#OK to post homework #Nicholas DiMarzio, 10/18/21, Assignment 13 # Problem 1 # Understand Maple code from lecture 13 # # #Problem 2 # F := [(x^2 + 8*x + 1)/(x^2 + x), (y^2 + 8*y + 1)/(y^2 + y + 1)]; [ 2 2 ] [x + 8 x + 1 y + 8 y + 1] F := [------------, ------------] [ 2 2 ] [ x + x y + y + 1 ] evalf(FP2(F, x, y)); [[2.888969401, 2.714478744]] SFP2(F, x, y); [[2.888969401, 2.714478744]] Orb2(F, x, y, [8.5, 1.5], 1000, 1010); [[2.888969405, 2.714478742], [2.888969397, 2.714478745], [2.888969405, 2.714478743], [2.888969397, 2.714478744], [2.888969405, 2.714478742], [2.888969397, 2.714478745], [2.888969405, 2.714478743], [2.888969397, 2.714478744], [2.888969405, 2.714478742], [2.888969397, 2.714478745]] # #Problem 3 # #1 F := RT2(x, y, 1, 100); [93 + 45 y + 96 x 44 + 100 y + 38 x] F := [----------------, -----------------] [6 + 98 y + 59 x 69 + 27 y + 96 x ] SFP2drz(F, x, y); [[1.624821324, 0.7307142101]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102], [1.624821323, 0.7307142102]] #2 F := RT2(x, y, 1, 100); [90 + 34 y + 18 x 83 + 25 y + 90 x] F := [----------------, ----------------] [52 + 56 y + 43 x 93 + 60 y + 93 x] SFP2drz(F, x, y); [[0.9692774235, 0.8208335066]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066], [0.9692774235, 0.8208335066]] #3 F := RT2(x, y, 1, 100); [14 + 50 y + 47 x 9 + 77 y + 59 x] F := [----------------, ---------------] [8 + 46 y + 44 x 16 + y + 70 x ] SFP2drz(F, x, y); [[1.106350715, 3.689085048]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048], [1.106350715, 3.689085048]] #4 F := RT2(x, y, 1, 100); [77 + 39 y + 92 x 51 + 53 y + 12 x] F := [----------------, ----------------] [71 + 67 y + 78 x 19 + 63 y + 40 x] SFP2drz(F, x, y); [[0.9648916375, 0.9608443733]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732], [0.9648916375, 0.9608443732]] #5 F := RT2(x, y, 1, 100); [2 + 73 y + 85 x 13 + 19 y + 10 x] F := [---------------, ----------------] [41 + 4 y + 44 x 15 + 64 y + 9 x ] SFP2drz(F, x, y); [[1.594818206, 0.5965048039]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.594818206, 0.5965048039], [1.594818206, 0.5965048038], [1.594818206, 0.5965048039], [1.594818206, 0.5965048038], [1.594818206, 0.5965048039], [1.594818206, 0.5965048038], [1.594818206, 0.5965048039], [1.594818206, 0.5965048038], [1.594818206, 0.5965048039], [1.594818206, 0.5965048038]] #6 F := RT2(x, y, 1, 100); [12 + 52 y + 25 x 43 + 55 y + 40 x] F := [----------------, ----------------] [72 + 90 y + 18 x 17 + 70 y + 52 x] SFP2drz(F, x, y); [[0.4449874645, 1.044140881]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.4449874742, 1.044140881], [0.4449874742, 1.044140882], [0.4449874742, 1.044140881], [0.4449874742, 1.044140882], [0.4449874742, 1.044140881], [0.4449874742, 1.044140882], [0.4449874742, 1.044140881], [0.4449874742, 1.044140882], [0.4449874742, 1.044140881], [0.4449874742, 1.044140882]] #7 F := RT2(x, y, 1, 100); [81 + 87 y + 34 x 83 + 63 y + 100 x] F := [----------------, -----------------] [85 + 9 y + 68 x 70 + 36 y + 36 x ] SFP2drz(F, x, y); [[1.446259999, 1.824628347]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.446260001, 1.824628347], [1.446260000, 1.824628348], [1.446260001, 1.824628347], [1.446260000, 1.824628348], [1.446260001, 1.824628347], [1.446260000, 1.824628348], [1.446260001, 1.824628347], [1.446260000, 1.824628348], [1.446260001, 1.824628347], [1.446260000, 1.824628348]] #8 F := RT2(x, y, 1, 100); [10 + 40 y + 66 x 43 + 53 y + 61 x] F := [----------------, ----------------] [87 + 16 y + 98 x 47 + 28 y + 75 x] SFP2drz(F, x, y); [[0.5777678089, 1.133027037]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037]] #9 F := RT2(x, y, 1, 100); [10 + 40 y + 66 x 43 + 53 y + 61 x] F := [----------------, ----------------] [87 + 16 y + 98 x 47 + 28 y + 75 x] SFP2drz(F, x, y); [[0.5777678089, 1.133027037]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037], [0.5777678083, 1.133027037]] #10 F := RT2(x, y, 1, 100); [11 + 30 y + 6 x 80 + 96 y + 11 x] F := [----------------, ----------------] [32 + 40 y + 24 x 23 + 41 y + 52 x] SFP2drz(F, x, y); [[0.5933472703, 2.054942274]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273], [0.5933472706, 2.054942273]] #11 F := RT2(x, y, 1, 100); [58 + 67 y + 81 x 36 + 61 y + 84 x] F := [----------------, ----------------] [65 + 69 y + 2 x 96 + 94 y + 31 x] SFP2drz(F, x, y); [[2.100384075, 1.062320488]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488], [2.100384074, 1.062320488]] #12 F := RT2(x, y, 1, 100); [81 + 31 y + 54 x 12 + 49 y + 90 x] F := [----------------, ----------------] [67 + 59 y + 66 x 35 + 15 y + 26 x] SFP2drz(F, x, y); [[0.7741723060, 2.138101856]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.7741723062, 2.138101856], [0.7741723059, 2.138101856], [0.7741723059, 2.138101855], [0.7741723062, 2.138101855], [0.7741723062, 2.138101856], [0.7741723059, 2.138101856], [0.7741723059, 2.138101855], [0.7741723062, 2.138101855], [0.7741723062, 2.138101856], [0.7741723059, 2.138101856]] #13 F := RT2(x, y, 1, 100); [100 + 24 y + 8 x 73 + 22 y + 32 x] F := [----------------, ----------------] [63 + 78 y + 23 x 98 + 9 y + 53 x ] SFP2drz(F, x, y); [[0.8732183590, 0.7806494748]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746], [0.8732183544, 0.7806494746]] #14 F := RT2(x, y, 1, 100); [3 + 98 y + 69 x 37 + 60 y + 94 x] F := [---------------, ----------------] [3 + 73 y + 88 x 52 + 16 y + 29 x] SFP2drz(F, x, y); [[1.133819230, 2.317348056]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055]] #15 F := RT2(x, y, 1, 100); [3 + 98 y + 69 x 37 + 60 y + 94 x] F := [---------------, ----------------] [3 + 73 y + 88 x 52 + 16 y + 29 x] SFP2drz(F, x, y); [[1.133819230, 2.317348056]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055], [1.133819230, 2.317348055]] #16 F := RT2(x, y, 1, 100); [ 1 + 83 y + 9 x 52 + 62 y + 46 x] F := [----------------, ----------------] [64 + 43 y + 57 x 76 + 9 y + 53 x ] SFP2drz(F, x, y); [[0.7244842333, 1.326294850]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851], [0.7244842345, 1.326294851]] #17 F := RT2(x, y, 1, 100); [37 + 88 y + 50 x 8 + 92 y + 92 x] F := [----------------, ---------------] [37 + 76 y + 95 x 2 + 97 y + 44 x] SFP2drz(F, x, y); [[0.8858304349, 1.258907254]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255], [0.8858304364, 1.258907255]] #18 F := RT2(x, y, 1, 100); [9 + 30 y + 14 x 78 + 49 y + 93 x] F := [----------------, ----------------] [79 + 73 y + 21 x 15 + 56 y + 69 x] SFP2drz(F, x, y); [[0.2986784818, 1.499119277]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277], [0.2986784848, 1.499119277]] #19 F := RT2(x, y, 1, 100); [17 + 21 y + 42 x 3 + 86 y + 55 x] F := [----------------, ---------------] [21 + 5 y + 58 x 97 + 4 y + 92 x] SFP2drz(F, x, y); [[0.8755023922, 0.5457490586]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586], [0.8755023920, 0.5457490586]] #20 F := RT2(x, y, 1, 100); [46 + 88 y + 34 x 21 + 86 y + 42 x] F := [----------------, ----------------] [68 + 49 y + 61 x 5 + 33 y + 77 x ] SFP2drz(F, x, y); [[1.028465691, 1.422412401]] Orb2(F, x, y, [0.5, 0.5], 1000, 1010); [[1.028465692, 1.422412402], [1.028465692, 1.422412401], [1.028465692, 1.422412402], [1.028465692, 1.422412401], [1.028465692, 1.422412402], [1.028465692, 1.422412401], [1.028465692, 1.422412402], [1.028465692, 1.422412401], [1.028465692, 1.422412402], [1.028465692, 1.422412401]] # #Problem 4 # RT3 := proc(x, y, z, d, K) local ra, i, j, k, f, g, h; ra := rand(1 .. K); f := add(add(add(ra()*x^i*y^j*z, z = 0 .. d - j), j = 0 .. d - i), i = 0 .. d)/add(add(add(ra()*x^i*y^j*z, z = 0 .. d - j), j = 0 .. d - i), i = 0 .. d); g := add(add(add(ra()*x^i*y^j*z, z = 0 .. d - j), j = 0 .. d - i), i = 0 .. d)/add(add(add(ra()*x^i*y^j*z, z = 0 .. d - j), j = 0 .. d - i), i = 0 .. d); h := add(add(add(ra()*x^i*y^j*z, z = 0 .. d - j), j = 0 .. d - i), i = 0 .. d)/add(add(add(ra()*x^i*y^j*z, z = 0 .. d - j), j = 0 .. d - i), i = 0 .. d); [f, g, h]; end proc Orb3 := proc(F, x, y, z, pt0, K1, K2) local pt, L, i; pt := pt0; for i to K1 do pt := subs({x = pt[1], y = pt[2], z = pt[3]}, F); end do; L := []; for i from K1 + 1 to K2 do L := [op(L), pt]; pt := subs({x = pt[1], y = pt[2], z = pt[3]}, F); end do; L; end proc FP3 := proc(F, x, y, z) local L, i; L := [solve({F[1] = x, F[2] = y, F[3] = z}, {x, y, z})]; [seq(subs(L[i], [x, y, z]), i = 1 .. nops(L))]; end proc SFP3 := proc(F, x, y, z) local L, J, S, J0, i, pt, EV; L := FP3(F, x, y, z); J := Matrix(normal([[diff(F[1], x), diff(F[2], x), diff(F[3], x)], [diff(F[1], y), diff(F[2], y), diff(F[3], y)], [diff(F[1], z), diff(F[2], z), diff(F[3], z)]])); S := []; for i to nops(L) do pt := L[i]; J0 := subs({x = pt[1], y = pt[2], z = pt[3]}, J); EV := Eigenvalues(J0); if abs(EV[1]) < 1 and abs(EV[2]) < 1 then S := [op(S), pt]; end if; end do; S; end proc # #Problem5 # #1 F := RT3(x, y, z, 1, 100); [104 x z + 96 y z + 138 z 124 x z + 17 y z + 123 z F := [-------------------------, ------------------------, [107 x z + 100 y z + 103 z 126 x z + 56 y z + 70 z 153 x z + 93 y z + 115 z] ------------------------] 54 x z + 47 y z + 64 z ] #2 F := RT3(x, y, z, 1, 100); [75 x z + 77 y z + 53 z 104 x z + 78 y z + 138 z F := [-----------------------, ------------------------, [131 x z + 77 y z + 71 z 130 x z + 63 y z + 31 z 141 x z + 49 y z + 52 z ] ------------------------] 170 x z + 27 y z + 164 z] #3 F := RT3(x, y, z, 1, 100); [45 x z + 85 y z + 75 z 77 x z + 12 y z + 73 z F := [----------------------, -----------------------, [25 x z + 19 y z + 57 z 98 x z + 18 y z + 162 z 133 x z + 70 y z + 57 z] -----------------------] 77 x z + 85 y z + 121 z] #4 F := RT3(x, y, z, 1, 100); [117 x z + 34 y z + 134 z 156 x z + 98 y z + 110 z F := [------------------------, ------------------------, [ 47 x z + 86 y z + 82 z 64 x z + 29 y z + 94 z 143 x z + 44 y z + 100 z ] -------------------------] 127 x z + 100 y z + 117 z] #5 F := RT3(x, y, z, 1, 100); [117 x z + 34 y z + 134 z 156 x z + 98 y z + 110 z F := [------------------------, ------------------------, [ 47 x z + 86 y z + 82 z 64 x z + 29 y z + 94 z 143 x z + 44 y z + 100 z ] -------------------------] 127 x z + 100 y z + 117 z] #6 F := RT3(x, y, z, 1, 100); [107 x z + 45 y z + 54 z 92 x z + y z + 63 z F := [------------------------, ------------------------, [129 x z + 49 y z + 145 z 114 x z + 57 y z + 107 z 90 x z + 9 y z + 122 z ] ------------------------] 171 x z + 37 y z + 138 z] #7 F := RT3(x, y, z, 1, 100); [106 x z + 100 y z + 146 z 114 x z + 43 y z + 114 z F := [-------------------------, ------------------------, [ 153 x z + 40 y z + 46 z 8 x z + 75 y z + 75 z 95 x z + 75 y z + 48 z ] -----------------------] 191 x z + 58 y z + 62 z] #8 F := RT3(x, y, z, 1, 100); [50 x z + 90 y z + 61 z 54 x z + 73 y z + 101 z F := [-----------------------, ------------------------, [71 x z + 24 y z + 126 z 101 x z + 53 y z + 107 z 125 x z + 73 y z + 72 z] -----------------------] 45 x z + 52 y z + 154 z] #9 F := RT3(x, y, z, 1, 100); [ 72 x z + 6 y z + 41 z 148 x z + 58 y z + 93 z F := [-----------------------, -----------------------, [34 x z + 96 y z + 104 z 97 x z + 2 y z + 134 z 112 x z + 94 y z + 180 z] ------------------------] 125 x z + 67 y z + 85 z ] #10 F := RT3(x, y, z, 1, 100); [99 x z + 92 y z + 100 z 142 x z + 78 y z + 94 z F := [-----------------------, -----------------------, [93 x z + 30 y z + 53 z 38 x z + 69 y z + 71 z 61 x z + 5 y z + 63 z ] -----------------------] 96 x z + 97 y z + 141 z]