{VERSION 5 0 "HP RISC UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b1 := [1,2,3]; b2 := [3,2,1] ; b3 := [2,3,1];" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 37 "Orthogonali sierung nach Gram-Schmidt:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c1 := b1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dotprod(b2 - lambda*c1, c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "mu[2,1] := solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c2 := b2 - mu[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "c3 := b3 - lambda*c1 - nu*c2;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "dotprod(c3, c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "mu[3, 1] := solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(c3, c2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "mu[3,2] := solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c3 := b3 - mu[3,1]*c1 - mu[3,2]*c2; " }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 6 "Probe:" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dotprod(c1, c2); dotpro d(c1, c3); dotprod(c2, c3);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 32 " Start der LLL-Reduktion mit k=1:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mu[2,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "b2 := b2 - b1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mu[2,1] := mu[2,1] - 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c2 = b2 - mu[2,1]*c1;" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 25 "In welchem Fall sind wir?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c2 + mu[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "dotprod(%, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3/4*dotprod(c1, c1);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 67 "Wir sind also in Fall 1 und m\374ssen die Vektoren b1, b2 vertauschen." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "b0 := b1: b1 := b2 ; b2 := b0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d1 := c2 + m u[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "nu[2,1] := mu [2,1]*dotprod(c1,c1)/(dotprod(c2,c2)+mu[2,1]^2*dotprod(c1,c1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d2 := c1 - nu[2,1]*d1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "nu[3,1] := mu[3,1]*nu[2,1] + mu[3,2]*dotprod(c2, c2)/dotprod(d1, d1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "nu[3,2] := mu[3,1] - mu[3,2]*mu[2,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "c1 := d1; c2 := d2;\nmu[2,1] := nu[ 2,1]; mu[3,1] := nu[3,1]; mu[3,2] := nu[3,2];" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c1 = b1; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c2 = b2 - mu[2,1]*c1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c3 = b3 - nu[3,1]*c1 - nu[3,2]*c2;" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 196 "Wir setzen nun k=0 und sind dam it automatisch im zweiten Fall.\nEs gibt nichts zu tun, also setzen wi r k=1 und beginnen mit einem neuen Iterationsschritt.\nAls erstes mu \337 sichergestellt werden, da\337 " }{XPPEDIT 18 0 "mu[2,1];" "6#&%#m uG6$\"\"#\"\"\"" }{TEXT -1 27 " h\366chstens Betrag 1/2 hat:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mu[2,1];" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "c2 + mu[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "dotprod(%, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3/4*dotprod(c1, c1);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 66 "Also sind wir im zweiten Fall.\nWir m\374ssen sicherstell en, da\337 alle " }{XPPEDIT 18 0 "mu[2,j];" "6#&%#muG6$\"\"#%\"jG" } {TEXT -1 152 " mit j < 1 h\366chstens Betrag 1/2 haben; solche j gibt es nicht.\nWir beginnen daher einen neuen Iterationsschritt mit k=2. \nAls erstes mu\337 der Betrag von " }{XPPEDIT 18 0 "mu[3,2];" "6#&%#m uG6$\"\"$\"\"#" }{TEXT -1 20 " betrachtet werden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mu[3,2];" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 47 "Das mu\337 also um eins reduziert werden und auch " }{XPPEDIT 18 0 "b[3];" "6#&%\"bG6#\"\"$" }{TEXT -1 35 " mu\337 entsprechend ang epa\337t werden:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "mu[3,1] := mu[3,1] - mu[2,1]; mu[3,2] := 0; b3 := b3 \+ - b2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c3 + mu[3,2]*c2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dotprod(%, %);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3/4*dotprod(c2, c2);" }}} {EXCHG {PARA 263 "" 0 "" {TEXT -1 40 "Wir sind also im ersten Fall und m\374ssen " }{XPPEDIT 18 0 "b[3];" "6#&%\"bG6#\"\"$" }{TEXT -1 6 " m it " }{XPPEDIT 18 0 "b[2];" "6#&%\"bG6#\"\"#" }{TEXT -1 14 " vertausc hen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "b0 := b2: b2 := b3; b3 := b0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d2 := c3 + mu [3,2]*c2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "nu[3,2] := mu[ 3,2]*dotprod(c2,c2)/(dotprod(c3,c3)+mu[3,2]^2*dotprod(c2,c2));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "nu[2,1] := mu[3,1]; nu[3,1] \+ := mu[2,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d3 := c2 - n u[3,2]*d2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "c2 := d2; c3 \+ := d3;\nmu[3,2] := nu[3,2]; mu[2,1] := nu[2,1]; mu[3,1] := nu[3,1];" } }}{EXCHG {PARA 269 "" 0 "" {TEXT -1 2 "Pr" }{MPLTEXT 1 0 0 "" }{TEXT -1 4 "obe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c1 = b1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c2 = b2 - mu[2,1]*c1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c3 = b3 - mu[3,1]*c1 - mu[3,2]*c2;" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 69 "Nun mu\337 k wieder um eins erniedrigt werden , d.h. k=1. Als erstes mu\337 " }{XPPEDIT 18 0 "mu[2,1] = 3/4;" "6#/&% #muG6$\"\"#\"\"\"*&\"\"$F(\"\"%!\"\"" }{TEXT -1 19 " reduziert werden :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "b2 \+ := b2 - b1; mu[2,1] := mu[2,1] - 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c2 + mu[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dotprod(%, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3/4*dotprod(c1, c1);" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 42 "Wir sind also wieder im Fall 1 und m\374ssen " }{XPPEDIT 18 0 "b[1]; " "6#&%\"bG6#\"\"\"" }{TEXT -1 6 " und " }{XPPEDIT 18 0 "b[2];" "6#&% \"bG6#\"\"#" }{TEXT -1 15 " vertauschen:." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "b0 := b1: b1 := b2; b2 := b0 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d1 := c2 + mu[2,1]*c1; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "nu[2,1] := mu[2,1]*dotp rod(c1,c1)/(dotprod(c2,c2)+mu[2,1]^2*dotprod(c1,c1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d2 := c1 - nu[2,1]*d1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "nu[3,1] := mu[3,1]*nu[2,1] + mu[3,2 ]*dotprod(c2, c2)/dotprod(d1, d1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "nu[3,2] := mu[3,1] - mu[3,2]*mu[2,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "c1 := d1; c2 := d2;\nmu[2,1] := nu[ 2,1]; mu[3,1] := nu[3,1]; mu[3,2] := nu[3,2];" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c1 = b1; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c2 = b2 - mu[2,1]*c1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c3 = b3 - nu[3,1]*c1 - nu[3,2]*c2;" }}} {EXCHG {PARA 267 "" 0 "" {TEXT -1 102 "Nun mu\337 k wieder auf 0 ernie drigt werden; es gibt nichts zu tun und wir erh\366hen gleich wieder a uf k=1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mu[2,1];" }}} {EXCHG {PARA 275 "" 0 "" {TEXT -1 32 "Das mu\337 auf 0 erh\366ht werde n und " }{XPPEDIT 18 0 "b[2];" "6#&%\"bG6#\"\"#" }{TEXT -1 35 " mu \337 entsprechend angepa\337t werden:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "b2 := b2 + b1; mu[2,1] := mu[2,1] + 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c2 + mu[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dotprod(%, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3/4*dotprod(c1, c1);" }}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 107 "Wir sind also im Fall 2, in dem es auch f\374r k=1 nichts zu tun \+ gibt.\nWir k\366nnen also gleich auf k=2 erh\366hen." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mu[3,2];" }}}{EXCHG {PARA 277 "" 0 "" {TEXT -1 17 "Das kann bleiben." }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c3 + mu[3,2]*c2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dotprod(%, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "3/4*dotprod(c2, c2);" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 36 "Auch hier sind wir wieder in Fall 2." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mu[3,1]; " }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 75 "Es gibt nichts zu tun, wir \+ setzen also k=3 und damit endet der Algorithmus." }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 264 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c1 = b1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c2 = b2 - mu[2,1]*c1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c3 = b3 - mu[3,1] *c1 - mu[3,2]*c2;" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 54 "Die LLL-re duzierte Basis besteht also aus den Vektoren" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "b1; b2; b3;" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 95 "Wir wollen direkt nachrechnen, da\337 dies eine LLL-reduzierte Basis \+ ist.\nDie Bedingungen f\374r die " }{XPPEDIT 18 0 "mu[i,j];" "6#&%#muG 6$%\"iG%\"jG" }{TEXT -1 72 " sind offensichtlich erf\374llt; bleiben \+ also noch die Lov\341sz-Bedingungen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c2 + mu[2,1]*c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dotprod(%, %) >= 3/4*dotprod(c1, c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c3 + mu[3,2]*c2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dotprod(%, %) >= 3/4*dotprod(c2, c2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "mu[2,1]; mu[3,1]; mu[3,2]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dotprod(b1, b1); dotpro d(b2, b2); dotprod(b3, b3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dotprod(c1, c1); dotprod(c2, c2); dotprod(c3, c3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalf([sqrt(2), sqrt(6), sqrt(12), \+ sqrt(14)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK " 0 0 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }