{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "Help Head ing" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Time s" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 7 "Advice:" }{TEXT -1 26 " R eal values for a RootOf" }}{PARA 0 "" 0 "" {TEXT -1 73 " Maple often r eturns solutions to various types of equations in terms of " } {MPLTEXT 0 21 6 "RootOf" }{TEXT -1 20 " a polynomial. The " } {MPLTEXT 0 21 9 "allvalues" }{TEXT -1 37 " function can be used to rep lace the " }{MPLTEXT 0 21 6 "RootOf" }{TEXT -1 401 " with the roots of the polynomial. By default this will use symbolic solutions if that \+ is possible (e.g. if the polynomial is of degree 4 or less). In many \+ cases that may be inconvenient because the symbolic solutions are very complicated. In previous releases, numerical values were used when s ymbolic solutions were not available, but in Maple 6 these are replace d by indexed roots. You can use " }{MPLTEXT 0 21 5 "evalf" }{TEXT -1 155 " to evaluate these. Note that this includes complex solutions. \+ An alternative, which always returns floating-point results for real s olutions, is to use " }{MPLTEXT 0 21 6 "fsolve" }{TEXT -1 27 " as foll ows. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "solve(\{ y^2+2*y+x=0, y \+ = x^2 - 1 \});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/%\"yG-%'RootOfG6$ ,,%#_ZG!\"\"*$)F*\"\"%\"\"\"F/*&F.F/)F*\"\"$F/F/*&F.F/)F*\"\"#F/F/F/F+ /%&labelG%$_L1G/%\"xG,$*&F&F/,&F&F/F5F/F/F+" }}}{PARA 0 "" 0 "" {TEXT -1 24 "Save this in a variable:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "q:= %:" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Isolate the " } {MPLTEXT 0 21 6 "RootOf" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ro:= op(indets(q,RootOf));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#roG-%'RootOfG6$,,%#_ZG!\"\"*$)F)\"\"%\"\"\"F.*&F-F.) F)\"\"$F.F.*&F-F.)F)\"\"#F.F.F.F*/%&labelG%$_L1G" }}}{PARA 0 "" 0 "" {TEXT -1 30 "Find a list of the real roots:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 " rts:= [ fsolve(op(1,ro)) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rtsG7$$!+8S6^Z!#5$\"+,7;-\\F(" }}}{PARA 0 "" 0 "" {TEXT -1 30 "Substitute into the solutions:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "map(t -> subs(ro=t,q), rts);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$<$/%\"yG$!+8S6^Z!#5/%\"xG$\"+\"f>\\C(F)<$/F&$\"+,7;- \\F)/F+$!+&3W2A\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 58 "You could obtai n complex solutions in the same way, using " }{MPLTEXT 0 21 6 "fsolve " }{TEXT -1 10 " with the " }{MPLTEXT 0 21 7 "complex" }{TEXT -1 8 " o ption:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "rts:= [ fsolve(op( 1,ro), _Z, complex) ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$rtsG7&^$$ !+fBb2?!\"*$!+cz:J^!#5^$F'$\"+cz:J^F,$!+8S6^ZF,$\"+,7;-\\F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "map(t -> subs(ro=t,q), rts); " }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&<$/%\"yG^$$!+fBb2 ?!\"*$!+cz:J^!#5/%\"xG^$$\"+O1E\"[#F-$!+h?)R.\"F*<$/F&^$F($\"+cz:J^F-/ F/^$F1$\"+h?)R.\"F*<$/F&$!+8S6^ZF-/F/$\"+\"f>\\C(F-<$/F&$\"+,7;-\\F-/F /$!+&3W2A\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Here is the same calcul ation, done using " }{MPLTEXT 0 21 9 "allvalues" }{TEXT -1 7 ". The \+ " }{MPLTEXT 0 21 8 "implicit" }{TEXT -1 93 " option prevents the use o f the very complicated explicit solution of the quartic polynomial." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "allvalues(q,implicit);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6&<$/%\"yG-%'RootOfG6$,,%#_ZG!\"\"*$)F* \"\"%\"\"\"F/*&F.F/)F*\"\"$F/F/*&F.F/)F*\"\"#F/F/F/F+/%&indexGF//%\"xG ,$*&F&F/,&F&F/F5F/F/F+<$/F%-F'6$F)/F7F5/F9,$*&F?F/,&F?F/F5F/F/F+<$/F%- F'6$F)/F7F2/F9,$*&FHF/,&FHF/F5F/F/F+<$/F%-F'6$F)/F7F./F9,$*&FQF/,&FQF/ F5F/F/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf([%]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7#7&<$/%\"yG$\"+,7;-\\!#5/%\"xG$!+&3W2 A\"!\"*<$/F'^$$!+fBb2?F/$\"+cz:J^F*/F,^$$\"+O1E\"[#F*$\"+h?)R.\"F/<$/F '$!+8S6^ZF*/F,$\"+\"f>\\C(F*<$/F'^$F3$!+cz:J^F*/F,^$F9$!+h?)R.\"F/" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 " See also:" }{TEXT -1 1 " " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" } {TEXT -1 2 ", " }{HYPERLNK 17 "RootOf" 2 "RootOf" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "allvalues" 2 "allvalues" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "allsolve" 2 "allsolve" "" }}}{PARA 0 "" 0 "" {TEXT 26 22 "Maple Advisor Database" }{TEXT -1 18 " R. Israel, 1997\n" }}} {MARK "2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }