##\"\"\"F%F'!\" \"I\"xG6\"F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 10 "E 11.10. P" } {TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "a:=36*x^2+1167*x+3 549/2; b:=x^3+23/30*x^2-2/15*x-2/15;\n" }{MPLTEXT 1 0 29 "resultant(a- z*diff(b,x),b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"#O\"\"\")I\"x G6\"\"\"#F%F%*&\"%n6F%F'F%F%#\"%\\NF)F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\"xG6\"\"\"$\"\"\"F(*&#\"#B\"#IF()F%\"\"#F(F(*&#F.\"#:F(F%F( !\"\"F0F2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&#\"#;\"$D'\"\"\")I\"zG6 \"\"\"$F'F'*&#\"$w&F&F')F)\"\"#F'!\"\"*&#\")4?(3#F%F'F)F'F1\"++z " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 ",$**#\"\"\"\"&++\"F%,&*&\"#;F%I\"zG6\"F%F%\"&^u$! \"\"F%,&F*F%\"%+!)F%F%,&F(F%\"&D6*F-F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "gcd(a+8000*diff(b,x),b); gcd(a-91125/16*diff(b,x),b); \n" }{MPLTEXT 1 0 28 "gcd(a-37451/16*diff(b,x),b);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"\"\"\"#F$I\"xG6\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"#\"\"$\"\"\"I\"xG6\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&# \"\"#\"\"&!\"\"I\"xG6\"\"\"\"" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 10 "E 11.11. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a :=7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3;\n" }{MPLTEXT 1 0 44 "b: =x^14-2*x^8-2*x^7-2*x^4-4*x^3-x^2+2*x+1;\n" }{MPLTEXT 1 0 29 "resultan t(a-z*diff(b,x),b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&\"\"(\"\"\" )I\"xG6\"\"#8F%F%*&\"#5F%)F'\"\")F%F%*&\"\"%F%)F'F$F%F%*&F$F%)F'\"\"'F %!\"\"*&F/F%)F'\"\"$F%F4*&F/F%)F'\"\"#F%F4*&F7F%F'F%F%F7F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*$)I\"xG6\"\"#9\"\"\"F(*&\"\"#F()F%\"\")F(!\" \"*&F*F()F%\"\"(F(F-*&F*F()F%\"\"%F(F-*&F3F()F%\"\"$F(F-*$)F%F*F(F-*&F *F(F%F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",@\"0Gx8-u5X\"\"\"\"*& \"1%Qc)fs+jSF$I\"zG6\"F$F$*&\"2C-Ue)zIpWF$)F'\"\"#F$F$*&\"3/v&>b1%GvAF $)F'\"\"$F$F$*&\"3Gty1zm'zt$F$)F'\"\"%F$F$*&\"4[k$4ouz806F$)F'\"\"&F$! \"\"*&\"4%yI)>c3&\\lRF$)F'\"\"'F$F9*&\"4c)px9N&pvT$F$)F'\"\"(F$F$*&\"5 OJKzCM!)>'e\"F$)F'\"\")F$F$*&\"5oJ)\\*[f2Ao " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"0Gx8-u5X\"\"\"\"),(*&\"\"%F%)I\"zG6\"\"\"#F%F%* &F)F%F+F%!\"\"F%F/\"\"(F%F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "alpha:=(1+2^(1/2))/2;\n" }{MPLTEXT 1 0 55 "gcd(a-alpha*diff(b,x),b ); gcd(a-(1-alpha)*diff(b,x),b);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&# \"\"\"\"\"#F$*&F#F$)F%F#F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I\" xG6\"\"\"(\"\"\"F(*&)\"\"##F(F+F()F%F+F(!\"\"F%F.*&F*F(F%F(F.F(F." }} {PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I\"xG6\"\"\"(\"\"\"F(*&)\"\"##F(F+F ()F%F+F(F(F%!\"\"*&F*F(F%F(F(F(F." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 20 "A 11.4. Algoritmus. " }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 10 "E 11.12. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a:=6*x^5+6*x^4-8*x^3-18*x^2+8*x+8;\n" } {MPLTEXT 1 0 33 "b:=x^6-5*x^4-8*x^3-2*x^2+2*x+1;\n" }{MPLTEXT 1 0 29 " resultant(a-z*diff(b,x),b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&\" \"'\"\"\")I\"xG6\"\"\"&F%F%*&F$F%)F'\"\"%F%F%*&\"\")F%)F'\"\"$F%!\"\"* &\"#=F%)F'\"\"#F%F1*&F.F%F'F%F%F.F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ", .*$)I\"xG6\"\"\"'\"\"\"F(*&\"\"&F()F%\"\"%F(!\"\"*&\"\")F()F%\"\"$F(F- *&\"\"#F()F%F3F(F-*&F3F(F%F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0 *&\"([KX\"\"\"\")I\"zG6\"\"\"'F%!\"\"*&\"()[>()F%)F'\"\"&F%F%*&F,F%)F' \"\"%F%F**&\")o>DBF%)F'\"\"$F%F**&\")w*Qu\"F%)F'\"\"#F%F%*&\")_z([$F%F 'F%F%\")%)fi6F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"([KX\"\"\"\"),(*$)I\"zG6\"\" \"#F%F%*&F,F%F*F%!\"\"F,F.\"\"$F%F." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 10 "E 11.13. P" }{TEXT 206 12 "\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a:=2*x^5-19*x^4+60*x^3-159*x^2+50*x+11;\n" }{MPLTEXT 1 0 44 "b:=x^ 6-13*x^5+58*x^4-85*x^3-66*x^2-17*x+1;\n" }{MPLTEXT 1 0 29 "resultant(a -z*diff(b,x),b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&\"\"#\"\"\")I \"xG6\"\"\"&F%F%*&\"#>F%)F'\"\"%F%!\"\"*&\"#gF%)F'\"\"$F%F%*&\"$f\"F%) F'F$F%F.*&\"#]F%F'F%F%\"#6F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*$)I\" xG6\"\"\"'\"\"\"F(*&\"#8F()F%\"\"&F(!\"\"*&\"#eF()F%\"\"%F(F(*&\"#&)F( )F%\"\"$F(F-*&\"#mF()F%\"\"#F(F-*&\"#\"\"\")I\"zG6\"\"\"'F%!\"\"*&\"0+gX\"H:-Q F%)F'\"\"&F%F%*&\"0+S=PHKq&F%)F'\"\"%F%F**&F$F%)F'\"\"#F%F%*&F,F%F'F%F *F$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"0+!Gdk2,>\"\"\"),**$)I\"zG6\"\"\"$ F%F%*$)F*\"\"#F%!\"\"F*F%F%F%F/F%F0" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 23 "12. A Risch-algoritmus." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }