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0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 259 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 344 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 346 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 347 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(29)" }{TEXT 290 1 " " }{TEXT 291 44 "Integration of ma trices comprising functions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 464 10 "OBJECTIVES" }{TEXT 465 1 ":" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 474 1 "\225" }{TEXT -1 97 " To state the condition necessary for a ma trix containing functions of one variable to have an " }{TEXT 475 8 " integral" }{TEXT -1 32 " with respect to this variable." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 476 1 "\225" }{TEXT -1 34 " To introduce the co ncept of an " }{TEXT 477 10 "integrable" }{TEXT -1 9 " matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 478 1 "\225" }{TEXT -1 34 " To provide a definition of the " }{TEXT 485 10 "indefinite" }{TEXT -1 60 " integ ral of a matrix in the form of a symbolic expression." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 479 1 "\225" }{TEXT -1 103 " To provide an exam ple of an indefinite integral of a rectangular matrix whose elements a re functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 483 1 "\225" }{TEXT -1 61 " To specify and illustrate properties of matrix integration." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 484 1 "\225" }{TEXT -1 34 " To provide a definition of the " }{TEXT 486 8 "definite" }{TEXT -1 60 " integral of a matrix in the form of a symbolic expression." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 491 1 "\225" }{TEXT -1 96 " To state the condition nece ssary for a matrix containing functions of one variable to have a " } {TEXT 492 6 "proper" }{TEXT -1 41 " integral with respect to this var iable." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 487 1 "\225" }{TEXT -1 30 " T o provide an example of a " }{TEXT 488 6 "proper" }{TEXT -1 59 " int egral of a square matrix whose elements are functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 489 1 "\225" }{TEXT -1 105 " To provide alterna tive methods of exact and floating-point evaluation of a proper integr al of a matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 490 1 "\225" }{TEXT -1 205 " To show how to evaluate a proper integral of a matrix whose \+ elements are functions but the integration limits have a symbolic form and their numerical values become known in a later phase of computati on." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 500 1 "\225" }{TEXT -1 20 " To p oint out that " }{TEXT 499 5 "Maple" }{TEXT -1 14 " can compute " } {TEXT 501 8 "improper" }{TEXT -1 45 " integrals of matrices comprisin g functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 493 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 494 8 "improper" }{TEXT -1 78 " integra l of a matrix and specify the two distinct cases of such an integral. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 495 1 "\225" }{TEXT -1 78 " To prov ide two examples of computation of improper matrix integrals due to \+ " }{TEXT 497 8 "infinite" }{TEXT -1 26 " interval of integration." }} }{EXCHG {PARA 0 "" 0 "" {TEXT 496 1 "\225" }{TEXT -1 78 " To provide \+ two examples of computation of improper matrix integrals due to " } {TEXT 498 13 "discontinuity" }{TEXT -1 50 " of the integrand in the i nterval of integration." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart : with(linalg, equ al, multiply) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "If some or all elements of a matrix [" } {TEXT 366 1 "A" }{TEXT -1 7 "] are " }{TEXT 359 9 "functions" }{TEXT -1 5 " of " }{TEXT 360 3 "one" }{TEXT -1 11 " variable " }{TEXT 361 1 "t" }{TEXT 362 1 "," }{TEXT -1 44 " it is convenient to denote the \+ matrix as [" }{TEXT 363 1 "A" }{TEXT 365 1 "(" }{XPPEDIT 18 0 "t" "6#% \"tG" }{TEXT 364 1 ")" }{TEXT -1 48 "] when considering integration of such a matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 357 3 "A. " }{TEXT 358 20 "Indefinite integrals " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "When those elements of a matrix [" }{TEXT 258 1 "A" } {TEXT 293 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 292 1 ")" }{TEXT -1 26 "] that are functions are " }{TEXT 259 3 "all" }{TEXT -1 34 " \+ differentiable with respect to " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 27 " in some common interval " }{XPPEDIT 18 0 "t[0]" "6#&%\"tG6# \"\"!" }{TEXT -1 1 " " }{TEXT 466 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\" \"\"" }{TEXT 467 1 "," }{TEXT -1 11 " then an " }{TEXT 261 19 "indef inite integral" }{TEXT -1 6 " of [" }{TEXT 260 1 "A" }{TEXT 295 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 294 1 ")" }{TEXT -1 19 "] with r espect to " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 41 " may be defin ed. Then, also the matrix [" }{TEXT 468 1 "A" }{TEXT 470 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 469 1 ")" }{TEXT -1 17 "] is said t o be " }{TEXT 473 10 "integrable" }{TEXT -1 19 " in the interval " }{XPPEDIT 18 0 "t[0]" "6#&%\"tG6#\"\"!" }{TEXT -1 1 " " }{TEXT 471 1 " <" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 3 " < " } {XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT 472 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "C onsider a " }{TEXT 314 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 262 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 315 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 263 1 "A" }{TEXT 297 1 "(" }{XPPEDIT 18 0 "t" "6#% \"tG" }{TEXT 296 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 102 "A(t) := matrix(2, 3, [a[11](t), a[12](t), a[1 3](t), a[21](t), a[22](t), a[23](t)]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%-&%\"aG6#\"#6F&-& F/6#\"#7F&-&F/6#\"#8F&7%-&F/6#\"#@F&-&F/6#\"#AF&-&F/6#\"#BF&" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The indefinite integral of [" }{TEXT 264 1 "A" }{TEXT 299 1 "( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 298 1 ")" }{TEXT -1 19 "] with r espect to " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 63 " is defined t o be the matrix of the same order with elements " }{XPPEDIT 18 0 "Int (a[ij](t), t)" "6#-%$IntG6$-&%\"aG6#%#ijG6#%\"tGF," }{TEXT -1 7 ", vi z." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 "`indef_int(A)` := map(Int, A(t), t) : Int('A(t)', t) = matrix(`indef_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ IntG6$-%\"AG6#%\"tGF*-%'matrixG6#7$7%-F%6$-&%\"aG6#\"#6F)F*-F%6$-&F46# \"#7F)F*-F%6$-&F46#\"#8F)F*7%-F%6$-&F46#\"#@F)F*-F%6$-&F46#\"#AF)F*-F% 6$-&F46#\"#BF)F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation can be displayed in \"like -in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " Int(A(t), t) = matrix(`indef_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'matrixG6#7$7%-&%\"aG6#\"#66#%\"tG-&F.6#\"#7F1-&F.6# \"#8F17%-&F.6#\"#@F1-&F.6#\"#AF1-&F.6#\"#BF1F2-F(6#7$7%-F%6$F,F2-F%6$F 3F2-F%6$F7F27%-F%6$F " 0 "" {MPLTEXT 1 0 105 "A(t) := matrix(2, 3, [2*exp(-3*t), sin(t/2), cosh(2*t), 3/sqrt(t), 2*t^2, 3*ln(2*t)]) : 'A(t)' = A(t) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%,$- %$expG6#,$F'!\"$\"\"#-%$sinG6#,$F'#\"\"\"F3-%%coshG6#,$F'F37%,$*&F9F9* $-%%sqrtG6#F'F9!\"\"\"\"$,$*$)F'F3F9F3,$-%#lnGF " 0 "" {MPLTEXT 1 0 81 "`indef_int(A)` := map(int, A(t), t) : Int('A(t)', t) = matrix(`indef_int(A)`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tGF*-%'matrixG6#7$7%,$-%$ex pG6#,$F*!\"$#!\"#\"\"$,$-%$cosG6#,$F*#\"\"\"\"\"#F7,$-%%sinhG6#,$F*F@F >7%,$*$-%%sqrtG6#F*F?\"\"',$*$)F*F8F?#F@F8,&*&-%#lnGFDF?F*F?F8*&F8F?F* F?!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 39 "Int(A(t), t) = matrix(`indef_int(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'matrixG6#7$7%,$-%$expG6#, $%\"tG!\"$\"\"#-%$sinG6#,$F1#\"\"\"F3-%%coshG6#,$F1F37%,$*&F9F9*$-%%sq rtG6#F1F9!\"\"\"\"$,$*$)F1F3F9F3,$-%#lnGF \+ " 0 "" {MPLTEXT 1 0 111 "A(t) := matrix(2, 3, [t, t^3, -1, t^2, 2*t, - 3*t^2]) : B(t) := matrix(2, 3, [2, t^2, t^3, -2*t, 3*t^2, -1]) :" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "'A(t)' = A(t) ; 'B(t)' = \+ B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$ 7%F'*$)F'\"\"$\"\"\"!\"\"7%*$)F'\"\"#F0,$F'F5,$F3!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7$7%\"\"#*$)F'F-\"\"\"*$ )F'\"\"$F07%,$F'!\"#,$F.F3!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "(a) The indefinite integral \+ of the sum of the two matrices is the following " }{TEXT 324 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 325 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 326 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`indef_int(A+B)` := map(sort, map(int, evalm( A(t)+B(t)), t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int([' A(t)' + 'B(t)'], t) = matrix(`indef_int(A+B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$7#,&-%\"AG6#%\"tG\"\"\"-%\"BGF+F-F,-%'matrixG 6#7$7%,&*$)F,\"\"#F-#F-F8*&F8F-F,F-F-,&*$)F,\"\"%F-#F-F>*&#F-\"\"$F-)F ,FBF-F-,&F " 0 "" {MPLTEXT 1 0 90 "`indef_int(A) + indef_int (B)` := map(sort, evalm(map(int, A(t), t) + map(int, B(t), t))) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Int('A(t)', t) + Int('B(t)', t) = matrix(`indef_int(A) + indef_int(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$IntG6$-%\"AG6#%\"tGF+\"\"\"-F&6$-%\"BGF*F+F,-%'ma trixG6#7$7%,&*$)F+\"\"#F,#F,F9*&F9F,F+F,F,,&*$)F+\"\"%F,#F,F?*&#F,\"\" $F,)F+FCF,F,,&F=F@F+!\"\"7%,&*$FDF,FBF7FF,&FIF,F7F,,&FIFFF+FF" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 356 5 "equal" }{TEXT -1 64 " function applied to th e resultant matrices of (a) and (b), i.e." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "equal(`indef_int(A+B)`, `indef_int(A) + indef_int(B )`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "returns th e " }{TEXT 502 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%% trueG" }{TEXT -1 47 ", which verifies that both matrices are equal." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 281 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 4 "N.B." }{TEXT -1 142 " The indefinite int egral of the product of a scalar (number) and a matrix is equal to the product of the scalar and the matrix antiderivative" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Int(k*A(t),t) = k*Int(A(t),t)" "6#/-%$IntG6 $*&%\"kG\"\"\"-%\"AG6#%\"tGF)F-*&F(F)-F%6$-F+6#F-F-F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "F or example, consider a " }{TEXT 330 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"# " }{TEXT 331 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 332 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 282 1 "A" }{TEXT 313 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 312 1 ")" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A(t) := matrix(2, 3, [t, t^3 , -1, t^2, 2*t, -3*t^2]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%F'*$)F'\"\"$\"\"\"!\"\"7% *$)F'\"\"#F0,$F'F5,$F3!\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and the scalar " }{XPPEDIT 18 0 "k=3" "6#/%\"kG\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k := 3 :" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "(a) The indefinite integral of the product of the scalar and th e matrix is the following " }{TEXT 333 1 "(" }{XPPEDIT 18 0 "2" "6#\" \"#" }{TEXT 334 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 335 1 " )" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "`indef_int(kA)` := map(int, evalm(k * A(t)), t) : Int('k*A(t)', \+ t) = matrix(`indef_int(kA)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$ IntG6$*&%\"kG\"\"\"-%\"AG6#%\"tGF)F--%'matrixG6#7$7%,$*$)F-\"\"#F)#\" \"$F6,$*$)F-\"\"%F)#F8F<,$F-!\"$7%*$)F-F8F),$F4F8,$FAF?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "( b) The product of the scalar and the matrix antiderivative is the foll owing " }{TEXT 336 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 337 3 " \+ \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 338 1 ")" }{TEXT -1 9 " ma trix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "`k indef_int(A)` \+ := evalm(k * map(int, A(t), t)) : 'k' * Int('A(t)', t) = matrix(`k i ndef_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\"-%$In tG6$-%\"AG6#%\"tGF-F&-%'matrixG6#7$7%,$*$)F-\"\"#F&#\"\"$F6,$*$)F-\"\" %F&#F8F<,$F-!\"$7%*$)F-F8F&,$F4F8,$FAF?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Both matrices of \+ (a) and (b) are equal by inspection." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 283 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 4 "N .B." }{TEXT -1 32 " The indefinite integral of a " }{TEXT 480 4 "zer o" }{TEXT -1 10 " matrix [" }{TEXT 285 1 "0" }{TEXT -1 15 "] is the s ame " }{TEXT 481 4 "zero" }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "Int(`0`,t)" "6#-%$IntG6$%\"0G%\"tG" }{TEXT 339 3 " = " }{TEXT -1 1 "[" }{TEXT 286 1 "0" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "F or example, consider the " }{TEXT 340 1 "(" }{XPPEDIT 18 0 "2" "6#\" \"#" }{TEXT 341 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 342 1 " )" }{TEXT -1 10 " matrix [" }{TEXT 287 1 "0" }{TEXT -1 1 "]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`0` := matrix(2, 3, [0, 0, 0 , 0, 0, 0]) : `0` = matrix(`0`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"0G-%'matrixG6#7$7%\"\"!F*F*F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The indefinite integral of the matrix [" }{TEXT 288 1 "0" }{TEXT -1 15 "] is the same " }{TEXT 482 4 "zero" }{TEXT -1 8 " matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "`indef_int(0)` := map(int, `0`, t) : Int(`0`, t) = \+ matrix(`indef_int(0)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$ %\"0G%\"tG-%'matrixG6#7$7%\"\"!F.F.F-" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 289 5 "* * *" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 367 3 "B. " }{TEXT 368 18 "Definite integrals" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "When those ele ments of a matrix [" }{TEXT 369 1 "A" }{TEXT 374 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 373 1 ")" }{TEXT -1 26 "] that are functions are \+ " }{TEXT 370 3 "all" }{TEXT -1 34 " differentiable with respect to \+ " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 24 " in a common interval \+ " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT 377 1 "," }{TEXT -1 10 " then a " }{TEXT 372 17 "definite integral" }{TEXT -1 6 " of [ " }{TEXT 371 1 "A" }{TEXT 376 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 375 1 ")" }{TEXT -1 36 "] over this interval may be defined." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 382 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 378 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 383 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 379 1 "A" }{TEXT 381 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 380 1 ")" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A(t) := matrix(2, 3, [a[11] (t), a[12](t), a[13](t), a[21](t), a[22](t), a[23](t)]) : 'A(t)' = A (t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7 %-&%\"aG6#\"#6F&-&F/6#\"#7F&-&F/6#\"#8F&7%-&F/6#\"#@F&-&F/6#\"#AF&-&F/ 6#\"#BF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 26 "The definite integral of [" }{TEXT 384 1 "A" } {TEXT 386 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 385 1 ")" }{TEXT -1 20 "] over an interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" } {TEXT -1 63 " is defined to be the matrix of the same order with elem ents " }{XPPEDIT 18 0 "Int(a[ij](t),t=a..b)" "6#-%$IntG6$-&%\"aG6#%#i jG6#%\"tG/F,;F(%\"bG" }{TEXT 387 1 "," }{TEXT -1 6 " viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`def_int(A)` := map(Int, A(t), t=a. .b) : Int('A(t)', t=a..b) = matrix(`def_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;%\"aG%\"bG-%'matrixG6#7$7 %-F%6$-&F-6#\"#6F)F+-F%6$-&F-6#\"#7F)F+-F%6$-&F-6#\"#8F)F+7%-F%6$-&F-6 #\"#@F)F+-F%6$-&F-6#\"#AF)F+-F%6$-&F-6#\"#BF)F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operati on can be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(A(t), t=a..b) = matrix(`def_int(A)`) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%'matrixG6#7$7%-&%\"aG 6#\"#66#%\"tG-&F.6#\"#7F1-&F.6#\"#8F17%-&F.6#\"#@F1-&F.6#\"#AF1-&F.6# \"#BF1/F2;F.%\"bG-F(6#7$7%-F%6$F,FH-F%6$F3FH-F%6$F7FH7%-F%6$F " 0 "" {MPLTEXT 1 0 86 "A(t) := matrix(2, 2, [9*exp(-2*t), 3/sqrt(t), sin(t/2 ), t*cos(t)]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"AG6#%\"tG-%'matrixG6#7$7$,$-%$expG6#,$F'!\"#\"\"*,$*&\"\"\"F6*$-%% sqrtG6#F'F6!\"\"\"\"$7$-%$sinG6#,$F'#F6\"\"#*&F'F6-%$cosGF&F6" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "and compute its integral over the interval " }{XPPEDIT 18 0 "[ 1,3]" "6#7$\"\"\"\"\"$" }{TEXT 394 1 "," }{TEXT -1 6 " i.e." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`prp_int(A)` := map(Int, A(t ), t=1..3) : Int('A(t)', t=1..3) = matrix(`prp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"\"\"\"$-%'mat rixG6#7$7$-F%6$,$-%$expG6#,$F*!\"#\"\"*F+-F%6$,$*&F-F-*$-%%sqrtG6#F*F- !\"\"F.F+7$-F%6$-%$sinG6#,$F*#F-\"\"#F+-F%6$*&F*F--%$cosGF)F-F+" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Exact evaluation of the resultant matrix may be performed using the function " }{TEXT 503 3 "map" }{TEXT -1 63 " together with the ar row-type procedure including the function " }{TEXT 504 5 "value" } {TEXT -1 6 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int(' A(t)', t=1..3) = map(x->value(x), `prp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$, &-%$expG6#!\"'#!\"*\"\"#*&#\"\"*F;F--F66#!\"#F-F-,&*$-%%sqrtG6#F.F-\" \"'FG!\"\"7$,&-%$cosG6##F.F;FA*&F;F--FL6##F-F;F-F-,*-FL6#F.F-*&F.F--%$ sinGFUF-F--FL6#F-FH-FXFZFH" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Floating-point evaluation of the exact resultant matrix may be done using any of the following alterna tive methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 396 8 "Method 1" }{TEXT -1 21 ". Using the funct ion " }{TEXT 395 5 "evalf" }{TEXT -1 26 " and any of the functions " } {TEXT 397 5 "evalm" }{TEXT -1 2 ", " }{TEXT 398 6 "matrix" }{TEXT -1 5 ", or " }{TEXT 399 2 "op" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 154 "Int('A(t)', t=1..3) = evalf(evalm(`prp_int(A)`)) \+ ; Int('A(t)', t=1..3) = evalf(matrix(`prp_int(A)`)) ; Int('A(t)', t =1..3) = evalf(op(`prp_int(A)`)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$IntG6$-%\"AG6#%\"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\" +X[I#R%!\"*7$$\"+?2p8;F9$!+jdS[>F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$IntG6$-%\"AG6#%\"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\" +X[I#R%!\"*7$$\"+?2p8;F9$!+jdS[>F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$IntG6$-%\"AG6#%\"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\" +X[I#R%!\"*7$$\"+?2p8;F9$!+jdS[>F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 402 8 "Method 2" }{TEXT -1 21 ". Using the function " }{TEXT 400 7 "convert" }{TEXT -1 31 " together with the form (type) " }{TEXT 401 5 "float" }{TEXT -1 26 " and any of the functions " }{TEXT 403 5 "evalm" }{TEXT -1 2 ", " }{TEXT 404 6 "m atrix" }{TEXT -1 5 ", or " }{TEXT 405 2 "op" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "Int('A(t)', t=1..3) = conve rt(evalm(`prp_int(A)`), float) ; Int('A(t)', t=1..3) = convert(matri x(`prp_int(A)`), float) ; Int('A(t)', t=1..3) = convert(op(`prp_int( A)`), float) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#% \"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\"+X[I#R%!\"*7$$\"+?2 p8;F9$!+jdS[>F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#% \"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\"+X[I#R%!\"*7$$\"+?2 p8;F9$!+jdS[>F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#% \"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\"+X[I#R%!\"*7$$\"+?2 p8;F9$!+jdS[>F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 407 8 "Method 3" }{TEXT -1 21 ". Using the funct ion " }{TEXT 406 3 "map" }{TEXT -1 63 " together with the arrow-type p rocedure including the function " }{TEXT 408 5 "evalf" }{TEXT -1 1 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int('A(t)', t=1..3) = m ap(x->evalf(x), `prp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$-%\"AG6#%\"tG/F*;\"\"\"\"\"$-%'matrixG6#7$7$$\"+)*Qayf!#5$\"+X[ I#R%!\"*7$$\"+?2p8;F9$!+jdS[>F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 409 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 411 4 "N.B." } {TEXT -1 24 " Where the end points " }{XPPEDIT 18 0 "a" "6#%\"aG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 31 " of t he integration interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" } {TEXT -1 35 " are not given in numerical form, " }{TEXT 412 5 "Maple " }{TEXT -1 61 " can evaluate the definite integral of a matrix symbol ically." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "For example, consider the same matrix [" }{TEXT 413 1 "A" }{TEXT 415 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 414 1 ")" } {TEXT -1 12 "] as before." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The integral of the matrix over th e interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 4 " i s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`prp_int(A)` := map(In t, A(t), t=a..b) : Int('A(t)', t=a..b) = matrix(`prp_int(A)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;%\"aG%\"bG- %'matrixG6#7$7$-F%6$,$-%$expG6#,$F*!\"#\"\"*F+-F%6$,$*&\"\"\"FA*$-%%sq rtG6#F*FA!\"\"\"\"$F+7$-F%6$-%$sinG6#,$F*#FA\"\"#F+-F%6$*&F*FA-%$cosGF )FAF+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Symbolic evaluation of the matrix integral over the in terval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 30 " yiel ds the following matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`prp_int(A)` := map(x->value(x), `prp_int(A)`) : Int('A(t)', t=a..b ) = matrix(`prp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$-%\"AG6#%\"tG/F*;%\"aG%\"bG-%'matrixG6#7$7$,&-%$expG6#,$F.!\"##!\"* \"\"#*&#\"\"*F<\"\"\"-F66#,$F-F9F@F@,&*$-%%sqrtG6#F.F@\"\"'*&FIF@-FG6# F-F@!\"\"7$,&-%$cosG6#,$F.#F@F " 0 "" {MPLTEXT 1 0 51 "a := Pi/4 : b := 3*Pi/2 : 'a' = a ; 'b' = b ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG,$%#PiG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG,$%#PiG#\"\"$\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "then exact evaluation of the \+ matrix may be performed as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "`prp_int(A)` := eval(subs('a'=a, 'b'=b, matrix(`prp_i nt(A)`))) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int('A(t)', \+ t=a..b) = matrix(`prp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %$IntG6$-%\"AG6#%\"tG/F*;,$%#PiG#\"\"\"\"\"%,$F.#\"\"$\"\"#-%'matrixG6 #7$7$,&-%$expG6#,$F.!\"$#!\"*F5*&#\"\"*F5F0-F=6#,$F.#!\"\"F5F0F0,&*(-% %sqrtG6#F4F0-FN6#F5F0-FN6#F.F0F4*&#F4F5F0*&-FN6#F1F0FRF0F0FJ7$,&-%$cos G6#,$F.#F4F1!\"#*&F5F0-Ffn6#,$F.#F0\"\")F0F0,*-Ffn6#F2F0*(F3F0F.F0-%$s inGFcoF0F0-Ffn6#F-FJ*&#F0F1F0*&F.F0-FfoFhoF0F0FJ" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Floating-p oint evaluation of the above matrix yields" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "`prp_int(A)` := evalf(matrix(`prp_int(A)`)) : Int ('A(t)', t=a..b) = matrix(`prp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;,$%#PiG#\"\"\"\"\"%,$F.#\"\" $\"\"#-%'matrixG6#7$7$$\"+c%\\4N*!#5$\"+F5Y2x!\"*7$$\"+FE(>E$F@$!+Gh&[ (fF@" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 410 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(2) " }{TEXT 462 18 "Improper integ rals" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 145 "When the integrand tends to infinity at some point in \+ the interval of integration or the interval itself is infinite in leng th, the integral is " }{TEXT 421 8 "improper" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 422 5 "Maple" }{TEXT -1 128 " is also able to compute integrals of mat rices if improper integrals are involved. Consider the two cases of im proper integrals." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 423 6 "CASE 1" }{TEXT -1 61 ". Improper integral due to infinite interval of integration " }{TEXT 416 1 "[" }{TEXT 417 1 "a" }{TEXT 418 2 ", " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT 419 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 433 9 "Example 1" }{TEXT -1 29 ". C ompute the integral of a " }{TEXT 431 1 "(" }{XPPEDIT 18 0 "2" "6#\" \"#" }{TEXT 427 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 432 1 " )" }{TEXT -1 10 " matrix [" }{TEXT 428 1 "A" }{TEXT 430 1 "(" } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 429 1 ")" }{TEXT -1 10 "] given as " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A(t) := matrix(2, 3) : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "A(t)[1,1] := exp(-5*t^ 2) : A(t)[1,2] := t^2*exp(-3*t^2) : A(t)[1,3] := exp(-2*t)*sin(t)/ t : A(t)[2,1] := t/(exp(t)+1) : A(t)[2,2] := exp(-2*t^2)*cos(3*t) \+ : A(t)[2,3] := t/(exp(t)-1) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A(t) := A(t) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%-%$expG6#,$*$)F'\"\"#\"\"\"!\"&*&F 2F4-F.6#,$F1!\"$F4*&*&-F.6#,$F'!\"#F4-%$sinGF&F4F4F'!\"\"7%*&F'F4,&-F. F&F4F4F4FC*&-F.6#,$F1F@F4-%$cosG6#,$F'\"\"$F4*&F'F4,&FGF4F4FCFC" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The integral of the matrix over the interval " }{TEXT 424 1 "[ " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 426 2 ", " }{XPPEDIT 18 0 "infin ity" "6#%)infinityG" }{TEXT 425 1 ")" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`imp_int(A)` := map(Int, A(t), t=0 ..infinity) : Int('A(t)', t=0..infinity) = matrix(`imp_int(A)`) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"!%)inf inityG-%'matrixG6#7$7%-F%6$-%$expG6#,$*$)F*\"\"#\"\"\"!\"&F+-F%6$*&F;F =-F76#,$F:!\"$F=F+-F%6$*&*&-F76#,$F*!\"#F=-%$sinGF)F=F=F*!\"\"F+7%-F%6 $*&F*F=,&-F7F)F=F=F=FPF+-F%6$*&-F76#,$F:FMF=-%$cosG6#,$F*\"\"$F=F+-F%6 $*&F*F=,&FVF=F=FPFPF+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Symbolic evaluation of the matrix \+ integral yields the following exact matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "`imp_int(A)` := map(x->value(x), `imp_int(A)`) : Int('A(t)', t=0..infinity) = matrix(`imp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"!%)infinityG-%'matr ixG6#7$7%,$*&-%%sqrtG6#\"\"&\"\"\"-F76#%#PiGF:#F:\"#5,$*&-F76#\"\"$F:F ;F:#F:\"#O-%'arctanG6##F:\"\"#7%,$*$)F=FKF:#F:\"#7,$*(F;F:-F76#FKF:-%$ expG6##!\"*\"\")F:#F:\"\"%,$FN#F:\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Floating-point ev aluation of the exact resultant matrix gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`imp_int(A)` := evalf(matrix(`imp_int(A)`)) : \+ Int('A(t)', t=0..infinity) = matrix(`imp_int(A)`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"!%)infinityG-%'matrixG 6#7$7%$\"+*HFL'R!#5$\"+qDsF&)!#6$\"+!4wkj%F67%$\"+O.nC#)F6$\"+NwXM?F6$ \"+oS$\\k\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 440 9 "Example 2" }{TEXT -1 29 ". Compute the in tegral of a " }{TEXT 438 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 434 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 439 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 435 1 "B" }{TEXT 437 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 436 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B(t) := matrix(2, 3) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "B(t)[1,1] := (sin(t)/t)^2 : B(t) [1,2] := sin(t)*cos(t/2)/t : B(t)[1,3] := 1/(1+t^2) : B(t)[2,1] := tan(t)/t : B(t)[2,2] := cos(t/4)*sin(t/2)/t : B(t)[2,3] := sin(2* t)/t :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "B(t) := B(t) : \+ 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'ma trixG6#7$7%*&*$)-%$sinGF&\"\"#\"\"\"F3*$)F'F2F3!\"\"*&*&F0F3-%$cosG6#, $F'#F3F2F3F3F'F6*&F3F3,&F3F3*$F5F3F3F67%*&-%$tanGF&F3F'F6*&*&-F:6#,$F' #F3\"\"%F3-F1F;F3F3F'F6*&-F16#,$F'F2F3F'F6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The integral of t he matrix over the interval " }{TEXT 441 1 "[" }{XPPEDIT 18 0 "0" "6# \"\"!" }{TEXT 443 2 ", " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT 442 1 ")" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "`imp_int(B)` := map(Int, B(t), t=0..infinity) : In t('B(t)', t=0..infinity) = matrix(`imp_int(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"BG6#%\"tG/F*;\"\"!%)infinityG-%'matrixG6# 7$7%-F%6$*&*$)-%$sinGF)\"\"#\"\"\"F<*$)F*F;FF " 0 "" {MPLTEXT 1 0 102 "`imp_int(B)` := map(x->value(x), `imp_int(B)`) : Int('B(t)', t= 0..infinity) = matrix(`imp_int(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"BG6#%\"tG/F*;\"\"!%)infinityG-%'matrixG6#7$7%,$%#PiG# \"\"\"\"\"#F4F4F3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Floating-point evaluation of the exact re sultant matrix gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`im p_int(B)` := evalf(matrix(`imp_int(B)`)) : Int('B(t)', t=0..infinity ) = matrix(`imp_int(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG 6$-%\"BG6#%\"tG/F*;\"\"!%)infinityG-%'matrixG6#7$7%$\"+Fjzq:!\"*F4F4F3 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 444 6 "CASE 2" }{TEXT -1 90 ". Improper integral due to disconti nuity of the integrand in the interval of integration " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 452 9 "Example 1" } {TEXT -1 42 ". Compute the integral over the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 9 " for a " }{TEXT 449 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 445 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 450 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 446 1 "A" } {TEXT 448 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 447 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A(t) : = matrix(2, 2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "A(t)[1 ,1] := ln(t)/(t-1) : A(t)[1,2] := ln(1+t)/t : A(t)[2,1] := ln(t)/( t^2-1) : A(t)[2,2] := (1+t)*ln(t)/(t-1) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A(t) := A(t) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7$*&-%#lnGF&\"\"\",&F'F0F0! \"\"F2*&-F/6#,&F0F0F'F0F0F'F27$*&F.F0,&*$)F'\"\"#F0F0F0F2F2*&*&F6F0F.F 0F0F1F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The integral of the matrix over the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "`imp_int(A)` := map(Int, A(t), t=0. .1) : Int('A(t)', t=0..1) = matrix(`imp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"!\"\"\"-%'matrixG6#7$ 7$-F%6$*&-%#lnGF)F.,&F*F.F.!\"\"F:F+-F%6$*&-F86#,&F.F.F*F.F.F*F:F+7$-F %6$*&F7F.,&*$)F*\"\"#F.F.F.F:F:F+-F%6$*&*&F@F.F7F.F.F9F:F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "N otice that each integrand has a singularity at either " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "t=1" "6#/ %\"tG\"\"\"" }{TEXT -1 56 " in the sense that the integrand tends to \+ infinity as " }{XPPEDIT 18 0 "t->0" "6#R6#%\"tG7\"6$%)operatorG%&arro wG6\"\"\"!F*F*F*" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "t->1" "6#R6#%\" tG7\"6$%)operatorG%&arrowG6\"\"\"\"F*F*F*" }{TEXT 420 1 "," }{TEXT -1 15 " respectively." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Symbolic evaluation of the matrix \+ integral yields the following exact matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`imp_int(A)` := map(x->value(x), `imp_int(A)`) : \+ Int('A(t)', t=0..1) = matrix(`imp_int(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F*;\"\"!\"\"\"-%'matrixG6#7$7$, $*$)%#PiG\"\"#F.#F.\"\"',$F5#F.\"#77$,$F5#F.\"\"),&!\"\"F.*&#F.\"\"$F. F6F.F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Floating-point evaluation of the exact resultant mat rix gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`imp_int(A)` : = evalf(matrix(`imp_int(A)`)) : Int('A(t)', t=0..1) = matrix(`imp_in t(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"AG6#%\"tG/F* ;\"\"!\"\"\"-%'matrixG6#7$7$$\"+oS$\\k\"!\"*$\"+O.nC#)!#57$$\"+]0qL7F6 $\"+M\"o)*G#F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 459 9 "Example 2" }{TEXT -1 42 ". Compute the in tegral over the interval " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7$\"\"!*&%#Pi G\"\"\"\"\"#!\"\"" }{TEXT -1 9 " for a " }{TEXT 457 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 453 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 458 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 454 1 "B" }{TEXT 456 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 455 1 ")" }{TEXT -1 10 " ] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B(t) := matri x(2, 2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "B(t)[1,1] := \+ (sin(4*t)/sin(t))^2 : B(t)[1,2] := ln(cos(t)^4) : B(t)[2,1] := 1/s qrt(t) : B(t)[2,2] := ln(1/cos(t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "B(t) := B(t) : 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7$7$*&*$)-%$sinG6#,$F'\"\"%\" \"#\"\"\"F6*$)-F1F&F5F6!\"\"-%#lnG6#*$)-%$cosGF&F4F67$*&F6F6*$-%%sqrtG 6#F'F6F:-F<6#*&F6F6F@F:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The integral of the matrix over th e interval " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7$\"\"!*&%#PiG\"\"\"\"\"#! \"\"" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 " `imp_int(B)` := map(Int, B(t), t=0..Pi/2) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$- %\"BG6#%\"tG/F*;\"\"!,$%#PiG#\"\"\"\"\"#-%'matrixG6#7$7$-F%6$*&*$)-%$s inG6#,$F*\"\"%F2F1F1*$)-F>F)F2F1!\"\"F+-F%6$-%#lnG6#*$)-%$cosGF)FAF1F+ 7$-F%6$*&F1F1*$-%%sqrtG6#F*F1FEF+-F%6$-FI6#*&F1F1FMFEF+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "N otice that each integrand has a singularity in the interval " } {XPPEDIT 18 0 "[0,Pi/2]" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 12 " at either " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 6 " \+ or " }{XPPEDIT 18 0 "t=Pi/2" "6#/%\"tG*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT -1 56 " in the sense that the integrand tends to infinity as \+ " }{XPPEDIT 18 0 "t->0" "6#R6#%\"tG7\"6$%)operatorG%&arrowG6\"\"\"!F*F *F*" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "t ->Pi/2" "6#R6#%\"tG7\"6$%) operatorG%&arrowG6\"*&%#PiG\"\"\"\"\"#!\"\"F*F*F*" }{TEXT 460 1 "," } {TEXT -1 15 " respectively." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Symbolic evaluation of the matr ix integral yields the following exact matrix:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 98 "`imp_int(B)` := map(x->value(x), `imp_int(B)`) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"BG6#%\"tG/F*;\"\"!,$%#PiG#\"\"\"\"\"#- %'matrixG6#7$7$,$F/F2,**&F/F1-%#lnG6#F2F1!\"#*&^#\"\"$F1)F/F2F1F1*(F2F 1F/F1-F<6#^$!\"\"F1F1FG*(F2F1F/F1-F<6#^$FGFGF1F17$*&-%%sqrtG6#F2F1-FO6 #F/F1,*F:F0*&^##!\"$\"\"%F1FBF1F1*(F0F1F/F1FDF1F1*&#F1F2F1*&F/F1FIF1F1 FG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Floating-point evaluation of the exact resultant matrix g ives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`imp_int(B)` := eva lf(matrix(`imp_int(B)`)) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int( B)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%\"BG6#%\"tG/F*; \"\"!,$%#PiG#\"\"\"\"\"#-%'matrixG6#7$7$$\"+3`=$G'!\"*^$$!+#=s^N%F:$F1 !\")7$$\"+u#Gm]#F:^$$\"+YIz)3\"F:$!\"$F:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Disregarding the \+ meaningless imaginary parts simplifies the above to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "`imp_int(B)` := evalm(Re(`imp_int(B)`)) \+ : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%\"BG6#%\"tG/F*;\"\"!,$%#PiG#\"\"\"\"\"#-% 'matrixG6#7$7$$\"+3`=$G'!\"*$!+#=s^N%F:7$$\"+u#Gm]#F:$\"+YIz)3\"F:" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 451 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (30) for \"" }{TEXT 349 27 "Application of the function" }{TEXT -1 2 " " }{TEXT 352 4 "exp(" }{TEXT -1 1 "[" }{TEXT 350 1 "A" }{TEXT -1 1 "]" }{TEXT 353 1 " " } {TEXT 354 1 "t" }{TEXT 355 1 ")" }{TEXT -1 2 " " }{TEXT 351 27 "in so lving matrix equations" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 348 67 "-------------------------------------------------------- -----------" }}}}{MARK "161 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }