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-----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 259 4 "(24)" }{TEXT -1 1 " " }{TEXT 257 49 "Functions of ma trices comprising linear functions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1224 10 "OBJECTIVES" }{TEXT 1225 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 1226 1 "\225" }{TEXT -1 221 " To provide five step-by-st ep solved examples for computation of matrix functions with real and c omplex elements using in each case all the three methods of computatio n of matrix functions that are employed in Unit (23)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1228 1 "\225" }{TEXT -1 145 " To provide a comp arison of the resultant matrix functions by computing the function of \+ each matrix for a given numerical value of the variable." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1229 1 "\225" }{TEXT -1 33 " To include computa tion of the " }{TEXT 1230 5 "limit" }{TEXT -1 105 " of the resultant matrix function having a discontinuity on the interval of convergence of the function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1231 1 "\225" } {TEXT -1 77 " To point out a disadvantage of computing matrix functio ns using truncated " }{TEXT 1232 10 "Maclaurin\222" }{TEXT -1 61 "s \+ series if a display of the resultant matrix was essential." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1233 1 "\225" }{TEXT -1 30 " To reveal a diffic ulty when " }{TEXT 1234 5 "Maple" }{TEXT -1 122 " does not return any \+ symbolic solutions to a system of transcendental equations and show ho w to get around such a problem." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "restart : with(lina lg, augment, coldim, diag, eigenvals, eigenvects, equal, exponential, \+ rowdim) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 85 "This Unit deals with computation of functions of s quare matrices whose elements are " }{TEXT 269 16 "linear functions" }{TEXT -1 11 " of type " }{XPPEDIT 18 0 "a*t+b" "6#,&*&%\"aG\"\"\"% \"tGF&F&%\"bGF&" }{TEXT 837 1 "," }{TEXT -1 9 " where " }{TEXT 1173 1 "t" }{TEXT -1 8 " is a " }{TEXT 1172 4 "real" }{TEXT -1 16 " vari able and " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 28 " are constant coefficients. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The exponential function of matrices whose elements are l inear functions of type " }{XPPEDIT 18 0 "a*t" "6#*&%\"aG\"\"\"%\"tGF %" }{TEXT 836 1 "," }{TEXT -1 6 " or " }{TEXT 831 5 "exp([" }{TEXT 832 1 "A" }{TEXT 833 2 "] " }{TEXT 834 1 "t" }{TEXT 835 2 ")," }{TEXT -1 126 " is a special case. It is treated separately in more detail i n Unit (30) together with specific applications of the function." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "A function " }{TEXT 273 1 "f" }{TEXT -1 16 " of a matrix [" }{TEXT 270 1 "A" }{TEXT 272 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 271 1 ")" }{TEXT -1 59 "] whose elements are linear functions of a re al variable " }{TEXT 274 1 "t" }{TEXT -1 43 " may be computed only i f the function is " }{TEXT 275 12 "well defined" }{TEXT -1 49 " for \+ the matrix. [ Refer to Unit (23) for the " }{TEXT 276 12 "well-defin ed" }{TEXT -1 13 " function. ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "A well-defined function of a matrix [" }{TEXT 277 1 "A" }{TEXT 279 1 "(" }{XPPEDIT 18 0 "t" "6#% \"tG" }{TEXT 278 1 ")" }{TEXT -1 31 "] may be computed using the " }{TEXT 280 15 "Cayley-Hamilton" }{TEXT -1 43 " theorem, similarity of matrices, or the " }{TEXT 260 9 "Maclaurin" }{TEXT -1 55 " series r epresentation 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 -1 306 "However, it should be borne in mind that none of the methods is universal since each has limitati ons of different nature \226 refer to Units (22) and (23). Whether a g iven method is suitable for computation of a matrix function, may be a ssessed only when both the sought-for function and the matrix are know n." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "In the examples provided hereunder, the matrices have be en carefully constructed such that all the three methods may be used f or computation of functions " }{TEXT 454 2 "f(" }{TEXT -1 1 "[" } {TEXT 450 1 "A" }{TEXT 452 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 451 1 ")" }{TEXT -1 1 "]" }{TEXT 453 1 ")" }{TEXT -1 164 " that are w ell defined for these matrices and can be determined without a great c omputational effort. This has been done for illustrating and comparati ve purposes." }}}{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 285 4 "cos(" }{TEXT -1 1 "[" }{TEXT 281 1 "A" }{TEXT 283 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 282 1 ")" }{TEXT -1 1 "]" }{TEXT 284 1 ")" } {TEXT -1 9 " for a " }{TEXT 287 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 286 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 288 1 ")" } {TEXT -1 11 " matrix [" }{TEXT 289 1 "A" }{TEXT 291 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 290 1 ")" }{TEXT -1 11 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A(t) := matrix(2, 2, [t+2, 0 , -t, 2*t-1]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"AG6#%\"tG-%'matrixG6#7$7$,&F'\"\"\"\"\"#F.\"\"!7$,$F'!\"\",&F'F/F. F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The power series of " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG 6#%\"xG" }{TEXT -1 21 " is convergent for " }{TEXT 334 3 "all" } {TEXT -1 2 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 44 " [refer to Unit (23)], which implies that " }{TEXT 331 4 "cos(" }{TEXT -1 1 "[ " }{TEXT 330 1 "A" }{TEXT -1 1 "]" }{TEXT 335 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 333 5 "every" }{TEXT -1 57 " square matri x. This, in turn, means that the function " }{TEXT 339 4 "cos(" } {TEXT -1 1 "[" }{TEXT 332 1 "A" }{TEXT 337 1 "(" }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT 336 1 ")" }{TEXT -1 1 "]" }{TEXT 338 1 ")" }{TEXT -1 38 " is well defined for every value of " }{TEXT 340 1 "t" }{TEXT 341 1 "," }{TEXT -1 82 " so it may be computed for this matrix. See a lso the note at the end of Method 1." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "For convenience in comp utations, let" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B(t) := A( t) : A(t) := 'A(t)' :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 293 8 "Method 1" }{TEXT -1 13 ". Using th e " }{TEXT 292 15 "Cayley-Hamilton" }{TEXT -1 9 " theorem" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 6 "Step 1" }{TEXT -1 23 ". Having in mind that " }{XPPEDIT 18 0 " n=2" "6#/%\"nG\"\"#" }{TEXT -1 51 " for this matrix, write equation ( 1) of Unit (23) " }{TEXT 295 1 "B" }{TEXT -1 7 " for [" }{TEXT 296 1 "A" }{TEXT 298 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 297 1 ")" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "`cos(A(t) )` := a[1] * A(t) + a[0]*U : cos(A(t)) = `cos(A(t))` ; Cos(A(t)) : = `cos(A(t))` :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#% \"tG,&*&&%\"aG6#\"\"\"F0F'F0F0*&&F.6#\"\"!F0%\"UGF0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 300 6 "S tep 2" }{TEXT -1 40 ". Bearing in mind that the unit matrix [" }{TEXT 299 1 "U" }{TEXT -1 38 "] appropriately sized for this case is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U := diag(1, 1) : U = matr ix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7$7$\"\" \"\"\"!7$F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "evaluate the matrix equation for " } {TEXT 305 4 "cos(" }{TEXT -1 1 "[" }{TEXT 301 1 "A" }{TEXT 303 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 302 1 ")" }{TEXT -1 1 "]" }{TEXT 304 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`cos(A(t))` := evalm(subs(A(t)=B(t), `cos(A(t))`)) : cos(A(t)) = matrix(`cos(A(t)) `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#%\"tG-%'matr ixG6#7$7$,&*&&%\"aG6#\"\"\"F5,&F*F5\"\"#F5F5F5&F36#\"\"!F5F:7$,$*&F2F5 F*F5!\"\",&*&F2F5,&F*F7F5F>F5F5F8F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 306 6 "Step 3" }{TEXT -1 38 ". Formulate equation (2) of Unit (23) " }{TEXT 307 1 "B" }{TEXT -1 28 " corresponding to this case:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "l := lambda : `r(l)` := subs(A(t)=l, U=1, Cos(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 308 6 "Step 4" }{TEXT -1 33 ". De termine the eigenvalues of [" }{TEXT 309 1 "A" }{TEXT 311 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 310 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$,&F*\" \"\"\"\"#F-,&F*F.F-!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Extracting the individual eigenval ues " }{XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " \+ yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "No_roots(A(t)) : = nops([charroots(A(t))]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "for i to No_roots(A(t)) do ch_r[i](A(t)) := charroots(A(t))[i] \+ : print(l[i] = ch_r[i](A(t))) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"\",&%\"tGF'\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6#\"\"#,&%\"tGF'\"\"\"!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 6 "S tep 5" }{TEXT -1 93 ". Taking notice of the fact that the roots are di stinct, formulate equation (3) of Unit (23) " }{TEXT 313 1 "B" }{TEXT -1 19 " in a general form:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Since the function " }{XPPEDIT 18 0 "f(lambda)" "6#-%\"fG6#%'lambdaG" }{TEXT -1 31 " corresponding t o this case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "`f(l)` := cos(l) : f(l) = `f(l)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%'lambdaG-%$cosGF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "therefore, equation (3) assumes th e 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#/-%$cosG6 #%'lambdaG,&*&&%\"aG6#\"\"\"F-F'F-F-&F+6#\"\"!F-" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 6 "Step 6" } {TEXT -1 80 ". Obtain a set of equations by substituting either eigenv alue into equation (3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i to No_roots(A(t)) do Eq3_f[i](l) := subs(l=ch_r[i](A(t)), Eq3_ f(l)) : print(Eq3_f[i](l)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%\"tG\"\"\"\"\"#F),&*&&%\"aG6#F)F)F'F)F)&F.6#\"\"!F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%\"tG\"\"#\"\"\"!\"\",&*& &%\"aG6#F*F*F'F*F*&F/6#\"\"!F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 315 6 "Step 7" }{TEXT -1 64 ". Solve the simultaneous equations for the unknown coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "solution := solve(\{Eq3_f[1](l), Eq 3_f[2](l)\}, \{a[0], a[1]\}) : solution ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/&%\"aG6#\"\"\"*&,&-%$cosG6#,&%\"tG\"\"#F(!\"\"F(-F,6 #,&F/F(F0F(F1F(,&F/F(\"\"$F1F1/&F&6#\"\"!,$*&,**&F2F(F/F(!\"#F2F(*&F/F (F+F(F(*&F0F(F+F(F(F(F5F1F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Extracting either unknown from t he solution set yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " assign(solution) : for i to No_roots(A(t)) do a[i-1] := a[i-1] : \+ print(evaln(a[i-1]) = a[i-1]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"!,$*&,**&-%$cosG6#,&%\"tG\"\"\"\"\"#F1F1F0F1!\"#F,F 1*&F0F1-F-6#,&F0F2F1!\"\"F1F1*&F2F1F5F1F1F1,&F0F1\"\"$F8F8F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\"*&,&-%$cosG6#,&%\"tG\"\"#F'! \"\"F'-F+6#,&F.F'F/F'F0F',&F.F'\"\"$F0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "After some re-arr angements, the expressions for both coefficients assume more compact f orms, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "a[0] := ((2*t -1)*cos(t+2) - (t+2)*cos(2*t-1))/(t-3) : a[1] := numer(a[1])/denom(a [1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for i to No_roots (A(t)) do print(evaln(a[i-1]) = a[i-1]) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"!*&,&*&,&%\"tG\"\"#\"\"\"!\"\"F.-%$cosG6#, &F,F.F-F.F.F.*&F3F.-F16#F+F.F/F.,&F,F.\"\"$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\"*&,&-%$cosG6#,&%\"tG\"\"#F'!\"\"F'-F+6#, &F.F'F/F'F0F',&F.F'\"\"$F0F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 6 "Step 8" }{TEXT -1 61 ". Substit ute the values of the coefficients into the matrix " }{TEXT 321 4 "co s(" }{TEXT -1 1 "[" }{TEXT 317 1 "A" }{TEXT 319 1 "(" }{XPPEDIT 18 0 " t" "6#%\"tG" }{TEXT 318 1 ")" }{TEXT -1 1 "]" }{TEXT 320 2 ")," } {TEXT -1 91 " simplify the elements of the resultant matrix, and add \+ a distinguishing subscript, say, " }{TEXT 322 3 "C_H" }{TEXT -1 2 " \+ " }{TEXT 429 1 "(" }{TEXT 430 1 "C" }{TEXT 431 6 "ayley-" }{TEXT 432 1 "H" }{TEXT 433 8 "amilton)" }{TEXT -1 46 " to its name for future c omparative purposes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "`co s(A(t))[C_H]` := map(simplify, 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#,&F+\"\"\"\"\"#F6\"\"!7$*&*&,&-F&6#, &F+F7F6!\"\"F@F3F6F6F+F6F6,&F+F6\"\"$F@F@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 329 4 "N.B." } {TEXT -1 28 " In general, the function " }{TEXT 327 4 "cos(" }{TEXT -1 1 "[" }{TEXT 323 1 "A" }{TEXT 325 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT 324 1 ")" }{TEXT -1 1 "]" }{TEXT 326 1 ")" }{TEXT -1 38 " is well defined for every value of " }{TEXT 328 1 "t" }{TEXT 343 1 "." }{TEXT -1 99 " This is true if the function elements of the matrix fu nction are continuous functions for every " }{TEXT 342 1 "t" }{TEXT 344 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 52 "It can be easily noticed that the function element " }{XPPEDIT 18 0 "a[21](t)" "6#-&%\"aG6#\"#@6#%\"tG" }{TEXT -1 46 " \+ in the above matrix has a discontinuity at " }{XPPEDIT 18 0 "t = 3" "6#/%\"tG\"\"$" }{TEXT 1205 1 "." }{TEXT -1 27 " Therefore, the funct ion " }{TEXT 1202 4 "cos(" }{TEXT -1 1 "[" }{TEXT 1198 1 "A" }{TEXT 1200 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1199 1 ")" }{TEXT -1 1 "]" }{TEXT 1201 1 ")" }{TEXT -1 23 " may be computed for " } {XPPEDIT 18 0 "t=3" "6#/%\"tG\"\"$" }{TEXT -1 50 " only if a limiting value of the matrix element " }{XPPEDIT 18 0 "a[21](t)" "6#-&%\"aG6# \"#@6#%\"tG" }{TEXT -1 15 " exists when " }{TEXT 1203 1 "t" }{TEXT -1 12 " tends to " }{TEXT 1204 1 "3" }{TEXT -1 69 ". Consequently, \+ this must checked by computing the limit as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`limit(a[21](t))` := limit(`cos(A(t))[C_ H]`[2,1], t=3) : Limit(a[21](t), t=3) = `limit(a[21](t))` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-&%\"aG6#\"#@6#%\"tG/F-\" \"$,$-%$sinG6#\"\"&F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "In computational practice, the ab ove check may be omitted and an attempt made to compute the value of t he function " }{TEXT 349 4 "cos(" }{TEXT -1 1 "[" }{TEXT 345 1 "A" } {TEXT 347 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 346 1 ")" }{TEXT -1 1 "]" }{TEXT 348 1 ")" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "t=3" " 6#/%\"tG\"\"$" }{TEXT -1 53 " directly as the limit of the matrix fun ction when " }{TEXT 350 1 "t" }{TEXT -1 12 " tends to " }{TEXT 351 1 "3" }{TEXT -1 8 ". Thus," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`lim cos(A(t))[C_H]` := map(limit, `cos(A(t))[C_H]`, t=3) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Limit(cos(A(t))[C_H], t=3) = matrix(`lim cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% &LimitG6$&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG/F.\"\"$-%'matrixG6#7$7$-F)6# \"\"&\"\"!7$,$-%$sinGF9F2F8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-point evaluation yields " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`lim cos(A(t))[C_H]` := evalf(matrix(`lim cos(A(t))[C_H]`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Limit(cos(A(t))[C_H], t=3) = matrix(`lim cos(A(t))[C_ H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$&-%$cosG6#-%\"AG 6#%\"tG6#%$C_HG/F.\"\"$-%'matrixG6#7$7$$\"+b=iOG!#5$\"\"!F<7$$!+CGxwG! \"*F8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "[ For computation of " }{TEXT 1227 6 "limits" }{TEXT -1 45 " of function matrices, refer to Unit (28). ]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 409 8 "Method \+ 2" }{TEXT -1 30 ". Using similarity of matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 352 6 "Step 1" } {TEXT -1 32 ". Check if the eigenvalues of [" }{TEXT 353 1 "A" } {TEXT 355 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 354 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 eigenva lues are found to be distinct." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 360 6 "Step 2" }{TEXT -1 10 ". Find \+ the" }{TEXT 356 1 " " }{TEXT -1 63 "sequence of lists containing eigen values and eigenvectors of [" }{TEXT 357 1 "A" }{TEXT 359 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 358 1 ")" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`roots&vectors(A(t))` := eig envects(subs(A(t)=B(t), A(t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "roots_and_vectors(A(t)) = `roots&vectors(A(t))` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#-%\"AG6#%\"tG6 $7%,&F*\"\"#\"\"\"!\"\"F/<#-%'vectorG6#7$\"\"!F/7%,&F*F/F.F/F/<#-F36#7 $*&,&F*F/\"\"$F0F/F*F0F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 361 6 "Step 3" }{TEXT -1 28 ". Extract th e eigenvectors " }{XPPEDIT 18 0 "v[lambda[i]]" "6#&%\"vG6#&%'lambdaG6 #%\"iG" }{TEXT -1 32 " corresponding to eigenvalues " }{XPPEDIT 18 0 "lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 7 " of [" }{TEXT 1209 1 "A" }{TEXT 1211 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1210 1 ") " }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for i to No_roots(A(t)) do e[i] := charroots(A(t))[i] : List[i] := `root s&vectors(A(t))`[i] : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "for j to No_roots(A(t)) do for i to No_roots(A(t)) do if Lis t[i][1] = e[j] then Lst[j] := `roots&vectors(A(t))`[i] : fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "for i to No_ro ots(A(t)) do ch_v[i](A(t)) := op(Lst[i][3]) : print(v[lambda[i]] = \+ eval(ch_v[i](A(t))), ` --> ` * lambda[i] = ch_r[i](A(t))) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"\"-%'vector G6#7$*&,&%\"tGF*\"\"$!\"\"F*F1F3F*/*&%(~~-->~~GF*F'F*,&F1F*\"\"#F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"#-%'vectorG6# 7$\"\"!\"\"\"/*&%(~~-->~~GF0F'F0,&%\"tGF*F0!\"\"" }}}{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#&%'la mbdaG6#%\"iG" }{TEXT -1 24 " into column matrices " }{XPPEDIT 18 0 " X[lambda[i]]" "6#&%\"XG6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "for i to No_roots(A(t)) do \+ X[l[i]] := convert(ch_v[i](A(t)), matrix) : print(X[l[i]] = matrix( X[l[i]])) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lamb daG6#\"\"\"-%'matrixG6#7$7#*&,&%\"tGF*\"\"$!\"\"F*F2F47#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"#-%'matrixG6#7$7#\" \"!7#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 363 6 "Step 4" }{TEXT -1 40 ". Construct the 'unique' \+ modal matrix [" }{TEXT 362 1 "M" }{TEXT 381 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 380 1 ")" }{TEXT -1 21 "] associated with [" } {TEXT 385 1 "A" }{TEXT 387 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 386 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "i := 'i' : `seq(X[l[i]])` := seq(X[l[i]], i=1..coldim(B(t))) : M(t) := augment(`seq(X[l[i]])`) : 'M(t)'=M(t) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"MG6#%\"tG-%'matrixG6#7$7$*&,&F'\"\"\"\"\"$!\"\"F /F'F1\"\"!7$F/F/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 6 "Step 5" }{TEXT -1 24 ". Construct the uni que " }{TEXT 364 6 "Jordan" }{TEXT -1 9 " form [" }{TEXT 382 1 "J" }{TEXT 384 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 383 1 ")" }{TEXT -1 8 "] of [" }{TEXT 373 1 "A" }{TEXT 375 1 "(" }{XPPEDIT 18 0 "t" " 6#%\"tG" }{TEXT 374 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "J_A := array(diagonal, 1..rowdim(B(t)), 1..coldim(B (t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "for i to rowdim (B(t)) do for j to coldim(B(t)) do if j = i then J_A[i,j] := ch_r [i](A(t)) fi : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J(t) := matrix(J_A) : 'J(t)' = J(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"JG6#%\"tG-%'matrixG6#7$7$,&F'\"\"\"\"\"#F.\"\"!7$F 0,&F'F/F.!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 367 6 "Step 6" }{TEXT -1 24 ". Compute the funct ion " }{TEXT 366 6 "cosine" }{TEXT -1 14 " of matrix [" }{TEXT 388 1 "J" }{TEXT 390 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 389 1 ")" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "J_A := ar ray(diagonal, 1..rowdim(J(t)), 1..coldim(J(t))) :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 114 "for i to rowdim(J(t)) do for j to coldim(J (t)) do if j = i then J_A[i,j] := cos(J(t)[i,j]) fi : od : od \+ :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`cos(J(t))` := matrix( J_A) : cos('J(t)') = matrix(`cos(J(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"JG6#%\"tG-%'matrixG6#7$7$-F%6#,&F*\"\"\" \"\"#F3\"\"!7$F5-F%6#,&F*F4F3!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 372 6 "Step 7" }{TEXT -1 11 ". Compute " }{TEXT 369 4 "cos(" }{TEXT -1 1 "[" }{TEXT 376 1 "A" } {TEXT 378 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 377 1 ")" }{TEXT -1 1 "]" }{TEXT 379 4 ") = " }{TEXT -1 1 "[" }{TEXT 394 1 "M" }{TEXT 396 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 395 1 ")" }{TEXT -1 2 "] " }{TEXT 370 4 "cos(" }{TEXT -1 1 "[" }{TEXT 391 1 "J" }{TEXT 393 1 " (" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 392 1 ")" }{TEXT -1 1 "]" } {TEXT 371 1 ")" }{TEXT -1 1 " " }{TEXT 368 3 "Inv" }{TEXT -1 1 "[" } {TEXT 397 1 "M" }{TEXT 399 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 398 1 ")" }{TEXT -1 1 "]" }{TEXT 400 1 "," }{TEXT -1 91 " simplify th e elements of the resultant matrix, and add a distinguishing subscript , say, " }{TEXT 401 3 "s_m" }{TEXT -1 3 " (" }{TEXT 427 1 "s" } {TEXT -1 13 "imilarity of " }{TEXT 428 1 "m" }{TEXT -1 53 "atrices) to its name for future comparative purposes:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 76 "`cos(A(t))[s_m]` := map(simplify, evalm(M(t) &* `co s(J(t))` &* M(t)^(-1))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "cos(A(t))[s_m] = matrix(`cos(A(t))[s_m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$-F&6#,& F+\"\"\"\"\"#F6\"\"!7$,$*&*&F+F6,&-F&6#,&F+F7F6!\"\"F6F3FAF6F6,&F+F6\" \"$FAFAFAF>" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The symbolic forms of matrices " }{XPPEDIT 18 0 "cos(A(t))[C_H]" "6#&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "cos(A(t))[s_m]" "6#&-%$cosG6#-%\"AG6#%\"tG6#%$s _mG" }{TEXT -1 12 " are equal." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 410 8 "Method 3" }{TEXT -1 19 ". Us ing truncated " }{TEXT 434 10 "Maclaurin\222" }{TEXT -1 9 "s series " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 402 6 "Step 1" }{TEXT -1 28 ". Find the eigenvalues of [" } {TEXT 579 1 "A" }{TEXT 581 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 580 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "This step was already done earlier to check if the eigenvalues of [" }{TEXT 758 1 "A" }{TEXT 760 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 759 1 ")" }{TEXT -1 71 "] are dis tinct, which was essential for both preceding methods. It is " }{TEXT 405 3 "not" }{TEXT -1 48 " necessary for this method since the series \+ of " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 21 " is co nvergent for " }{TEXT 407 3 "all" }{TEXT -1 2 " " }{XPPEDIT 18 0 "x " "6#%\"xG" }{TEXT 404 1 "," }{TEXT -1 22 " which implies that " } {TEXT 403 4 "cos(" }{TEXT -1 1 "[" }{TEXT 412 1 "A" }{TEXT 414 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 413 1 ")" }{TEXT -1 1 "]" }{TEXT 408 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 406 5 "every" }{TEXT -1 16 " square matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 415 6 "Step 2" }{TEXT -1 28 ". Write the expression for " }{TEXT 416 10 "Maclaurin\222" }{TEXT -1 37 "s \+ series representing the function " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG 6#%\"xG" }{TEXT -1 37 " and substitute the matrix name for " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 104 "`cos(A(t))` := U + subs(x=A(t), Sum((-1)^n*x^(2*n) /(2*n)!, n=1..infinity)) : cos(A(t)) = `cos(A(t))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#%\"tG,&%\"UG\"\"\"-%$SumG6$*&*&) !\"\"%\"nGF-)F',$F5\"\"#F-F--%*factorialG6#F7F4/F5;F-%)infinityGF-" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 411 6 "Step 3" }{TEXT -1 73 ". Substitute the matrix names with their \+ corresponding matrix structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`cos(A(t))` := subs(A(t)=B(t), U=matrix(U), `cos(A(t) )`) : cos(A(t)) = `cos(A(t))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$cosG6#-%\"AG6#%\"tG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F2F1F1-%$SumG6$*& *&)!\"\"%\"nGF1)-F-6#7$7$,&F*F1\"\"#F1F27$,$F*F:,&F*FBF1F:,$F;FBF1F1-% *factorialG6#FFF:/F;;F1%)infinityGF1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 417 6 "Step 4" }{TEXT -1 23 ". Evaluate the matrix " }{TEXT 422 4 "cos(" }{TEXT -1 1 "[" }{TEXT 418 1 "A" }{TEXT 420 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 419 1 " )" }{TEXT -1 1 "]" }{TEXT 421 1 ")" }{TEXT -1 71 " with its symbolic \+ elements after truncating the series to its first " }{XPPEDIT 18 0 "5 0" "6#\"#]" }{TEXT -1 8 " terms." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "At this point, a relative disadvantage of this method appears in comparison with the previous m ethods: displaying of the matrix for a large number of terms of the \+ " }{TEXT 423 9 "Maclaurin" }{TEXT -1 29 " series becomes impractical. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Exemplarily, adopt at this point only " }{XPPEDIT 18 0 " n=2" "6#/%\"nG\"\"#" }{TEXT 604 1 "," }{TEXT -1 89 " simplify the elem ent polynomials, and sort the terms of the matrix elements. (Note that " }{TEXT 424 5 "Maple" }{TEXT -1 89 " sorts the terms of polynomials \+ into descending order.) These operations yield the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "`cos(A(t))` := map(sort, map(simpl ify, evalm(subs(A(t)=B(t), U + sum((-1)^n*A(t)^(2*n)/(2*n)!, n=1..2))) )) : cos(A(t)) = matrix(`cos(A(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#-%\"AG6#%\"tG-%'matrixG6#7$7$,,*$)F*\"\"%\"\"\"#F4\"#C *&#F4\"\"$F4)F*F9F4F4*&#F4\"\"#F4)F*F=F4F4*&#F=F9F4F*F4!\"\"#F4F9FA\" \"!7$,*F1#!\"&\"\")*&#\"\"&F6F4*$F:F4F4FA*&#\"\"(FHF4F>F4F4*&#FOF6F4F* F4F4,,F1#F=F9*&#F3F9F4FLF4FA*$F>F4FA*&#FKF9F4F*F4F4#\"#8F6F4" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "However, computational experiments show that the number of seri es terms adopted, " }{TEXT 444 1 "n" }{TEXT 445 1 "," }{TEXT -1 19 " \+ should be about " }{XPPEDIT 18 0 "50" "6#\"#]" }{TEXT -1 254 " to e nsure a satisfactory accuracy. On the other hand, it is rather unlikel y that a display of such matrices with their symbolic elements would b e of any use. Practically, the result of numerical computation of the \+ matrix function for a given value of " }{TEXT 425 1 "t" }{TEXT -1 19 " is required only." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Consequently, " }{XPPEDIT 18 0 "n =50" "6#/%\"nG\"#]" }{TEXT -1 81 " is adopted and the resultant matri x is added a distinguishing subscript, say, " }{TEXT 426 3 "M_s" } {TEXT -1 2 " " }{TEXT 435 1 "(" }{TEXT 446 1 "M" }{TEXT 447 9 "aclaur in\222" }{TEXT -1 3 "s " }{TEXT 448 1 "s" }{TEXT -1 5 "eries" }{TEXT 436 1 ")" }{TEXT -1 71 " to its name for future comparative purposes. Moreover, the functions " }{TEXT 437 8 "simplify" }{TEXT -1 5 " and \+ " }{TEXT 449 4 "sort" }{TEXT -1 72 " are dropped, which results in a s ignificant saving of computation time." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Thus, the final e xpression for " }{TEXT 812 4 "cos(" }{TEXT -1 1 "[" }{TEXT 808 1 "A" }{TEXT 810 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 809 1 ")" }{TEXT -1 1 "]" }{TEXT 811 1 ")" }{TEXT -1 48 " computed by this method is a dopted as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`cos( A(t))[M_s]`:= evalm(subs(A(t)=B(t), U + sum((-1)^n*A(t)^(2*n)/(2*n)!, \+ n=1..50))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "cos(A(t))[M _s] = subs(x=B(t), matrix(U) + Sum((-1)^n*x^(2*n)/(2*n)!, n=1..50)) ; B(t) := 'B(t)' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"A G6#%\"tG6#%$M_sG,&-%'matrixG6#7$7$\"\"\"\"\"!7$F5F4F4-%$SumG6$*&*&)!\" \"%\"nGF4)-F06#7$7$,&F+F4\"\"#F4F57$,$F+F=,&F+FEF4F=,$F>FEF4F4-%*facto rialG6#FIF=/F>;F4\"#]F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 438 51 "Comparison of numerical results o f computation of " }{TEXT 461 4 "cos(" }{TEXT 462 1 "[" }{TEXT 457 1 "A" }{TEXT 459 1 "(" }{XPPEDIT 463 0 "t" "6#%\"tG" }{TEXT 458 1 ")" } {TEXT 464 1 "]" }{TEXT 460 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 439 4 "cos( " }{TEXT -1 1 "[" }{TEXT 440 1 "A" }{TEXT 442 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 441 1 ")" }{TEXT -1 1 "]" }{TEXT 443 1 ")" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "t=4" "6#/%\"tG\"\"%" }{TEXT -1 30 " usi ng all the three methods." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "t := 4 :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 28 "(a) For Method 1 using the " }{TEXT 455 15 "Cayle y-Hamilton" }{TEXT -1 10 " theorem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "`cos(A(t))[C_H]` := map(x->eval(x), `cos(A(t))[C_H]` ) : cos(A('t'))[C_H] = matrix(`cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6#7$7$-F&6# \"\"'\"\"!7$,&-F&6#\"\"(!\"%*&\"\"%\"\"\"F3F?F?F9" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-p oint evaluation yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 " `cos(A(t))[C_H]` := evalf(matrix(`cos(A(t))[C_H]`)) : cos(A('t'))[C_ H] = matrix(`cos(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&- %$cosG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6#7$7$$\"+nGq,'*!#5$\"\"!F77$$ \"*I@2D)!\"*$\"+VD-RvF5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "(b) For Method 2 using similarity \+ of matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "`cos(A(t)) [s_m]` := map(x->eval(x), `cos(A(t))[s_m]`) : cos(A('t'))[s_m] = mat rix(`cos(A(t))[s_m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6# -%\"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$-F&6#\"\"'\"\"!7$,&-F&6#\"\"(!\"% *&\"\"%\"\"\"F3F?F?F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-point evaluation yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`cos(A(t))[s_m]` := eval f(matrix(`cos(A(t))[s_m]`)) : cos(A('t'))[s_m] = matrix(`cos(A(t))[s _m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#% $s_mG-%'matrixG6#7$7$$\"+nGq,'*!#5$\"\"!F77$$\"*I@2D)!\"*$\"+VD-RvF5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(c) For Method 3 using truncated " }{TEXT 456 10 "Maclau rin\222" }{TEXT -1 10 "s series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`cos(A(t))[M_s]` := evalf(map(x->eval(x), `cos(A(t))[ M_s]`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "cos(A('t'))[M_ s] = matrix(`cos(A(t))[M_s]`) ; t := 't' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%$cosG6#-%\"AG6#%\"tG6#%$M_sG-%'matrixG6#7$7$$\"+mGq ,'*!#5$\"\"!F77$$\"+#H@2D)F5$\"+VD-RvF5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "The matrices of \+ (a) and (b) are precisely equal and the matrix of (c) may be considere d practically equal to the former ones." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 806 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 465 9 "Example 2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 470 5 "sinh(" } {TEXT -1 1 "[" }{TEXT 466 1 "A" }{TEXT 468 1 "(" }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT 467 1 ")" }{TEXT -1 1 "]" }{TEXT 469 1 ")" }{TEXT -1 9 " for a " }{TEXT 472 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 471 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 473 1 ")" }{TEXT -1 11 " matrix [" }{TEXT 474 1 "A" }{TEXT 476 1 "(" }{XPPEDIT 18 0 "t" "6# %\"tG" }{TEXT 475 1 ")" }{TEXT -1 11 "] given as" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "A(t) := matrix(2, 2, [6*t, -5*t, 2*t, 4*t]) \+ : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG- %'matrixG6#7$7$,$F'\"\"',$F'!\"&7$,$F'\"\"#,$F'\"\"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The pow er series of " }{XPPEDIT 18 0 "sinh(x)" "6#-%%sinhG6#%\"xG" }{TEXT -1 21 " is convergent for " }{TEXT 481 3 "all" }{TEXT -1 2 " " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 44 " [refer to Unit (23)], whic h implies that " }{TEXT 478 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 477 1 "A " }{TEXT -1 1 "]" }{TEXT 482 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 480 5 "every" }{TEXT -1 57 " square matrix. This, in turn, means that the function " }{TEXT 486 5 "sinh(" }{TEXT -1 1 "[" } {TEXT 479 1 "A" }{TEXT 484 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 483 1 ")" }{TEXT -1 1 "]" }{TEXT 485 1 ")" }{TEXT -1 38 " is well def ined for every value of " }{TEXT 487 1 "t" }{TEXT 488 1 "," }{TEXT -1 40 " so it may be computed for this matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "For convenie nce 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 490 8 "Method 1" }{TEXT -1 13 ". Using the " }{TEXT 489 15 "Cayley-Hamilton" }{TEXT -1 9 " theorem " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 491 6 "Step 1" }{TEXT -1 23 ". Having in mind that " }{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"#" }{TEXT -1 51 " for this matrix, write equa tion (1) of Unit (23) " }{TEXT 492 1 "B" }{TEXT -1 7 " for [" }{TEXT 493 1 "A" }{TEXT 495 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 494 1 " )" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`sinh (A(t))` := a[1] * A(t) + a[0]*U : sinh(A(t)) = `sinh(A(t))` ; Sinh (A(t)) := `sinh(A(t))` :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6 #-%\"AG6#%\"tG,&*&&%\"aG6#\"\"\"F0F'F0F0*&&F.6#\"\"!F0%\"UGF0F0" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 497 6 "Step 2" }{TEXT -1 24 ". With the unit matrix [" }{TEXT 496 1 "U " }{TEXT -1 53 "] as in Example 1, evaluate the matrix equation for \+ " }{TEXT 502 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 498 1 "A" }{TEXT 500 1 " (" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 499 1 ")" }{TEXT -1 1 "]" } {TEXT 501 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`sinh(A( t))` := evalm(subs(A(t)=B(t), `sinh(A(t))`)) : sinh(A(t)) = matrix(` sinh(A(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#-%\"AG6#% \"tG-%'matrixG6#7$7$,&*&&%\"aG6#\"\"\"F5F*F5\"\"'&F36#\"\"!F5,$F1!\"&7 $,$F1\"\"#,&F1\"\"%F7F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 503 6 "Step 3" }{TEXT -1 38 ". Formulate \+ equation (2) of Unit (23) " }{TEXT 504 1 "B" }{TEXT -1 28 " correspond ing to this case:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`r(l)` := subs(A(t)=l, U=1, Sinh(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 505 6 "Step 4" }{TEXT -1 32 ". Determine the eigenvalues of [" }{TEXT 506 1 "A" }{TEXT 508 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 507 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "charroots(A(t)) := eigenvals(subs(A(t)=B(t), A(t))) : char_r oots(A(t)) = charroots(A(t)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%+ char_rootsG6#-%\"AG6#%\"tG6$*&^$\"\"&\"\"$\"\"\"F*F0*&^$F.!\"$F0F*F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Extracting the individual eigenvalues " }{XPPEDIT 18 0 " lambda[i]" "6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "No_roots(A(t)) := nops([charroots(A (t))]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "for i to No_ro ots(A(t)) do ch_r[i](A(t)) := charroots(A(t))[i] : print(l[i] = ch_ r[i](A(t))) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'lambdaG6# \"\"\"*&^$\"\"&\"\"$F'%\"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'l ambdaG6#\"\"#*&^$\"\"&!\"$\"\"\"%\"tGF," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Note that the eig envalues are complex numbers." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 509 6 "Step 5" }{TEXT -1 93 ". Takin g notice of the fact that the roots are distinct, formulate equation ( 3) of Unit (23) " }{TEXT 510 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 37 "`f(l)` := sinh(l) : f(l) = `f(l)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%'lambdaG-%%sinh GF&" }}}{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#/-%%sinhG6#%'lambdaG,&*&&% \"aG6#\"\"\"F-F'F-F-&F+6#\"\"!F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 511 6 "Step 6" }{TEXT -1 80 ". Ob tain a set of equations by substituting either eigenvalue into equatio n (3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for i to No_root s(A(t)) do Eq3_f[i](l) := subs(l=ch_r[i](A(t)), Eq3_f(l)) : print(E q3_f[i](l)) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#*&^ $\"\"&\"\"$\"\"\"%\"tGF+,&*(F(F+&%\"aG6#F+F+F,F+F+&F06#\"\"!F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#*&^$\"\"&!\"$\"\"\"%\"tGF+, &*(F(F+&%\"aG6#F+F+F,F+F+&F06#\"\"!F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 512 6 "Step 7" }{TEXT -1 64 ". Solve the simultaneous equations for the unknown coefficients :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "solution := solve(\{Eq 3_f[1](l), Eq3_f[2](l)\}, \{a[0], a[1]\}) : solution ;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#<$/&%\"aG6#\"\"!,*-%%sinhG6#*&^$\"\"&!\"$\"\"\"% \"tGF1#F1\"\"#*&^##F/\"\"'F1-F+6#*&^$F/\"\"$F1F2F1F1F1*&^##!\"&F8F1F*F 1F1*&F3F1F9F1F1/&F&6#F1*&*&^##!\"\"F8F1,&F9F1F*FJF1F1F2FJ" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "E xtracting either unknown from the solution set yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "assign(solution) : for i to No_r oots(A(t)) do a[i-1] := a[i-1] : print(evaln(a[i-1]) = a[i-1]) : \+ od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"!,*-%%sinhG6#*&^$ \"\"&!\"$\"\"\"%\"tGF0#F0\"\"#*&^##F.\"\"'F0-F*6#*&^$F.\"\"$F0F1F0F0F0 *&^##!\"&F7F0F)F0F0*&F2F0F8F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#\"\"\"*&*&^##!\"\"\"\"'F',&-%%sinhG6#*&^$\"\"&\"\"$F'%\"tGF'F'-F 06#*&^$F4!\"$F'F6F'F,F'F'F6F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "After some re-arrangements, \+ the expressions for both coefficients are written in more compact form s, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "a[0] := (1/2+5/ 6*I)*sinh((5+3*I)*t) + (1/2-5/6*I)*sinh((5-3*I)*t) : a[1] := numer(a [1])/denom(a[1]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for i to No_roots(A(t)) do print(evaln(a[i-1]) = a[i-1]) : od :" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"!,&*&^$#\"\"\"\"\"##\"\"& \"\"'F,-%%sinhG6#*&^$F/\"\"$F,%\"tGF,F,F,*&^$F+#!\"&F0F,-F26#*&^$F/!\" $F,F7F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#\"\"\"*&*&^##! \"\"\"\"'F',&-%%sinhG6#*&^$\"\"&\"\"$F'%\"tGF'F'-F06#*&^$F4!\"$F'F6F'F ,F'F'F6F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 513 6 "Step 8" }{TEXT -1 61 ". Substitute the values of t he coefficients into the matrix " }{TEXT 517 5 "sinh(" }{TEXT -1 1 "[ " }{TEXT 514 1 "A" }{TEXT 516 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 515 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`sinh(A(t))` := map(x->eval(x), `sinh(A(t))`) : sinh(A(t)) = matrix(`sinh(A(t))`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6# -%\"AG6#%\"tG-%'matrixG6#7$7$,(*&^#!\"\"\"\"\",&-F%6#*&^$\"\"&\"\"$F4F *F4F4-F%6#*&^$F:!\"$F4F*F4F3F4F4*&^$#F4\"\"##F:\"\"'F4F6F4F4*&^$FC#!\" &FFF4F " 0 "" {MPLTEXT 1 0 76 "`sinh(A(t))`[1,1] := (1/2-I/6)*sinh ((5+3*I)*t) + (1/2+I/6)*sinh((5-3*I)*t) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "`sinh(A(t))`[1,2] := 5*I/6*(sinh((5+3*I)*t) - sinh( (5-3*I)*t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`sinh(A(t) )`[2,1] := -I/3*(sinh((5+3*I)*t) - sinh((5-3*I)*t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "`sinh(A(t))`[2,2] := (1/2+I/6)*sinh ((5+3*I)*t) + (1/2-I/6)*sinh((5-3*I)*t) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 126 "for i to rowdim(`sinh(A(t))`) do for j to coldim( `sinh(A(t))`) do print(sinh(A(t))[i,j] = `sinh(A(t))`[i,j]) : od : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6$ \"\"\"F-,&*&^$#F-\"\"##!\"\"\"\"'F--F&6#*&^$\"\"&\"\"$F-F+F-F-F-*&^$F1 #F-F5F--F&6#*&^$F:!\"$F-F+F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& -%%sinhG6#-%\"AG6#%\"tG6$\"\"\"\"\"#*&^##\"\"&\"\"'F-,&-F&6#*&^$F2\"\" $F-F+F-F--F&6#*&^$F2!\"$F-F+F-!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6$\"\"#\"\"\"*&^##!\"\"\"\"$F.,&-F&6#*&^$ \"\"&F3F.F+F.F.-F&6#*&^$F9!\"$F.F+F.F2F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6$\"\"#F-,&*&^$#\"\"\"F-#F2\"\"'F2-F&6# *&^$\"\"&\"\"$F2F+F2F2F2*&^$F1#!\"\"F4F2-F&6#*&^$F9!\"$F2F+F2F2F2" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Substitute the above elements into the matrix " }{TEXT 1179 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 1175 1 "A" }{TEXT 1177 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1176 1 ")" }{TEXT -1 1 "]" }{TEXT 1178 1 ") " }{TEXT -1 44 " and add a distinguishing subscript, say, " }{TEXT 518 3 "C_H" }{TEXT -1 2 " " }{TEXT 519 1 "(" }{TEXT 520 1 "C" }{TEXT 521 6 "ayley-" }{TEXT 522 1 "H" }{TEXT 523 8 "amilton)" }{TEXT -1 70 " to the name of the resultant matrix for future comparative purposes: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "`sinh(A(t))[C_H]` := ma trix(`sinh(A(t))`) : sinh(A(t))[C_H] = matrix(`sinh(A(t))[C_H]`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6#%$C_HG-%' matrixG6#7$7$,&*&^$#\"\"\"\"\"##!\"\"\"\"'F7-F&6#*&^$\"\"&\"\"$F7F+F7F 7F7*&^$F6#F7F;F7-F&6#*&^$F@!\"$F7F+F7F7F7*&^##F@F;F7,&F " 0 "" {MPLTEXT 1 0 59 "for i to No _roots(A(t)) do a[i-1] := evaln(a[i-1]) : od :" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 524 8 "Method 2" }{TEXT -1 30 ". Using similarity of matrices" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 525 6 "Step 1" }{TEXT -1 32 ". Check if the eigenvalues of [" }{TEXT 526 1 "A" }{TEXT 528 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 527 1 ")" }{TEXT -1 16 "] a re distinct:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "See Step 4 above where the eigenvalues are foun d to be distinct." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 533 6 "Step 2" }{TEXT -1 10 ". Find the" }{TEXT 529 1 " " }{TEXT -1 63 "sequence of lists containing eigenvalues and e igenvectors of [" }{TEXT 530 1 "A" }{TEXT 532 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 531 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`roots&vectors(A(t))` := eigenvects(subs(A(t)=B( t), A(t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "roots_and_v ectors(A(t)) = `roots&vectors(A(t))` ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%2roots_and_vectorsG6#-%\"AG6#%\"tG6$7%*&^$\"\"&\"\"$\"\"\"F*F 1F1<#-%'vectorG6#7$,$*&,&*&^$!\"&!\"$F1F*F1F1*&\"\"%F1F*F1F1F1F*!\"\"# F@\"\"#F17%*&^$F/F=F1F*F1F1<#-F46#7$,$*&,&*&^$F " 0 "" {MPLTEXT 1 0 103 "for i to No_roots(A (t)) do e[i] := charroots(A(t))[i] : List[i] := `roots&vectors(A(t) )`[i] : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "for j to No_roots(A(t)) do for i to No_roots(A(t)) do if List[i][1] = e[j] \+ then Lst[j] := `roots&vectors(A(t))`[i] : fi : od : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "for i to No_roots(A(t)) do \+ ch_v[i](A(t)) := op(Lst[i][3]) : print(v[lambda[i]] = eval(ch_v[i]( A(t))), ` --> ` * lambda[i] = ch_r[i](A(t))) : od :" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$/&%\"vG6#&%'lambdaG6#\"\"\"-%'vectorG6#7$,$*&,&* &^$!\"&!\"$F*%\"tGF*F**&\"\"%F*F6F*F*F*F6!\"\"#F9\"\"#F*/*&%(~~-->~~GF *F'F**&^$\"\"&\"\"$F*F6F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"vG6# &%'lambdaG6#\"\"#-%'vectorG6#7$,$*&,&*&^$!\"&\"\"$\"\"\"%\"tGF6F6*&\" \"%F6F7F6F6F6F7!\"\"#F:F*F6/*&%(~~-->~~GF6F'F6*&^$\"\"&!\"$F6F7F6" }}} {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 column matrices \+ " }{XPPEDIT 18 0 "X[lambda[i]]" "6#&%\"XG6#&%'lambdaG6#%\"iG" }{TEXT -1 8 " yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "for i to No_roots(A(t)) do X[l[i]] := convert(ch_v[i](A(t)), matrix) : prin t(X[l[i]] = matrix(X[l[i]])) : od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"\"-%'matrixG6#7$7#,$*&,&*&^$!\"&!\"$F*%\"t GF*F**&\"\"%F*F7F*F*F*F7!\"\"#F:\"\"#7#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"XG6#&%'lambdaG6#\"\"#-%'matrixG6#7$7#,$*&,&*&^$!\" &\"\"$\"\"\"%\"tGF7F7*&\"\"%F7F8F7F7F7F8!\"\"#F;F*7#F7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 536 6 "S tep 4" }{TEXT -1 40 ". Construct the 'unique' modal matrix [" }{TEXT 535 1 "M" }{TEXT 538 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 537 1 " )" }{TEXT -1 21 "] associated with [" }{TEXT 539 1 "A" }{TEXT 541 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 540 1 ")" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "i := 'i' : `seq(X[l[i]])` := seq(X[l[i]], i=1..coldim(B(t))) : M(t) := augment(`seq(X[l[i]])` ) : 'M(t)'=M(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"MG6#%\"tG- %'matrixG6#7$7$,$*&,&*&^$!\"&!\"$\"\"\"F'F4F4*&\"\"%F4F'F4F4F4F'!\"\"# F7\"\"#,$*&,&*&^$F2\"\"$F4F'F4F4*&F6F4F'F4F4F4F'F7F87$F4F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 543 6 "S tep 5" }{TEXT -1 24 ". Construct the unique " }{TEXT 542 6 "Jordan" } {TEXT -1 9 " form [" }{TEXT 547 1 "J" }{TEXT 549 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 548 1 ")" }{TEXT -1 8 "] of [" }{TEXT 544 1 " A" }{TEXT 546 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 545 1 ")" } {TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "J_A := ar ray(diagonal, 1..rowdim(B(t)), 1..coldim(B(t))) :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 112 "for i to rowdim(B(t)) do for j to coldim(B (t)) do if j = i then J_A[i,j] := ch_r[i](A(t)) fi : od : od : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J(t) := matrix(J_A) : \+ 'J(t)' = J(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"JG6#%\"tG-%'m atrixG6#7$7$*&^$\"\"&\"\"$\"\"\"F'F1\"\"!7$F2*&^$F/!\"$F1F'F1" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 551 6 "Step 6" }{TEXT -1 24 ". Compute the function " }{TEXT 550 15 " hyperbolic sine" }{TEXT -1 14 " of matrix [" }{TEXT 552 1 "J" } {TEXT 554 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 553 1 ")" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "J_A := array(di agonal, 1..rowdim(J(t)), 1..coldim(J(t))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for i to rowdim(J(t)) do for j to coldim(J(t)) d o if j = i then J_A[i,j] := sinh(J(t)[i,j]) fi : od : od :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`sinh(J(t))` := matrix(J_A) \+ : sinh('J(t)') = matrix(`sinh(J(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#-%\"JG6#%\"tG-%'matrixG6#7$7$-F%6#*&^$\"\"& \"\"$\"\"\"F*F6\"\"!7$F7-F%6#*&^$F4!\"$F6F*F6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 559 6 "Step 7" } {TEXT -1 11 ". Compute " }{TEXT 556 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 560 1 "A" }{TEXT 562 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 561 1 " )" }{TEXT -1 1 "]" }{TEXT 563 4 ") = " }{TEXT -1 1 "[" }{TEXT 567 1 "M " }{TEXT 569 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 568 1 ")" } {TEXT -1 2 "] " }{TEXT 557 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 564 1 "J" }{TEXT 566 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 565 1 ")" }{TEXT -1 1 "]" }{TEXT 558 1 ")" }{TEXT -1 1 " " }{TEXT 555 3 "Inv" }{TEXT -1 1 "[" }{TEXT 570 1 "M" }{TEXT 572 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT 571 1 ")" }{TEXT -1 53 "] and simplify the elements of the r esultant matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "`sinh( A(t))`:=map(simplify, evalm(M(t) &* `sinh(J(t))` &* M(t)^(-1))) : sinh (A(t))=matrix(`sinh(A(t))`) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%%s inhG6#-%\"AG6#%\"tG-%'matrixG6#7$7$*&^##!\"\"\"\"'\"\"\",*-F%6#*&^$\" \"&\"\"$F5F*F5F5*&^#FF5F?F5F5F5*&^ ##!\"&F4F5,&F7F3F?F5F57$*&^##F5F \+ " 0 "" {MPLTEXT 1 0 127 "for i to rowdim(`sinh(A(t))`) do for j to co ldim(`sinh(A(t))`) do `sinh(A(t))`[i,j] := `sinh(A(t))[C_H]`[i,j] : \+ od : od :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Substitute the above elements into the matrix \+ " }{TEXT 1180 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 1181 1 "A" }{TEXT 1183 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1182 1 ")" }{TEXT -1 1 "]" } {TEXT 1184 2 ") " }{TEXT -1 43 " and add a distinguishing subscript, s ay, " }{TEXT 573 3 "s_m" }{TEXT -1 3 " (" }{TEXT 574 1 "s" }{TEXT -1 13 "imilarity of " }{TEXT 575 1 "m" }{TEXT -1 77 "atrices) to the n ame of the resultant matrix for future comparative purposes:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "`sinh(A(t))[s_m]` := matrix( `sinh(A(t))`) : sinh(A(t))[s_m] = matrix(`sinh(A(t))[s_m]`) ;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6#%$s_mG-%'ma trixG6#7$7$,&*&^$#\"\"\"\"\"##!\"\"\"\"'F7-F&6#*&^$\"\"&\"\"$F7F+F7F7F 7*&^$F6#F7F;F7-F&6#*&^$F@!\"$F7F+F7F7F7*&^##F@F;F7,&F " 0 "" {MPLTEXT 1 0 100 "`sinh(A( t))` := subs(x=A(t), Sum(x^(2*n+1)/(2*n+1)!, n=0..infinity)) : sinh( A(t)) = `sinh(A(t))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#- %\"AG6#%\"tG-%$SumG6$*&)F',&%\"nG\"\"#\"\"\"F3F3-%*factorialG6#F0!\"\" /F1;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 596 6 "Step 3" }{TEXT -1 69 ". Substitute the matrix name with its corresponding matrix structure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`sinh(A(t))` := subs(A(t)=B(t), `si nh(A(t))`) : sinh(A(t)) = `sinh(A(t))` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#-%\"AG6#%\"tG-%$SumG6$*&)-%'matrixG6#7$7$,$F *\"\"',$F*!\"&7$,$F*\"\"#,$F*\"\"%,&%\"nGF;\"\"\"F@F@-%*factorialG6#F> !\"\"/F?;\"\"!%)infinityG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 597 6 "Step 4" }{TEXT -1 23 ". Evaluate t he matrix " }{TEXT 602 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 598 1 "A" } {TEXT 600 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 599 1 ")" }{TEXT -1 1 "]" }{TEXT 601 1 ")" }{TEXT -1 71 " with its symbolic elements a fter truncating the series to its first " }{XPPEDIT 18 0 "50" "6#\"#] " }{TEXT -1 8 " terms." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Since displaying of the matrix for a large number of terms of the " }{TEXT 603 9 "Maclaurin" }{TEXT -1 31 " series is impractical, only " }{XPPEDIT 18 0 "n=3" "6#/%\"nG\" \"$" }{TEXT -1 142 " is adopted at this point and the matrix thus obt ained is displayed for instructive purposes upon simplification and so rting of its elements:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 " `sinh(A(t))`:=map(sort, map(simplify, evalm(subs(A(t)=B(t), sum(A(t)^( 2*n+1)/(2*n+1)!, n=0..3))))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sinh(A(t)) = matrix(`sinh(A(t))`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%sinhG6#-%\"AG6#%\"tG-%'matrixG6#7$7$,**$)F*\"\"(\" \"\"#!%yZ\"$0\"*&#\"$V'\"#:F4*$)F*\"\"&F4F4!\"\"*&#\"#G\"\"$F4)F*FCF4F 4*&\"\"'F4F*F4F4,*F1#\"%8d\"$E\"*&#\"$R#FFF4FF4 F*F4F?7$,*F1#!%8d\"$:$*&#FMF;F4F=F4F4*&\"#AF4FDF4F4*&\"\"#F4F*F4F4,*F1 #!%@')FU*&#\"$%HF>F4F " 0 "" {MPLTEXT 1 0 81 "`sinh(A(t))[M_s]`:= e valm(subs(A(t)=B(t), sum(A(t)^(2*n+1)/(2*n+1)!, n=0..50))) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "sinh(A(t))[M_s] = subs(x=B(t ), Sum(x^(2*n+1)/(2*n+1)!, n=0..50)) ; B(t) := 'B(t)' :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6#%$M_sG-%$SumG6$*&)-% 'matrixG6#7$7$,$F+\"\"',$F+!\"&7$,$F+\"\"#,$F+\"\"%,&%\"nGF>\"\"\"FCFC -%*factorialG6#FA!\"\"/FB;\"\"!\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 613 51 "Comparison of numerica l results of computation of " }{TEXT 618 5 "sinh(" }{TEXT 619 1 "[" } {TEXT 614 1 "A" }{TEXT 616 1 "(" }{XPPEDIT 620 0 "t" "6#%\"tG" }{TEXT 615 1 ")" }{TEXT 621 1 "]" }{TEXT 617 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 622 5 "sinh(" }{TEXT -1 1 "[" }{TEXT 623 1 "A" }{TEXT 625 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 624 1 ")" }{TEXT -1 1 "]" }{TEXT 626 1 ")" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "t=0.5" "6#/%\"tG$\"\"&! \"\"" }{TEXT -1 30 " using all the three methods." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "t := 0.5 :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(a) For Method 1 \+ using the " }{TEXT 627 15 "Cayley-Hamilton" }{TEXT -1 10 " theorem: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "`sinh(A(t))[C_H]` := m ap(x->eval(x), `sinh(A(t))[C_H]`) : sinh(A('t'))[C_H] = matrix(`sinh(A (t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#% \"tG6#%$C_HG-%'matrixG6#7$7$^$$\"+Q0&pY#!\"*$\"\"!F8^$$!+-!)[>5!\")F77 $^$$\"+1?&z2%F6$!\"!F8^$$!+q9+6;F6F7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Disregarding the meanin gless imaginary parts simplifies the above to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`sinh(A(t))[C_H]` := evalm(Re(`sinh(A(t))[C_H]` )) : sinh(A('t'))[C_H] = matrix(`sinh(A(t))[C_H]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6#%$C_HG-%'matrixG6#7$7$$ \"+Q0&pY#!\"*$!+-!)[>5!\")7$$\"+1?&z2%F5$!+q9+6;F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "(b) For M ethod 2 using similarity of matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`sinh(A(t))[s_m]`:=map(x->eval(x), `sinh(A(t))[s_m]` ) : sinh(A('t'))[s_m]=matrix(`sinh(A(t))[s_m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$^$$\"+ Q0&pY#!\"*$\"\"!F8^$$!+-!)[>5!\")F77$^$$\"+1?&z2%F6$!\"!F8^$$!+q9+6;F6 F7" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Disregarding the meaningless imaginary parts simplifies t he above to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`sinh(A(t)) [s_m]` := evalm(Re(`sinh(A(t))[s_m]`)) : sinh(A('t'))[s_m] = matrix( `sinh(A(t))[s_m]`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-% \"AG6#%\"tG6#%$s_mG-%'matrixG6#7$7$$\"+Q0&pY#!\"*$!+-!)[>5!\")7$$\"+1? &z2%F5$!+q9+6;F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(c) For Method 3 using truncated " } {TEXT 628 10 "Maclaurin\222" }{TEXT -1 10 "s series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`sinh(A(t))[M_s]` := evalf(map(x->e val(x), `sinh(A(t))[M_s]`)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sinh(A('t'))[M_s] = matrix(`sinh(A(t))[M_s]`) ; t := 't' :" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&-%%sinhG6#-%\"AG6#%\"tG6#%$M_sG-%'ma trixG6#7$7$$\"+S0&pY#!\"*$!+-!)[>5!\")7$$\"+3?&z2%F5$!+o9+6;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 the matrix of (c) may be considered practically equal to the former ones." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 807 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 629 9 "Example 3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{TEXT 634 4 "exp(" }{TEXT -1 1 "[" }{TEXT 630 1 "A" }{TEXT 632 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 631 1 ")" }{TEXT -1 1 "]" }{TEXT 633 1 ")" }{TEXT -1 9 " for a " }{TEXT 636 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 635 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 637 1 ")" }{TEXT -1 11 " matrix [" }{TEXT 638 1 "A" }{TEXT 640 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 639 1 ")" }{TEXT -1 11 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A(t) := matrix(2, 2, [0, t, -t, 0]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"AG6#%\"tG-%'matrixG6#7$7$\"\"!F'7$,$F'!\"\"F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The pow er series of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 21 " is convergent for " }{TEXT 645 3 "all" }{TEXT -1 2 " " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 44 " [refer to Unit (23)], whic h implies that " }{TEXT 642 4 "exp(" }{TEXT -1 1 "[" }{TEXT 641 1 "A " }{TEXT -1 1 "]" }{TEXT 646 1 ")" }{TEXT -1 23 " is well defined for " }{TEXT 644 5 "every" }{TEXT -1 57 " square matrix. This, in turn, means that the function " }{TEXT 650 4 "exp(" }{TEXT -1 1 "[" } {TEXT 643 1 "A" }{TEXT 648 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 647 1 ")" }{TEXT -1 1 "]" }{TEXT 649 1 ")" }{TEXT -1 38 " is well def ined for every value of " }{TEXT 651 1 "t" }{TEXT 652 1 "," }{TEXT -1 40 " so it may be computed for this matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 1239 5 "* * *" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 1243 4 "N.B." }{TEXT -1 21 " The above matrix [" }{TEXT 1240 1 "A" } {TEXT 1242 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 1241 1 ")" } {TEXT -1 52 "] may be written equivalently as the product of a " } {TEXT 1236 8 "constant" }{TEXT -1 10 " matrix [" }{TEXT 1235 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " A := matrix(2, 2, [0, 1, -1, 0]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and the scalar " }{TEXT 1237 1 "t" }{TEXT 1238 1 "," }{TEXT -1 6 " viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A*t = matri x(A) * t ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"tGF&*&- %'matrixG6#7$7$\"\"!F&7$!\"\"F.F&F'F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "since the product [" }{TEXT 1249 1 "A" }{TEXT -1 2 "] " }{TEXT 1250 1 "t" }{TEXT -1 25 " evaluates to the matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`At` := evalm(A * t) : A*t = matrix(`At`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"tGF&-%'matrixG6#7$7$\"\"!F'7$,$F'!\" \"F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 86 "This easy observation has a deeper meaning when special applications of the function " }{TEXT 1244 5 "exp([" }{TEXT 1245 1 " A" }{TEXT 1246 2 "] " }{TEXT 1247 1 "t" }{TEXT 1248 1 ")" }{TEXT -1 28 " are examined in Unit (30)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A := 'A' : A(t) := matrix(`At`) :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 1251 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The exponential f unction of a square matrix may be computed in " }{TEXT 823 5 "Maple" } {TEXT -1 1 " " }{TEXT 824 8 "directly" }{TEXT -1 12 ", using the " } {TEXT 825 11 "exponential" }{TEXT -1 74 " function \226 refer also to \+ Units (23) and (24). Therefore, computation of " }{TEXT 830 4 "exp(" }{TEXT -1 1 "[" }{TEXT 826 1 "A" }{TEXT 828 1 "(" }{XPPEDIT