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2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Normal" -1 260 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 261 39 "MATRICES AND MATRIX OPE RATIONS: Unit 25" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 259 "" 0 "" {TEXT 263 23 "Dr. Wlodzislaw Kostecki" }}{PARA 260 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 260 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 260 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 260 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 262 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 259 "" 0 "" {TEXT 264 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 265 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 259 4 "(25)" }{TEXT -1 1 " " }{TEXT 257 56 "Functions of ma trices comprising non-algebraic functions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1347 10 "OBJECTIVES" } {TEXT 1348 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1349 1 "\225" }{TEXT -1 40 " To include a state ment based on author" }{TEXT 1370 1 "\222" }{TEXT -1 118 "s own comput ational experiments that computation of functions of matrices comprisi ng non-algebraic functions becomes " }{TEXT 1350 4 "very" }{TEXT -1 47 " extensive for matrices of order higher than " }{TEXT 1352 6 "se cond" }{TEXT 1351 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1354 1 "\225" }{TEXT -1 45 " To stress that the method using truncated " }{TEXT 1353 10 "Maclaurin\222" }{TEXT -1 84 "s series is completely impracti cal for this kind of matrices of order higher than " }{TEXT 1355 6 "s econd" }{TEXT -1 46 " because of the prohibitive computation time." } }}{EXCHG {PARA 0 "" 0 "" {TEXT 1356 1 "\225" }{TEXT -1 169 " To inclu de an observation that computation of functions of second-order matric es having four non-algebraic functions, or three such functions and a \+ number (including " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 38 " at an 'i mproper' location) is also " }{TEXT 1360 4 "very" }{TEXT -1 12 " ext ensive." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1357 1 "\225" }{TEXT -1 166 " To include an observation that computation of functions of second-or der matrices with non-algebraic functions is practically tractable onl y if a matrix contains a " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 60 " e lement at either of two specific locations in the matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1358 1 "\225" }{TEXT -1 144 " To substantiate t he preceding observation by inclusion of an analysis of eigenvalues of second-order matrices with symbolic-function elements." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1359 1 "\225" }{TEXT -1 64 " To demonstrate tha t the eigenvalues of a matrix containing a " }{XPPMATH 20 "6#%%zeroG " }{TEXT -1 111 " element at either of the two specific locations are equal to the elements on the main diagonal of the matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1361 1 "\225" }{TEXT -1 150 " To provide four s tep-by-step solved examples for computation of functions of second-ord er matrices with at least two non-algebraic functions and a " } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 140 " element at a 'proper' locatio n using in each case all the three methods of computation of matrix fu nctions that are employed in Unit (23)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1363 1 "\225" }{TEXT -1 140 " To provide one step-by-step solve d example for computation of a function of a second-order matrix with \+ two non-algebraic functions and a " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 65 " element at an 'improper' location using the same three method s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1362 1 "\225" }{TEXT -1 145 " To \+ provide a comparison of the resultant matrix functions by computing th e function of each matrix for a given numerical value of the variable. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1369 1 "\225" }{TEXT -1 25 " To sho w how to use the " }{TEXT 1368 10 "eigenvects" }{TEXT -1 66 " function for matrices whose elements are non-algebraic functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1364 1 "\225" }{TEXT -1 34 " To include computa tions of the " }{TEXT 1365 6 "limits" }{TEXT -1 106 " of the resulta nt matrix functions having a discontinuity on the interval of converge nce of the function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1367 1 "\225" } {TEXT -1 249 " To stress that, and explain why, the numerical values \+ of the elements on the main diagonal of the limit of a matrix function having an element with discontinuity are equal when the variable tend s to the value at which the element is discontinuous." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "restart : interface(warnlevel=0) : with(linalg, augment, ch arpoly, coldim, diag, eigenvals, eigenvects, exponential, rowdim) :" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "This Unit deals with computation of functions of square \+ matrices in which at least one element is a " }{TEXT 268 15 "non-algeb raic " }{TEXT -1 4 "or " }{TEXT 1135 23 "transcendental function" } {TEXT -1 10 " of one " }{TEXT 863 4 "real" }{TEXT -1 12 " variable \+ " }{TEXT 1134 1 "t" }{TEXT 1136 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "A function " } {TEXT 272 1 "f" }{TEXT -1 16 " of a matrix [" }{TEXT 269 1 "A" } {TEXT 271 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 270 1 ")" }{TEXT -1 66 "] whose elements are non-algebraic functions of a real variabl e " }{TEXT 273 1 "t" }{TEXT -1 43 " may be computed only if the func tion is " }{TEXT 274 12 "well defined" }{TEXT -1 49 " for the matrix . [ Refer to Unit (23) for the " }{TEXT 275 12 "well-defined" } {TEXT -1 13 " function. ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "It was stated in the preceding Un its (23 and 24) that a well-defined function of a matrix [" }{TEXT 276 1 "A" }{TEXT 278 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 277 1 " )" }{TEXT -1 30 "] may be computed using the " }{TEXT 279 15 "Cayley -Hamilton" }{TEXT -1 43 " theorem, similarity of matrices, or the " }{TEXT 260 9 "Maclaurin" }{TEXT -1 55 " series representation of the \+ corresponding function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "However, it appears that computation of \+ functions of matrices with non-algebraic functions becomes " }{TEXT 1144 4 "very" }{TEXT -1 47 " extensive for matrices of order higher t han " }{TEXT 1146 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1145 3 " \+ \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1147 2 ")." }{TEXT -1 30 " \+ The method using truncated " }{TEXT 1148 10 "Maclaurin\222" }{TEXT -1 111 "s series turns out to be completely impractical for such matr ices because of the prohibitive computation time." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Even if ma trices of order " }{TEXT 1150 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 1149 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1151 1 ")" } {TEXT -1 151 " only are involved, computations of a matrix function a re very extensive irrespective of the method used. The exception are m atrices that comprise a " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 30 " el ement either at location " }{TEXT 1152 5 "(1,2)" }{TEXT -1 6 " or \+ " }{TEXT 1153 6 "(2,1)." }{TEXT -1 82 " In this case, eigenvalues of \+ the matrix are equal to its elements at locations " }{TEXT 1154 5 "(1 ,1)" }{TEXT -1 7 " and " }{TEXT 1172 6 "(2,2)," }{TEXT -1 123 " i.e . on the main diagonal. This ensures that a function of the matrix may be found without a great computational effort. " }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "These obse rvations are briefly analysed hereunder." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Consider " } {TEXT 1156 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1155 3 " \327 " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1157 1 ")" }{TEXT -1 13 " matrices [" }{TEXT 1158 1 "A" }{TEXT 1160 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1159 1 ")" }{TEXT -1 9 "] and [" }{TEXT 1161 1 "B" }{TEXT 1163 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1162 1 ")" }{TEXT -1 17 "] comprising a " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 23 " elemen t at location " }{TEXT 1164 5 "(1,2)" }{TEXT -1 7 " and " }{TEXT 1165 6 "(2,1)," }{TEXT -1 15 " respectively." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "A(t) := matrix(2, 2, [a[11](t), 0, a[21](t), a[ 22](t)]) : B(t) := matrix(2, 2, [b[11](t), b[12](t), 0, b[22](t)]) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "'A(t)' = A(t) ; 'B(t) ' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG 6#7$7$-&%\"aG6#\"#6F&\"\"!7$-&F/6#\"#@F&-&F/6#\"#AF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7$7$-&%\"bG6#\"#6F&-&F/6# \"#7F&7$\"\"!-&F/6#\"#AF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The characteristic equation of eit her matrix has the same form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "l := lambda : char_eq('A(t)') := charpoly(A(t), l[ A]) = 0 : char_eq('B(t)') := charpoly(B(t), l[B]) = 0 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "char_eq('A(t)') ; char_eq('B(t)') ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&&%'lambdaG6#%\"AG\"\"\"-&% \"aG6#\"#66#%\"tG!\"\"F*,&F&F*-&F-6#\"#AF0F2F*\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&,&&%'lambdaG6#%\"BG\"\"\"-&%\"bG6#\"#66#%\"tG!\"\" F*,&F&F*-&F-6#\"#AF0F2F*\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Inspection of the characteri stic equations shows clearly that the eigenvalues are equal to the ele ments at locations " }{TEXT 1173 5 "(1,2)" }{TEXT -1 7 " and " } {TEXT 1174 6 "(2,1)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The solution of either equation yi elds the eigenvalues, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "solution(char_eq('A(t)')) := solve(\{char_eq('A(t)')\}, \{l[A]\}) \+ :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solution(char_eq('B(t) ')) := solve(\{char_eq('B(t)')\},\{l[B]\}) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solution(char_eq('A(t)')) ; solution(char_eq(' B(t)')) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/&%'lambdaG6#%\"AG-&%\" aG6#\"#A6#%\"tG<#/F%-&F+6#\"#6F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<# /&%'lambdaG6#%\"BG-&%\"bG6#\"#A6#%\"tG<#/F%-&F+6#\"#6F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "wh ence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "l1A := rhs(op(solut ion(char_eq('A(t)'))[1])) : l2A := rhs(op(solution(char_eq('A(t)'))[ 2])) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "l1B := rhs(op(sol ution(char_eq('B(t)'))[1])) : l2B := rhs(op(solution(char_eq('B(t)') )[2])) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "l[1][A] = l1A \+ ; l[2][A] = l2A ; `` ; l[1][B] = l1B ; l[2][B] = l2B ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"\"6#%\"AG-&%\"aG6#\" #A6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"#6#%\"A G-&%\"aG6#\"#66#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"\"6#%\"BG-&%\"bG6#\"#A6#% \"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"#6#%\"BG-&% \"bG6#\"#66#%\"tG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "The above analysis has been done for ins tructive purposes to show why the eigenvalues are equal to the element s at the specified locations of a matrix. Naturally, the eigenvalues m ay be computed 'directly' using the " }{TEXT 1166 9 "eigenvals" } {TEXT -1 94 " function, which is not that instructive but gives the sa me result with simpler commands, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "charroots('A(t)') := eigenvals(A(t)) : charroots('B (t)') := eigenvals(B(t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "l1A := charroots('A(t)')[1] : l2A := charroots('A(t)')[2] :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "l1B := charroots('B(t)')[1] \+ : l2B := charroots('B(t)')[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "l[1][A] = l1A ; l[2][A] = l2A ; `` ; l[1][B] = \+ l1B ; l[2][B] = l2B ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambda G6#\"\"\"6#%\"AG-&%\"aG6#\"#66#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&&%'lambdaG6#\"\"#6#%\"AG-&%\"aG6#\"#A6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\" \"\"6#%\"BG-&%\"bG6#\"#66#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&& %'lambdaG6#\"\"#6#%\"BG-&%\"bG6#\"#A6#%\"tG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "If a matrix con tains a " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 50 " element, which is \+ at a location different than " }{TEXT 1167 5 "(1,2)" }{TEXT -1 6 " o r " }{TEXT 1168 6 "(2,1)," }{TEXT -1 41 " i.e. on the main diagonal, for example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "A(t) := m atrix(2, 2, [0, a[12](t), a[21](t), a[22](t)]) : B(t) := matrix(2, 2 , [b[11](t), b[12](t), b[21](t), 0]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "'A(t)' = A(t) ; 'B(t)' = B(t) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7$\"\"!-&%\"aG6#\"#7F&7$- &F06#\"#@F&-&F06#\"#AF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#% \"tG-%'matrixG6#7$7$-&%\"bG6#\"#6F&-&F/6#\"#7F&7$-&F/6#\"#@F&\"\"!" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "then the expressions of eigenvalues become more complicated sin ce the square root is involved, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "charroots('A(t)') := eigenvals(A(t)) : charroots('B (t)') := eigenvals(B(t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "l1A := charroots('A(t)')[1] : l2A := charroots('A(t)')[2] :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "l1B := charroots('B(t)')[1] \+ : l2B := charroots('B(t)')[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "l[1][A] = l1A ; l[2][A] = l2A ; `` ; l[1][B] = \+ l1B ; l[2][B] = l2B ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambda G6#\"\"\"6#%\"AG,&-&%\"aG6#\"#A6#%\"tG#F(\"\"#*&F3F(-%%sqrtG6#,&*$)F,F 4F(F(*(\"\"%F(-&F.6#\"#7F1F(-&F.6#\"#@F1F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"#6#%\"AG,&-&%\"aG6#\"#A6#%\"tG#\"\"\" F(*&#F4F(F4*$-%%sqrtG6#,&*$)F,F(F4F4*(\"\"%F4-&F.6#\"#7F1F4-&F.6#\"#@F 1F4F4F4F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"\"6#%\"BG,&-&%\"bG6#\"#66#%\"tG# F(\"\"#*&F3F(-%%sqrtG6#,&*$)F,F4F(F(*(\"\"%F(-&F.6#\"#7F1F(-&F.6#\"#@F 1F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"#6#%\"B G,&-&%\"bG6#\"#66#%\"tG#\"\"\"F(*&#F4F(F4*$-%%sqrtG6#,&*$)F,F(F4F4*(\" \"%F4-&F.6#\"#7F1F4-&F.6#\"#@F1F4F4F4F4!\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "If a matrix con tains no " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 22 " element, for exam ple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "A(t) := matrix(2, 2, [a[11](t), a[12](t), a[21](t), a[22](t)]) : 'A(t)' = A(t) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7$-&%\"aG6 #\"#6F&-&F/6#\"#7F&7$-&F/6#\"#@F&-&F/6#\"#AF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "then the expre ssions of eigenvalues also include the square root and are yet more co mplex, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "charroots(' A(t)') := eigenvals(A(t)) : l1A := charroots('A(t)')[1] : l2A := c harroots('A(t)')[2] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "l[ 1][A] = l1A ; l[2][A] = l2A ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&& %'lambdaG6#\"\"\"6#%\"AG,(-&%\"aG6#\"#66#%\"tG#F(\"\"#*&F3F(-&F.6#\"#A F1F(F(*&F3F(-%%sqrtG6#,**$)F,F4F(F(*(F4F(F,F(F6F(!\"\"*$)F6F4F(F(*(\" \"%F(-&F.6#\"#7F1F(-&F.6#\"#@F1F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"#6#%\"AG,(-&%\"aG6#\"#66#%\"tG#\"\"\"F(*&F3F4- &F.6#\"#AF1F4F4*&#F4F(F4*$-%%sqrtG6#,**$)F,F(F4F4*(F(F4F,F4F6F4!\"\"*$ )F6F(F4F4*(\"\"%F4-&F.6#\"#7F1F4-&F.6#\"#@F1F4F4F4F4FD" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "e ven if re-arranged to assume more compact forms" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 86 "l1A := a[11](t)/2 + a[22](t)/2 + sqrt((a[11](t )-a[22](t))^2 + 4*a[12](t)*a[21](t))/2 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "l2A := a[11](t)/2 + a[22](t)/2 - sqrt((a[11](t)-a[22] (t))^2 + 4*a[12](t)*a[21](t))/2 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "l[1][A] = l1A ; l[2][A] = l2A ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&&%'lambdaG6#\"\"\"6#%\"AG,(-&%\"aG6#\"#66#%\"tG#F( \"\"#*&F3F(-&F.6#\"#AF1F(F(*&F3F(-%%sqrtG6#,&*$),&F,F(F6!\"\"F4F(F(*( \"\"%F(-&F.6#\"#7F1F(-&F.6#\"#@F1F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&&%'lambdaG6#\"\"#6#%\"AG,(-&%\"aG6#\"#66#%\"tG#\"\"\" F(*&F3F4-&F.6#\"#AF1F4F4*&#F4F(F4*$-%%sqrtG6#,&*$),&F,F4F6!\"\"F(F4F4* (\"\"%F4-&F.6#\"#7F1F4-&F.6#\"#@F1F4F4F4F4FD" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "In view of the above analysis, it is stated that mathematically tractable are those \+ matrices with transcendental-function elements whose order does not ex ceed " }{TEXT 1345 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1344 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 1346 1 ")" }{TEXT -1 23 " \+ and which contain a " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 30 " eleme nt either at location " }{TEXT 1169 5 "(1,2)" }{TEXT -1 6 " or " } {TEXT 1170 6 "(2,1)." }{TEXT -1 243 " The first four examples involve such matrices. Their function elements have also been carefully selec ted so that all the three methods may be used for computation of a wel l-defined function of the matrix without a great computational effort. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "The last, fifth, example is associated with Example 1 in asmuch as the same function is computed and the matrix contains the sa me function elements, but the " }{XPPMATH 20 "6#%%zeroG" }{TEXT -1 26 " element is at location " }{TEXT 1171 5 "(2,2)" }{TEXT -1 119 ". The intent of this example is to illustrate the aforementioned much \+ greater effort in computing the matrix function." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 258 9 "Example \+ 1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 284 4 "cos(" }{TEXT -1 1 "[" }{TEXT 280 1 "A" }{TEXT 282 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 281 1 ")" } {TEXT -1 1 "]" }{TEXT 283 1 ")" }{TEXT -1 8 " of a " }{TEXT 286 1 "( " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 285 3 " \327 " }{XPPEDIT 18 0 "2 " "6#\"\"#" }{TEXT 287 1 ")" }{TEXT -1 11 " matrix [" }{TEXT 288 1 " A" }{TEXT 290 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 289 1 ")" } {TEXT -1 11 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A(t) := matrix(2, 2, [sin(t), t+1, 0, cos(t)]) : 'A(t)' = A(t) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7$-%$sinG F&,&F'\"\"\"F0F07$\"\"!-%$cosGF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The power series of " } {XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 21 " is convergen t for " }{TEXT 333 3 "all" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x" "6#%\" xG" }{TEXT -1 44 " [refer to Unit (23)], which implies that " } {TEXT 330 4 "cos(" }{TEXT -1 1 "[" }{TEXT 329 1 "A" }{TEXT -1 1 "]" } {TEXT 334 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 332 5 "e very" }{TEXT -1 57 " square matrix. This, in turn, means that the fun ction " }{TEXT 338 4 "cos(" }{TEXT -1 1 "[" }{TEXT 331 1 "A" }{TEXT 336 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 335 1 ")" }{TEXT -1 1 "] " }{TEXT 337 1 ")" }{TEXT -1 38 " is well defined for every value of \+ " }{TEXT 339 1 "t" }{TEXT 340 1 "," }{TEXT -1 82 " so it may be comp uted for this matrix. See also the note at the end of Method 1." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "For convenience in computations, let" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B(t) := A(t) : A(t) := 'A(t)' :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 8 "M ethod 1" }{TEXT -1 13 ". Using the " }{TEXT 291 15 "Cayley-Hamilton" }{TEXT -1 9 " theorem" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 293 6 "Step 1" }{TEXT -1 23 ". Having in \+ mind that " }{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"#" }{TEXT -1 51 " for \+ this matrix, write equation (1) of Unit (23) " }{TEXT 294 1 "B" } {TEXT -1 7 " for [" }{TEXT 295 1 "A" }{TEXT 297 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 296 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 94 "`cos(A(t))` := a[1] * A(t) + a[0]*U : cos(A( t)) = `cos(A(t))` ; Cos(A(t)) := `cos(A(t))` :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#%\"tG,&*&&%\"aG6#\"\"\"F0F'F0F0*&&F.6 #\"\"!F0%\"UGF0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 6 "Step 2" }{TEXT -1 40 ". Bearing in mind t hat the unit matrix [" }{TEXT 298 1 "U" }{TEXT -1 38 "] appropriately \+ sized for this case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U := diag(1, 1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"UG-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "evaluate the matr ix equation for " }{TEXT 304 4 "cos(" }{TEXT -1 1 "[" }{TEXT 300 1 "A " }{TEXT 302 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 301 1 ")" } {TEXT -1 1 "]" }{TEXT 303 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`cos(A(t))` := evalm(subs(A(t)=B(t), `cos(A(t))`)) : cos(A(t )) = matrix(`cos(A(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG 6#-%\"AG6#%\"tG-%'matrixG6#7$7$,&*&&%\"aG6#\"\"\"F5-%$sinGF)F5F5&F36# \"\"!F5*&F2F5,&F*F5F5F5F57$F:,&*&F2F5-F%F)F5F5F8F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 305 6 "Step 3" } {TEXT -1 38 ". Formulate equation (2) of Unit (23) " }{TEXT 306 1 "B" }{TEXT -1 28 " corresponding to this case:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "`r(l)` := subs(A(t)=l, U=1, Cos(A(t))) : r(l) = ` r(l)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"rG6#%'lambdaG,&*&&%\"a G6#\"\"\"F-F'F-F-&F+6#\"\"!F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 307 6 "Step 4" }{TEXT -1 32 ". Deter mine the eigenvalues of [" }{TEXT 308 1 "A" }{TEXT 310 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 309 1 ")" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "charroots(A(t)) := eigenvals (subs(A(t)=B(t), A(t))) : char_roots(A(t)) = charroots(A(t)) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#-%\"AG6#%\"tG6$-%$sin GF)-%$cosGF)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Extracting the individual eigenvalues " } {XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "No_roots(A(t)) := nops( [charroots(A(t))]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "fo r i to No_roots(A(t)) do ch_r[i](A(t)) := charroots(A(t))[i] : prin t(l[i] = ch_r[i](A(t))) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %'lambdaG6#\"\"\"-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% 'lambdaG6#\"\"#-%$cosG6#%\"tG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 311 6 "Step 5" }{TEXT -1 93 ". Takin g notice of the fact that the roots are distinct, formulate equation ( 3) of Unit (23) " }{TEXT 312 1 "B" }{TEXT -1 19 " in a general form:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Since the function " }{XPPEDIT 18 0 "f(lambda)" "6#-%\"f G6#%'lambdaG" }{TEXT -1 31 " corresponding to this case is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`f(l)` := cos(l) : f(l) = \+ `f(l)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%'lambdaG-%$cosGF &" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "therefore, equation (3) assumes the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Eq3_f(l) := `f(l)` = `r(l)` : Eq3 _f(l) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%'lambdaG,&*&&%\" aG6#\"\"\"F-F'F-F-&F+6#\"\"!F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 6 "Step 6" }{TEXT -1 80 ". Obtai n a set of equations by substituting either eigenvalue into equation ( 3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i to No_roots(A (t)) do Eq3_f[i](l) := subs(l=ch_r[i](A(t)), Eq3_f(l)) : print(Eq3_ f[i](l)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%$sinG 6#%\"tG,&*&&%\"aG6#\"\"\"F0F'F0F0&F.6#\"\"!F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-F%6#%\"tG,&*&&%\"aG6#\"\"\"F/F'F/F/&F-6#\"\" !F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 314 6 "Step 7" }{TEXT -1 64 ". Solve the simultaneous equation s for the unknown coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "solution := solve(\{Eq3_f[1](l), Eq3_f[2](l)\}, \{a[0], a[1]\} ) : solution ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"aG6#\"\"!*& ,&*&-%$cosG6#-%$sinG6#%\"tG\"\"\"-F-F1F3!\"\"*&F/F3-F-6#F4F3F3F3,&F4F5 F/F3F5/&F&6#F3,$*&,&F7F3F,F5F3F9F5F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Extracting either unkno wn from the solution set yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "assign(solution) : for i to No_roots(A(t)) do a[i-1] := a[ i-1] : print(evaln(a[i-1]) = a[i-1]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"!*&,&*&-%$cosG6#-%$sinG6#%\"tG\"\"\"-F,F0F 2!\"\"*&F.F2-F,6#F3F2F2F2,&F3F4F.F2F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\",$*&,&-%$cosG6#-F,6#%\"tGF'-F,6#-%$sinGF/!\"\"F',&F. F5F3F'F5F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 315 6 "Step 8" }{TEXT -1 61 ". Substitute the values of \+ the coefficients into the matrix " }{TEXT 320 4 "cos(" }{TEXT -1 1 "[ " }{TEXT 316 1 "A" }{TEXT 318 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 317 1 ")" }{TEXT -1 1 "]" }{TEXT 319 2 ")," }{TEXT -1 91 " simp lify the elements of the resultant matrix, and add a distinguishing su bscript, say, " }{TEXT 321 3 "C_H" }{TEXT -1 2 " " }{TEXT 394 1 "(" }{TEXT 395 1 "C" }{TEXT 396 6 "ayley-" }{TEXT 397 1 "H" }{TEXT 398 8 " amilton)" }{TEXT -1 46 " to its name for future comparative purposes: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`cos(A(t))[C_H]` := map (normal, map(x->eval(x), `cos(A(t))`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "cos(A(t))[C_H] = matrix(`cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6# 7$7$-F&6#-%$sinGF*,$*&*&,&-F&6#-F&F*\"\"\"F3!\"\"F>,&F+F>F>F>F>F>,&F=F ?F5F>F?F?7$\"\"!F;" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i to No_roots(A(t)) do a[i-1] : = evaln(a[i-1]) : od :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 328 4 "N.B." }{TEXT -1 28 " In general, \+ the function " }{TEXT 326 4 "cos(" }{TEXT -1 1 "[" }{TEXT 322 1 "A" } {TEXT 324 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 323 1 ")" }{TEXT -1 1 "]" }{TEXT 325 1 ")" }{TEXT -1 38 " is well defined for every va lue of " }{TEXT 327 1 "t" }{TEXT 342 1 "." }{TEXT -1 90 " This is tr ue if the elements of the matrix function are continuous functions for every " }{TEXT 341 1 "t" }{TEXT 343 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "It can be easily \+ noticed that the function element at location " }{TEXT 893 5 "(1,2)" }{TEXT -1 67 " in the above matrix is discontinuous if the denominato r becomes " }{XPPMATH 20 "6#%%zeroG" }{TEXT 892 1 "." }{TEXT -1 31 " \+ Equating the denominator to " }{XPPMATH 20 "6#%%zeroG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Eq_denom(el12) := denom(`cos(A(t))[ C_H]`[1,2]) = 0 : Eq_denom(el12) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/,&-%$cosG6#%\"tG!\"\"-%$sinGF'\"\"\"\"\"!" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "and solving the resultant equation yield" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "solution := solve(\{Eq_denom(el12)\}) : solution ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<#/%\"tG,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Let the nu merical part of the solution be assigned the name " }{XPPEDIT 18 0 "t [d_s]" "6#&%\"tG6#%$d_sG" }{TEXT 894 1 "." }{TEXT -1 7 " Thus," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "t[d_s] := rhs(solution[1]) \+ : 't[d_s]' = t[d_s] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#%$d _sG,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The denominator becomes " } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 12 " at every " }{XPPEDIT 18 0 "Pi /4+2*Pi*n" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&*(\"\"#F&F%F&%\"nGF&F&" } {TEXT 897 1 "," }{TEXT -1 9 " where " }{TEXT 895 1 "n" }{TEXT 896 17 " = 0, \2611, \2612, \2613." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "If the function " }{TEXT 868 4 "cos(" }{TEXT -1 1 "[" }{TEXT 864 1 "A" }{TEXT 866 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 865 1 ")" }{TEXT -1 1 "]" }{TEXT 867 1 ")" }{TEXT -1 25 " is to be computed for " }{XPPEDIT 18 0 "t=t [d_s]" "6#/%\"tG&F$6#%$d_sG" }{TEXT 899 1 "," }{TEXT -1 79 " this can be done only if a limiting value of the matrix element exists when \+ " }{TEXT 869 1 "t" }{TEXT -1 35 " tends to the numerical value of " }{XPPEDIT 18 0 "t[d_s]" "6#&%\"tG6#%$d_sG" }{TEXT 898 1 "." }{TEXT -1 71 " Consequently, this must be checked by computing the limit as fol lows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`limit(el[12](cos( A(t))[C_H]))` := limit(`cos(A(t))[C_H]`[1,2], t=t[d_s]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Limit(element[12](cos(A(t))[C_H]), \+ t=t[d_s]) = `limit(el[12](cos(A(t))[C_H]))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-&%(elementG6#\"#76#&-%$cosG6#-%\"AG6#%\"tG 6#%$C_HG/F4,$%#PiG#\"\"\"\"\"%,&*&-%$sinG6#,$*$-%%sqrtG6#\"\"#F;#F;FGF ;F9F;#!\"\"F " 0 "" {MPLTEXT 1 0 76 "`limit(el[12](cos(A(t))[C_H]))` := \+ factor(`limit(el[12](cos(A(t))[C_H]))`) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 80 "Limit(element[12](cos(A(t))[C_H]), t=t[d_s]) = `lim it(el[12](cos(A(t))[C_H]))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&L imitG6$-&%(elementG6#\"#76#&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG/F4,$%#PiG# \"\"\"\"\"%,$*&-%$sinG6#,$*$-%%sqrtG6#\"\"#F;#F;FGF;,&F9F;F " 0 "" {MPLTEXT 1 0 77 "`lim cos(A(t))[C_H]` := map(factor, map(limit, `cos(A(t))[C_H]`, t =t[d_s])) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Limit(cos(A( t))[C_H], t=t[d_s]) = matrix(`lim cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG/F.,$%#Pi G#\"\"\"\"\"%-%'matrixG6#7$7$-F)6#,$*$-%%sqrtG6#\"\"#F5#F5FC,$*&-%$sin GF=F5,&F3F5F6F5F5#!\"\"F67$\"\"!F<" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Floating-point evaluati on of the above gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Li mit(cos(A(t))[C_H], t=t[d_s]) = evalf(matrix(`lim cos(A(t))[C_H]`)) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$&-%$cosG6#-%\"AG6#%\"t G6#%$C_HG/F.,$%#PiG#\"\"\"\"\"%-%'matrixG6#7$7$$\"+sfW-w!#5$!+)fg)f6! \"*7$$\"\"!FDF<" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "[ For computation of " }{TEXT 1366 6 "li mits" }{TEXT -1 45 " of function matrices, refer to Unit (28). ]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 377 8 "Method 2" }{TEXT -1 30 ". Using similarity of matrices" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 350 6 "Step 1" }{TEXT -1 32 ". Check if the eigenvalues of [" }{TEXT 351 1 "A" }{TEXT 353 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 352 1 " )" }{TEXT -1 16 "] are distinct:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "See Step 4 above where the eigenvalues are found to be distinct." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 358 6 "Step 2" }{TEXT -1 10 ". Find the" }{TEXT 354 1 " " }{TEXT -1 63 "sequence of lists co ntaining eigenvalues and eigenvectors of [" }{TEXT 355 1 "A" }{TEXT 357 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 356 1 ")" }{TEXT -1 2 "] :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 900 7 "WARNING" }{TEXT 901 1 ":" }{TEXT -1 6 " The " }{TEXT 902 10 "eigenvects" }{TEXT -1 142 " function works only for a matrix w hose elements are rationals, rational functions, algebraic numbers, or algebraic functions. Since matrix [" }{TEXT 904 1 "A" }{TEXT 906 1 " (" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 905 1 ")" }{TEXT -1 38 "] comp rises non-algebraic functions, " }{TEXT 903 10 "eigenvects" }{TEXT -1 22 " cannot be used for [" }{TEXT 907 1 "A" }{TEXT 909 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 908 1 ")" }{TEXT -1 63 "]. To get \+ around this problem, construct an auxiliary matrix [" }{TEXT 910 1 "C " }{TEXT -1 13 "] as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C := matrix(2, 2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "C[1,1] := f[11] : C[1,2] := f[12] : C[2,1] := B(t)[2,1] : C[ 2,2] := f[22] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "C := mat rix(C) : C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-% 'matrixG6#7$7$&%\"fG6#\"#6&F+6#\"#77$\"\"!&F+6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "whose e lements " }{XPPEDIT 18 0 "f[11]" "6#&%\"fG6#\"#6" }{TEXT 1025 1 "," } {TEXT -1 2 " " }{XPPEDIT 18 0 "f[12]" "6#&%\"fG6#\"#7" }{TEXT 1026 1 "," }{TEXT -1 7 " and " }{XPPEDIT 18 0 "f[22]" "6#&%\"fG6#\"#A" } {TEXT -1 23 " denote, respectively," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f[11] = B(t)[1,1] ; f[12] = B(t)[1,2] ; f[22] = B (t)[2,2] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"fG6#\"#6-%$sinG6#% \"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"fG6#\"#7,&%\"tG\"\"\"F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"fG6#\"#A-%$cosG6#%\"tG" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Determine the eigenvalues of [" }{TEXT 958 1 "C" }{TEXT -1 2 "] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "charroots(C) := eigenv als(C) : char_roots(C) = charroots(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#%\"CG6$&%\"fG6#\"#6&F*6#\"#A" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "No_roots(C) := nops([charroots(C)]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for i to No_roots(C) do ch_r[i](C) := charroots(C)[i] : print(l [i] = ch_r[i](C)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lamb daG6#\"\"\"&%\"fG6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG 6#\"\"#&%\"fG6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "The" }{TEXT 911 1 " " }{TEXT -1 62 "sequence of lists containing eigenvalues and eigenvectors of [" } {TEXT 912 1 "C" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "`roots&vectors(C)` := eigenvects(C) : roots_and_vect ors(C) = `roots&vectors(C)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2r oots_and_vectorsG6#%\"CG6$7%&%\"fG6#\"#A\"\"\"<#-%'vectorG6#7$F.,$*&,& F*!\"\"&F+6#\"#6F.F.&F+6#\"#7F7F77%F8F.<#-F16#7$F.\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 359 6 "S tep 3" }{TEXT -1 28 ". Extract the eigenvectors " }{XPPEDIT 18 0 "v[l ambda[i]]" "6#&%\"vG6#&%'lambdaG6#%\"iG" }{TEXT -1 32 " corresponding to eigenvalues " }{XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" } {TEXT -1 6 " of [" }{TEXT 1339 1 "C" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "for i to No_roots(C) do e[i] := ch arroots(C)[i] : List[i] := `roots&vectors(C)`[i] : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "for j to No_roots(C) do fo r i to No_roots(C) do if List[i][1] = e[j] then Lst[j] := `roots&v ectors(C)`[i] : fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "for i to No_roots(C) do ch_v[i](C) := op(Lst[i][3]) : print(v[lambda[i]] = eval(ch_v[i](C)), ` --> ` * lambda[i] = ch _r[i](C)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lamb daG6#\"\"\"-%'vectorG6#7$F*\"\"!/*&%(~~-->~~GF*F'F*&%\"fG6#\"#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"#-%'vectorG6# 7$\"\"\",$*&,&&%\"fG6#\"#A!\"\"&F46#\"#6F/F/&F46#\"#7F7F7/*&%(~~-->~~G F/F'F/F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 32 "Conversion of the eigenvectors " }{XPPEDIT 18 0 " v[lambda[i]]" "6#&%\"vG6#&%'lambdaG6#%\"iG" }{TEXT -1 24 " into colum n matrices " }{XPPEDIT 18 0 "X[lambda[i]]" "6#&%\"XG6#&%'lambdaG6#%\" iG" }{TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "for i to No_roots(C) do X[l[i]] := convert(ch_v[i](C), matrix) \+ : print(X[l[i]] = matrix(X[l[i]])) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"\"-%'matrixG6#7$7#F*7#\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"#-%'matrixG6 #7$7#\"\"\"7#,$*&,&&%\"fG6#\"#A!\"\"&F66#\"#6F0F0&F66#\"#7F9F9" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 361 6 "Step 4" }{TEXT -1 39 ". Construct the 'unique' modal matrix [" }{TEXT 360 1 "M" }{TEXT -1 19 "] associated with [" }{TEXT 368 1 "C" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "i := 'i' : `seq(X[l[i]])` := seq(X[l[i]], i=1..coldim(C)) : M := augment(` seq(X[l[i]])`) : M=matrix(M) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"MG-%'matrixG6#7$7$\"\"\"F*7$\"\"!,$*&,&&%\"fG6#\"#A!\"\"&F16#\"#6F*F *&F16#\"#7F4F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 363 6 "Step 5" }{TEXT -1 24 ". Construct the uni que " }{TEXT 362 6 "Jordan" }{TEXT -1 9 " form [" }{TEXT 367 1 "J" }{TEXT -1 7 "] of [" }{TEXT 366 1 "C" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "J_C := array(diagonal, 1..rowdim(C) , 1..coldim(C)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for i to rowdim(C) do for j to coldim(C) do if j = i then J_C[i,j] := \+ ch_r[i](C) fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "J := matrix(J_C) : J = matrix(J) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"JG-%'matrixG6#7$7$&%\"fG6#\"#6\"\"!7$F.&F+6#\"#A" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 6 "Step 6" }{TEXT -1 24 ". Compute the function " }{TEXT 364 6 "cosine" }{TEXT -1 13 " of matrix [" }{TEXT 369 1 "J" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "J_C := array(di agonal, 1..rowdim(J), 1..coldim(J)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i to rowdim(J) do for j to coldim(J) do if j \+ = i then J_C[i,j] := cos(J[i,j]) fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`cos(J)` := matrix(J_C) : cos(J) \+ = matrix(`cos(J)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"J G-%'matrixG6#7$7$-F%6#&%\"fG6#\"#6\"\"!7$F3-F%6#&F06#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 886 6 "S tep 7" }{TEXT -1 11 ". Compute " }{TEXT 883 4 "cos(" }{TEXT -1 1 "[" }{TEXT 887 1 "C" }{TEXT -1 1 "]" }{TEXT 888 4 ") = " }{TEXT -1 1 "[" } {TEXT 890 1 "M" }{TEXT -1 2 "] " }{TEXT 884 4 "cos(" }{TEXT -1 1 "[" } {TEXT 889 1 "J" }{TEXT -1 1 "]" }{TEXT 885 1 ")" }{TEXT -1 1 " " } {TEXT 882 3 "Inv" }{TEXT -1 1 "[" }{TEXT 891 1 "M" }{TEXT -1 53 "] an d simplify the elements of the resultant matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`cos(C)` := map(normal, evalm(M &* `cos(J)` & * M^(-1))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "cos(C) = ma trix(`cos(C)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"CG-%' matrixG6#7$7$-F%6#&%\"fG6#\"#6*&*&&F06#\"#7\"\"\",&F-F8-F%6#&F06#\"#A! \"\"F8F8,&F " 0 "" {MPLTEXT 1 0 95 "`co s(A(t))[s_m]` := subs(f[11]=B(t)[1,1], f[12]=B(t)[1,2], f[22]=B(t)[2,2 ], matrix(`cos(C)`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "c os(A(t))[s_m] = matrix(`cos(A(t))[s_m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$-F&6#-% $sinGF**&*&,&F+\"\"\"F:F:F:,&F3F:-F&6#-F&F*!\"\"F:F:,&F>F?F5F:F?7$\"\" !F<" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 32 "The symbolic forms of matrices " }{XPPEDIT 18 0 "cos(A (t))[C_H]" "6#&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "cos(A(t))[s_m]" "6#&-%$cosG6#-%\"AG6#%\"tG6#%$s_mG" } {TEXT -1 12 " are equal." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 378 8 "Method 3" }{TEXT -1 19 ". Using tr uncated " }{TEXT 399 10 "Maclaurin\222" }{TEXT -1 9 "s series" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 370 6 "Step 1" }{TEXT -1 28 ". Find the eigenvalues of [" }{TEXT 506 1 "A" }{TEXT 508 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 507 1 ")" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "This step was already done earlier to che ck if the eigenvalues of [" }{TEXT 664 1 "A" }{TEXT 666 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 665 1 ")" }{TEXT -1 71 "] are dist inct, which was essential for both preceding methods. It is " }{TEXT 373 3 "not" }{TEXT -1 48 " necessary for this method since the series \+ of " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 21 " is co nvergent for " }{TEXT 375 3 "all" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x " "6#%\"xG" }{TEXT 372 1 "," }{TEXT -1 22 " which implies that " } {TEXT 371 4 "cos(" }{TEXT -1 1 "[" }{TEXT 380 1 "A" }{TEXT 382 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 381 1 ")" }{TEXT -1 1 "]" }{TEXT 376 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 374 5 "every" }{TEXT -1 16 " square matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 383 6 "Step 2" }{TEXT -1 28 ". Write the expression for " }{TEXT 384 10 "Maclaurin\222" }{TEXT -1 37 "s \+ series representing the function " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG 6#%\"xG" }{TEXT -1 38 " and substitute the matrix name for " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 104 "`cos(A(t))` := U + subs(x=A(t), Sum((-1)^n*x^(2*n) /(2*n)!, n=1..infinity)) : cos(A(t)) = `cos(A(t))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#%\"tG,&%\"UG\"\"\"-%$SumG6$*&*&) !\"\"%\"nGF-)F',$F5\"\"#F-F--%*factorialG6#F7F4/F5;F-%)infinityGF-" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 379 6 "Step 3" }{TEXT -1 73 ". Substitute the matrix names with their \+ corresponding matrix structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`cos(A(t))` := subs(A(t)=B(t), U=matrix(U), `cos(A(t) )`) : cos(A(t)) = `cos(A(t))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$cosG6#-%\"AG6#%\"tG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F2F1F1-%$SumG6$*& *&)!\"\"%\"nGF1)-F-6#7$7$-%$sinGF),&F*F1F1F17$F2-F%F),$F;\"\"#F1F1-%*f actorialG6#FFF:/F;;F1%)infinityGF1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 385 6 "Step 4" }{TEXT -1 23 ". Evaluate the matrix " }{TEXT 390 4 "cos(" }{TEXT -1 1 "[" }{TEXT 386 1 "A" }{TEXT 388 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 387 1 " )" }{TEXT -1 1 "]" }{TEXT 389 1 ")" }{TEXT -1 71 " with its symbolic \+ elements after truncating the series to its first " }{XPPEDIT 18 0 "5 0" "6#\"#]" }{TEXT -1 8 " terms." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "At this point, a relative disadvantage of this method appears in comparison with the previous m ethods: displaying of the matrix for a large number of terms of the \+ " }{TEXT 391 9 "Maclaurin" }{TEXT -1 29 " series becomes impractical. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Exemplarily, adopt at this point only " }{XPPEDIT 18 0 " n=2" "6#/%\"nG\"\"#" }{TEXT 531 1 "," }{TEXT -1 53 " simplify the matr ix elements, and display the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "`cos(A(t))` := map(combine, evalm(subs(A(t)=B(t), U + sum((-1)^n*A(t)^(2*n)/(2*n)!, n=1..2)))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "cos(A(t)) = matrix(`cos(A(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#%\"tG-%'matrixG6#7$7$,(#\"#\\\"# k\"\"\"*&#\"#6\"#[F4-F%6#,$F*\"\"#F4F4*&#F4\"$#>F4-F%6#,$F*\"\"%F4F4,* *&-%$sinGF)F4F*F4#!#6\"#C*&#F7FJF4FFF4!\"\"*&#F7FJF4*&-F%F)F4F*F4F4FM* &#F7FJF4FQF4FM7$\"\"!,(F1F4*&#F7F8F4F9F4FM*&F>F4F@F4F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "H owever, computational experiments show that the number of series terms adopted, " }{TEXT 409 1 "n" }{TEXT 410 1 "," }{TEXT -1 19 " should \+ be about " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 254 " to ensure a \+ satisfactory accuracy. On the other hand, it is rather unlikely that a display of such matrices with their symbolic elements would be of any use. Practically, the result of numerical computation of the matrix f unction for a given value of " }{TEXT 392 1 "t" }{TEXT -1 19 " is re quired only." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Consequently, " }{XPPEDIT 18 0 "n=50" "6#/%\"n G\"#]" }{TEXT -1 81 " is adopted and the resultant matrix is added a \+ distinguishing subscript, say, " }{TEXT 393 3 "M_s" }{TEXT -1 2 " " }{TEXT 400 1 "(" }{TEXT 411 1 "M" }{TEXT 412 9 "aclaurin\222" }{TEXT -1 3 "s " }{TEXT 413 1 "s" }{TEXT -1 5 "eries" }{TEXT 401 1 ")" } {TEXT -1 70 " to its name for future comparative purposes. Moreover, \+ the function " }{TEXT 402 7 "combine" }{TEXT -1 62 " is dropped, which results in some saving of computation time." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Thus, the final expression for " }{TEXT 712 4 "cos(" }{TEXT -1 1 "[" }{TEXT 708 1 "A " }{TEXT 710 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 709 1 ")" } {TEXT -1 1 "]" }{TEXT 711 1 ")" }{TEXT -1 48 " computed by this metho d is adopted as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`cos(A(t))[M_s]`:= evalm(subs(A(t)=B(t), U + sum((-1)^n*A(t)^(2*n)/(2 *n)!, n=1..50))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "cos(A (t))[M_s] = subs(x=B(t), matrix(U) + Sum((-1)^n*x^(2*n)/(2*n)!, n=1..5 0)) ; B(t) := 'B(t)' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6 #-%\"AG6#%\"tG6#%$M_sG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F5F4F4-%$SumG6$*& *&)!\"\"%\"nGF4)-F06#7$7$-%$sinGF*,&F+F4F4F47$F5-F&F*,$F>\"\"#F4F4-%*f actorialG6#FIF=/F>;F4\"#]F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 403 51 "Comparison of numerical results of computation of " }{TEXT 420 4 "cos(" }{TEXT 421 1 "[" }{TEXT 416 1 "A" }{TEXT 418 1 "(" }{XPPEDIT 422 0 "t" "6#%\"tG" }{TEXT 417 1 ")" }{TEXT 423 1 "]" }{TEXT 419 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 404 4 "co s(" }{TEXT -1 1 "[" }{TEXT 405 1 "A" }{TEXT 407 1 "(" }{XPPEDIT 18 0 " t" "6#%\"tG" }{TEXT 406 1 ")" }{TEXT -1 1 "]" }{TEXT 408 1 ")" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 30 " usi ng all the three methods." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "t := 2 :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 28 "(a) For Method 1 using the " }{TEXT 414 15 "Cayle y-Hamilton" }{TEXT -1 10 " theorem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "`cos(A(t))[C_H]` := map(x->eval(x), `cos(A(t))[C_H]` ) : cos(A('t'))[C_H] = matrix(`cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6#7$7$-F&6# -%$sinG6#\"\"#,$*&,&-F&6#-F&F7\"\"\"F3!\"\"F?,&F>F@F5F?F@!\"$7$\"\"!F< " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-point evaluation yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`cos(A(t))[C_H]` := evalf(matrix(`cos(A(t))[C_H ]`)) : cos(A('t'))[C_H] = matrix(`cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6#7$7$$ \"+@G+Vh!#5$!+**o;)z'F57$$\"\"!F:$\"+fK`Y\"*F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "(b) For Meth od 2 using similarity of matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "`cos(A(t))[s_m]` := map(x->eval(x), `cos(A(t))[s_m]` ) : cos(A('t'))[s_m] = matrix(`cos(A(t))[s_m]`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$-F&6# -%$sinG6#\"\"#,$*&,&F3\"\"\"-F&6#-F&F7!\"\"F<,&F?F@F5F " 0 "" {MPLTEXT 1 0 101 "`cos(A(t))[s_m]` := evalf(matrix(`cos(A(t))[s_m ]`)) : cos(A('t'))[s_m] = matrix(`cos(A(t))[s_m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$$ \"+@G+Vh!#5$!+**o;)z'F57$$\"\"!F:$\"+fK`Y\"*F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(c) For Meth od 3 using truncated " }{TEXT 415 10 "Maclaurin\222" }{TEXT -1 10 "s \+ series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`cos(A(t))[M_s] ` := evalf(map(x->eval(x), `cos(A(t))[M_s]`)) :" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "cos(A('t'))[M_s] = matrix(`cos(A(t))[M_s]`) ; t := 't' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\" tG6#%$M_sG-%'matrixG6#7$7$$\"+AG+Vh!#5$!+'*o;)z'F57$$\"\"!F:$\"+fK`Y\" *F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 125 "The matrices of (a) and (b) are precisely equal and th e matrix of (c) may be considered practically equal to the former ones ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 " " {TEXT 706 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 259 "" 0 "" {TEXT 424 9 "Example 2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " } {TEXT 429 4 "sin(" }{TEXT -1 1 "[" }{TEXT 425 1 "A" }{TEXT 427 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 426 1 ")" }{TEXT -1 1 "]" }{TEXT 428 1 ")" }{TEXT -1 8 " of a " }{TEXT 431 1 "(" }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 430 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 432 1 ")" }{TEXT -1 11 " matrix [" }{TEXT 433 1 "A" }{TEXT 435 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 434 1 ")" }{TEXT -1 11 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A(t) := matrix(2, 2, [4*t^2+3*t, 0, 3*t*sin(2*t), -5*t+2]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7$,&*$)F'\"\"#\" \"\"\"\"%*&\"\"$F1F'F1F1\"\"!7$,$*&F'F1-%$sinG6#,$F'F0F1F4,&F'!\"&F0F1 " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The power series of " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG 6#%\"xG" }{TEXT -1 21 " is convergent for " }{TEXT 440 3 "all" } {TEXT -1 2 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 44 " [refer to Unit (23)], which implies that " }{TEXT 437 4 "sin(" }{TEXT -1 1 "[ " }{TEXT 436 1 "A" }{TEXT -1 1 "]" }{TEXT 441 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 439 5 "every" }{TEXT -1 57 " square matri x. This, in turn, means that the function " }{TEXT 445 4 "sin(" } {TEXT -1 1 "[" }{TEXT 438 1 "A" }{TEXT 443 1 "(" }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT 442 1 ")" }{TEXT -1 1 "]" }{TEXT 444 1 ")" }{TEXT -1 38 " is well defined for every value of " }{TEXT 446 1 "t" }{TEXT 447 1 "," }{TEXT -1 82 " so it may be computed for this matrix. See a lso the note at the end of Method 1." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "For convenience in comp utations, let" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B(t) := A( t) : A(t) := 'A(t)' :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 449 8 "Method 1" }{TEXT -1 13 ". Using th e " }{TEXT 448 15 "Cayley-Hamilton" }{TEXT -1 9 " theorem" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 450 6 "Step 1" }{TEXT -1 23 ". Having in mind that " }{XPPEDIT 18 0 " n=2" "6#/%\"nG\"\"#" }{TEXT -1 51 " for this matrix, write equation ( 1) of Unit (23) " }{TEXT 451 1 "B" }{TEXT -1 7 " for [" }{TEXT 452 1 "A" }{TEXT 454 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 453 1 ")" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "`sin(A(t) )` := a[1] * A(t) + a[0]*U : sin(A(t)) = `sin(A(t))` ; Sin(A(t)) : = `sin(A(t))` :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#-%\"AG6#% \"tG,&*&&%\"aG6#\"\"\"F0F'F0F0*&&F.6#\"\"!F0%\"UGF0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 456 6 "S tep 2" }{TEXT -1 24 ". With the unit matrix [" }{TEXT 455 1 "U" } {TEXT -1 53 "] as in Example 1, evaluate the matrix equation for " } {TEXT 461 4 "sin(" }{TEXT -1 1 "[" }{TEXT 457 1 "A" }{TEXT 459 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 458 1 ")" }{TEXT -1 1 "]" }{TEXT 460 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`sin(A(t))` := evalm(subs(A(t)=B(t), `sin(A(t))`)) : sin(A(t)) = matrix(`sin(A(t)) `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#-%\"AG6#%\"tG-%'matr ixG6#7$7$,&*&&%\"aG6#\"\"\"F5,&*$)F*\"\"#F5\"\"%*&\"\"$F5F*F5F5F5F5&F3 6#\"\"!F5F?7$,$*(F2F5F*F5-F%6#,$F*F9F5F<,&*&F2F5,&F*!\"&F9F5F5F5F=F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 462 6 "Step 3" }{TEXT -1 38 ". Formulate equation (2) of Unit (2 3) " }{TEXT 463 1 "B" }{TEXT -1 28 " corresponding to this case:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "`r(l)` := subs(A(t)=l, U=1, \+ Sin(A(t))) : r(l) = `r(l)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"rG6#%'lambdaG,&*&&%\"aG6#\"\"\"F-F'F-F-&F+6#\"\"!F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 464 6 "S tep 4" }{TEXT -1 32 ". Determine the eigenvalues of [" }{TEXT 465 1 "A " }{TEXT 467 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 466 1 ")" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "charroots (A(t)) := eigenvals(subs(A(t)=B(t), A(t))) : char_roots(A(t)) = char roots(A(t)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+char_rootsG6#-%\" AG6#%\"tG6$,&*$)F*\"\"#\"\"\"\"\"%*&\"\"$F0F*F0F0,&F*!\"&F/F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Extracting the individual eigenvalues " }{XPPEDIT 18 0 "lambda [i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "No_roots(A(t)) := nops([charroots(A(t))]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "for i to No_roots(A(t)) \+ do ch_r[i](A(t)) := charroots(A(t))[i] : print(l[i] = ch_r[i](A(t)) ) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"\",&*$) %\"tG\"\"#F'\"\"%*&\"\"$F'F+F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %'lambdaG6#\"\"#,&%\"tG!\"&F'\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 468 6 "Step 5" }{TEXT -1 93 ". Taking notice of the fact that the roots are distinct, formulate equa tion (3) of Unit (23) " }{TEXT 469 1 "B" }{TEXT -1 19 " in a general f orm:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 20 "Since the function " }{XPPEDIT 18 0 "f(lambda)" "6#-% \"fG6#%'lambdaG" }{TEXT -1 31 " corresponding to this case is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`f(l)` := sin(l) : f(l) = \+ `f(l)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%'lambdaG-%$sinGF &" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "therefore, equation (3) assumes the form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Eq3_f(l) := `f(l)` = `r(l)` : Eq3 _f(l) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%'lambdaG,&*&&%\" aG6#\"\"\"F-F'F-F-&F+6#\"\"!F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 470 6 "Step 6" }{TEXT -1 80 ". Obtai n a set of equations by substituting either eigenvalue into equation ( 3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i to No_roots(A (t)) do Eq3_f[i](l) := subs(l=ch_r[i](A(t)), Eq3_f(l)) : print(Eq3_ f[i](l)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&*$)% \"tG\"\"#\"\"\"\"\"%*&\"\"$F,F*F,F,,&*&&%\"aG6#F,F,F'F,F,&F36#\"\"!F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,&%\"tG!\"&\"\"#\"\"\",& *&&%\"aG6#F+F+F'F+F+&F/6#\"\"!F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 471 6 "Step 7" }{TEXT -1 64 ". So lve the simultaneous equations for the unknown coefficients:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "solution := solve(\{Eq3_f[1] (l), Eq3_f[2](l)\}, \{a[0], a[1]\}) : solution ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$/&%\"aG6#\"\"!,$*&,**&-%$sinG6#,&%\"tG\"\"&\"\"#!\" \"\"\"\")F1F3F5!\"%*(\"\"$F5F-F5F1F5F4*(F2F5F1F5-F.6#,&*$F6F5\"\"%*&F9 F5F1F5F5F5F5*&F3F5F;F5F4F5,(F>F3*&F?F5F1F5F5F5F4F4#F5F3/&F&6#F5,$*&,&F ;F4F-F4F5FBF4#F4F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Extracting either unknown from the soluti on set yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "assign(so lution) : for i to No_roots(A(t)) do a[i-1] := a[i-1] : print(eva ln(a[i-1]) = a[i-1]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6#\"\"!,$*&,**&-%$sinG6#,&%\"tG\"\"&\"\"#!\"\"\"\"\")F0F2F4!\"%*(\" \"$F4F,F4F0F4F3*(F1F4F0F4-F-6#,&*$F5F4\"\"%*&F8F4F0F4F4F4F4*&F2F4F:F4F 3F4,(F=F2*&F>F4F0F4F4F4F3F3#F4F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6#\"\"\",$*&,&-%$sinG6#,&*$)%\"tG\"\"#F'\"\"%*&\"\"$F'F1F'F'!\"\" -F,6#,&F1\"\"&F2F6F6F',(F/F2*&F3F'F1F'F'F'F6F6#F6F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "After s ome re-arrangements, both coefficients are written in more compact for ms, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "a[0]:=((5*t-2) *sin(4*t^2+3*t) - (4*t^2+3*t)*sin(5*t-2))/(4*t^2+8*t-2) : a[1]:=nume r(a[1])/denom(a[1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "fo r i to No_roots(A(t)) do print(evaln(a[i-1]) = a[i-1]) : od :" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"!*&,&*&,&%\"tG\"\"&\"\"#! \"\"\"\"\"-%$sinG6#,&*$)F,F.F0\"\"%*&\"\"$F0F,F0F0F0F0*&F4F0-F26#F+F0F /F0,(F5F7*&\"\")F0F,F0F0F.F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#\"\"\"*&,&-%$sinG6#,&*$)%\"tG\"\"#F'\"\"%*&\"\"$F'F0F'F'F'-F+6#, &F0\"\"&F1!\"\"F'F',(F.F2*&\"\")F'F0F'F'F1F9F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 881 6 "Step 8" } {TEXT -1 61 ". Substitute the values of the coefficients into the matr ix " }{TEXT 874 4 "sin(" }{TEXT -1 1 "[" }{TEXT 870 1 "A" }{TEXT 872 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 871 1 ")" }{TEXT -1 1 "]" } {TEXT 873 2 ")," }{TEXT -1 91 " simplify the elements of the resultan t matrix, and add a distinguishing subscript, say, " }{TEXT 875 3 "C_ H" }{TEXT -1 2 " " }{TEXT 876 1 "(" }{TEXT 877 1 "C" }{TEXT 878 6 "ay ley-" }{TEXT 879 1 "H" }{TEXT 880 8 "amilton)" }{TEXT -1 46 " to its \+ name for future comparative purposes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`sin(A(t))[C_H]` := map(normal, map(x->eval(x), `sin( A(t))`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sin(A(t))[C_H ] = matrix(`sin(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-% $sinG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6#7$7$-F&6#,&*$)F+\"\"#\"\"\"\" \"%*&\"\"$F9F+F9F9\"\"!7$,$*&*(,&F3F9-F&6#,&F+\"\"&F8!\"\"F9F9F+F9-F&6 #,$F+F8F9F9,(F6F8*&F:F9F+F9F9F9FGFG#F " 0 "" {MPLTEXT 1 0 59 "for i \+ to No_roots(A(t)) do a[i-1] := evaln(a[i-1]) : od :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 933 4 "N .B." }{TEXT -1 28 " In general, the function " }{TEXT 931 4 "sin(" } {TEXT -1 1 "[" }{TEXT 927 1 "A" }{TEXT 929 1 "(" }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT 928 1 ")" }{TEXT -1 1 "]" }{TEXT 930 1 ")" }{TEXT -1 3