Baixe Exercícios de Cálculo Numérico: Métodos de Resolução de Equações e outras Exercícios em PDF para Cálculo Avançado, somente na Docsity!
ü 1)
aL 381928 =
RND Æ 0.382 ¥ 106
bL 78.457 =
RND Æ 0.785 ¥ 102
cL - 9142.683 =
ü 2)Escreva os seguintes números que estão no sistema binário no sistema de base 10
aL
11 = 1 ◊ 21 + 1 ◊ 20 = 3 d
0.11 = 1 ◊ 2 -^1 + 1 ◊ 2 -^2 =
= 0.75 d
11.11 = 3.75 d
bL 0.1011 = 1 ◊ 2 -^1 + 0 ◊ 2 -^2 + 1 ◊ 2 -^3 + 1 ◊ 2 -^4 0.1011 = 0.6875 d
cL 1.0011 = 1 + 0 ◊ 2 -^1 + 0 ◊ 2 -^2 + 1 ◊ 2 -^3 + 1 ◊ 2 -^4 1.0011 = 1.1875 d
dL 110101 = 53
eL 0.111101101 =
ü 3)Escreva os seguintes números que estão no sistema decimal no sistema de binário
aL 13.25 d = D +
h = 1101.01 b
bL 0.
0.10125 * 2 = 0.2025 Æ 0
0.2025 * 2 = 0.4050 Æ 0
0.4050 * 2 = 0.8100 Æ 0
0.8100 * 2 = 1.6200 Æ 1
0.6200 * 2 = 1.2400 Æ 1
0.2400 * 2 = 0.4800 Æ 0
0.4800 * 2 = 0.96 Æ 0
0.96 * 2 = 1.92 Æ 1
0.92 * 2 = 1.84 Æ 1
0.84 * 2 = 1.68 Æ 1
0.68 * 2 = 1.36 Æ 1
0.36 * 2 = 0.72 Æ 0
0.72 * 2 = 1.44 Æ 1
= 0.00011001111010111000011 b
dL 13 d = D h = 1101 b
eL 12.03135 = 1100.0000100000000111 b
ü 4)
ü a) x = cosH x L
xn + 1 = xn -
f H xn L f ¢H xn L f H x L = cosH x L - x
f ¢H x L = -sinH x L - 1
x 0 = 0.
0.5 1.0 1.5 2.
n x xn+ 1 »xn+ 1 - xn» 1 0.5 0.7552224171 0. 2 0.7552224171 0.7391416661 0. 3 0.7391416661 0.7390851339 0. 4 0.7390851339 0.7390851332 7.056460971 ¥ 10 -^10
x = 0.
ü b) 5 LogH x L - 2 + 0.4 x = 0
xn = 0.5;
xn = 1.0;
5
10
15
xn = 2.0;
x = 2.2360 ± 0.
ü 8 L 26
5
fn@xD = x -
x^5 - 26 5 x^4
xn = 3.0;
n x xn+ 1 »FHxL» »xn+ 1 - xn» 1 3. 2.464197531 64.86101634 0. 2 2.464197531 2.112384701 16.05959316 0. 3 2.112384701 1.951070545 2.272542303 0. 4 1.951070545 1.919705199 0.07190142514 0. 5 1.919705199 1.918646362 0.00007927253425 0. 6 1.918646362 1.918645192 9.667999734 ¥ 10 -^11 1.169965351 ¥ 10 -^6 7 1.918645192 1.918645192 0. 1.426858631 ¥ 10 -^12 8 1.918645192 1.918645192 0. 0.
x = 1.91864 ± 1.2 ¥ 10 -^6
Obs: Na máquina, fazendo com precisao de 10 temos:
NB 26
5 , 10F
ü 10) J x 2 M
2
- SinH x L=
In[16]:= fn@x_D := x -
I x 2 M^2 - Sin@xD x 2 -^ Cos@xD
In[23]:= xn^ =^
Out[23]= 1.
In[24]:= Calculos^ =^ TableB:n, xnn^ =^ xn, xn^ =^ fn@xnD, AbsB^
xn 2
2
- Sin@xnDF, Abs@xnn - xnD>, 8 n, 5<F
Out[24]= 98 1, 1.75, 1.957321875, 0.03155280944, 0.2073218748<, 8 2, 1.957321875, 1.934046551, 0.0003871027911, 0.02327532417<, 9 3, 1.934046551, 1.933753809, 6.147857079 ¥ 10 -^8 , 0.0002927412905=, 9 4, 1.933753809, 1.933753763, 1.33226763 ¥ 10 -^15 , 4.65071166 ¥ 10 -^8 =, 9 5, 1.933753763, 1.933753763, 1.110223025 ¥ 10 -^16 , 1.110223025 ¥ 10 -^15 ==
In[25]:= Insert@Calculos, 8 "n", "x", "xn+ 1 ", "»FHxL»", "»Erro»"<, 1D êê TableForm
Out[25]//TableForm= n x xn+ 1 »FHxL» »Erro» 1 1.75 1.957321875 0.03155280944 0. 2 1.957321875 1.934046551 0.0003871027911 0. 3 1.934046551 1.933753809 6.147857079 ¥ 10 -^8 0. 4 1.933753809 1.933753763 1.33226763 ¥ 10 -^15 4.65071166 ¥ 10 -^8 5 1.933753763 1.933753763 1.110223025 ¥ 10 -^16 1.110223025 ¥ 10 -^15
ü 9) x^3 - 2 x^2 + 2 x - 5 = 0
x 0 = 2; x- 1 = - 2
Out[59]=
xn xn+ 1 erro
- 0.3421052632 6. 0.3421052632 0.4578018321 0. 0.4578018321 1.224238104 0. 1.224238104 0.6488285049 0. 0.6488285049 0.7408028831 0. 0.7408028831 0.8187751087 0. 0.8187751087 0.8033058267 0. 0.8033058267 0.8044336475 0. 0.8044336475 0.8044534159 0. 0.8044534159 0.8044533884 3.419458367 ¥ 10 -^8 0.8044533884 0.8044533884 8.188110462 ¥ 10 -^13
x > 0.
ü 12) Ln(x) - x + 2=0 Ε [3,4]
In[60]:= xn@nD^ =^ 3;^ xn@n^ -^1 D^ =^ 4;
Out[68]=
xn xn+ 1 erro 3 3.138438589 0. 3.138438589 3.146281039 0. 3.146281039 3.14619317 0. 3.14619317 3.146193221 1.606294995 ¥ 10 -^8 3.146193221 3.146193221 1.044519495 ¥ 10 -^13
x > 3.
ü 13)
x^2 - 3 x + „x^ = 2
2
4
6
8
In[160]:= xn@nD = 2; xn@n - 1 D = 1;
Método das secantes :
Out[167]=
xn xn+ 1 FHxL erro 2 1.274412356 3.389056099 0. 1.274412356 1.387008355 - 0.6225111896 0. 1.387008355 1.454999563 - 0.2343758921 0. 1.454999563 1.445836439 0.03650663372 0. 1.445836439 1.446236039 - 0.00166462923 0. 1.446236039 1.446238687 - 0.00001095912569 1.831098053 ¥ 10 -^6
x = 1.
Método de Newton:
In[176]:= xn^ =^ 1.00;
Out[178]=
n xn xn+ 1 FHxL erro 0 1. 1.745930121 1.541711356 0. 1 1.745930121 1.498189631 0.2235861771 0. 2 1.498189631 1.448169929 0.008006202531 0. 3 1.448169929 1.446241495 0.00001162624072 0. 4 1.446241495 1.446238686 2.463851345 ¥ 10 -^11 2.808538916 ¥ 10 -^6 5 1.446238686 1.446238686 8.881784197 ¥ 10 -^16 5.951905635 ¥ 10 -^12
x = 1.
Metodo bissecção
In[195]:= a^ =^ 1.0; b^ =^ 2.0;
Out[197]=
n a x b FHxL Ε 0 1. 1.5 2. 0.2316890703 0. 1 1. 1.25 1.5 - 0.6971570425 0. 2 1.25 1.375 1.5 - 0.2792982771 0. 3 1.375 1.4375 1.5 - 0.03593649386 0. 4 1.4375 1.46875 1.5 0.09477855606 0. 5 1.4375 1.453125 1.46875 0.02865485134 0. 6 1.4375 1.4453125 1.453125 - 0.003831348585 0. 7 1.4453125 1.44921875 1.453125 0.01236399297 0. 8 1.4453125 1.447265625 1.44921875 0.004254398448 0. 9 1.4453125 1.446289063 1.447265625 0.0002085459752 0. 10 1.4453125 1.445800781 1.446289063 - 0.001812145797 0.
x = 1.
Método ponto fixo:
In[157]:= LogLinearPlotBAbsB^
Vm
J (^) ¸ 1 Ω c + R + ¸ Ω LN
F, 8 Ω, 1, 1000<, GridLines Æ AutomaticF
Out[157]=
5 10 50 100 500 1000
Podemos fazer x- 1 = 10 e x 0 = 100
im ä
Vm
R^2 + JH 2 Π fL L - (^) H 2 Π^1 fL c N 2
im R^2 + H 2 Π fL L -
H 2 Π fL c
2
2
ä Vm^2
A função fica: (lembre que frequencia negativa não é válido fisicamente, apenas pode-se considerá-la quando levar em
consideracao um atraso de fase de -Π)
im^2 R^2 + H 2 Π fL L -
H 2 Π fL c
- Vm^2 ä 0
Out[155]=
xn xn+ 1 F@xD erro 100 29.52962646 465.4662638 2. 29.52962646 40.1546866 - 82.63983873 0. 40.1546866 49.27955311 - 38.18128086 0. 49.27955311 47.26449216 10.82133788 0. 47.26449216 47.41288755 - 0.8602701653 0. 47.41288755 47.41581332 - 0.01663320285 0. 47.41581332 47.41580865 0.00002658455128 9.846398778 ¥ 10 -^8
f = 47.4158086 Hz
ü 15)
vo = 15.2; x1 = 18.2; h = 1.82; y = 2.1; g = 9.0;
- 1 1. 1.109320061 0.15315055 0. n x xn+ 1 »FHxL» »xn+ 1 - xn»
- 2 1.109320061 1.20854264 0.1222798506 0.
- 3 1.20854264 1.290274358 0.08763025042 0.
- 4 1.290274358 1.350741518 0.05698526485 0.
- 5 1.350741518 1.391109964 0.03432372084 0.
- 6 1.391109964 1.415880902 0.0195983603 0.
- 7 1.415880902 1.430191939 0.01081944996 0.
- 8 1.430191939 1.438147169 0.00585587258 0.
- 9 1.438147169 1.44246951 0.003134388443 0.
- 10 1.44246951 1.444787957 0.001667549188 0.
- 11 1.444787957 1.446022812 0.0008842741638 0.
- 12 1.446022812 1.446678032 0.0004681004545 0.
- 13 1.446678032 1.447024991 0.0002475652294 0.
- 14 1.447024991 1.44720852 0.0001308662066 0.
- 15 1.44720852 1.447305544 0.00006915965757 0.
- x = 1.
- ü d L x^3 - x - 5 = - 0.5 1.0 1.5 2.0 2.5 3.
- 1 2. 1.909090909 0.04883546206 0. n x xn+ 1 »FHxL» »xn+ 1 - xn»
- 2 1.909090909 1.90417486 0.0001382952717 0.
- 3 1.90417486 1.904160859 1.11978693 ¥ 10 -^9 0.
- x = 1.
- ü 5) „ x - x^2 +
- ü 6)
- x? 2. - g x - + h + x tanHΘL - y á 2 I v 02 cos^2 HΘLM
- 0.01 - 6. Tabela de valores a função:
- 0.3241592654 - 1.
- 0.6383185307 3.
- 0.9524777961 6.
- 1.266637061 - 14. - 1 0.3 0.47 0.64 0.8486644367 0. a = 0.3; b = 0.64; - 2 0.3 0.385 0.47 - 0.4159551941 0. - 3 0.385 0.4275 0.47 0.2211237631 0. - 4 0.385 0.40625 0.4275 - 0.09623513818 0. - 5 0.40625 0.416875 0.4275 0.06273976922 0. - 6 0.40625 0.4115625 0.416875 - 0.01667392707 0. - 7 0.4115625 0.41421875 0.416875 0.02305136942 0. - 8 0.4115625 0.412890625 0.41421875 0.003193331869 0. - 9 0.4115625 0.4122265625 0.412890625 - 0.006739145077 0.
- 10 0.4122265625 0.4125585938 0.412890625 - 0.001772618456 0.
- 11 0.4125585938 0.4127246094 0.412890625 0.0007104287462 0.
- 12 0.4125585938 0.4126416016 0.4127246094 - 0.0005310768451 0.
- 13 0.4126416016 0.4126831055 0.4127246094 0.00008968045299 0.
- 14 0.4126416016 0.4126623535 0.4126831055 - 0.0002206970705 0.
- 15 0.4126623535 0.4126727295 0.4126831055 - 0.00006550802733 0.
- 16 0.4126727295 0.4126779175 0.4126831055 0.00001208628318 5.187988281 ¥ 10 -
- 17 0.4126727295 0.4126753235 0.4126779175 - 0.00002671085449 2.593994141 ¥ 10 -
- 18 0.4126753235 0.4126766205 0.4126779175 - 7.312281258 ¥ 10 -^6 1.29699707 ¥ 10 -
- 19 0.4126766205 0.412677269 0.4126779175 2.387002058 ¥ 10 -^6 6.484985352 ¥ 10 -
- 20 0.4126766205 0.4126769447 0.412677269 - 2.462639326 ¥ 10 -^6 3.242492676 ¥ 10 -
- 21 0.4126769447 0.4126771069 0.412677269 - 3.781856406 ¥ 10 -^8 1.621246338 ¥ 10 -
- 22 0.4126771069 0.4126771879 0.412677269 1.174591764 ¥ 10 -^6 8.106231691 ¥ 10 -
- 23 0.4126771069 0.4126771474 0.4126771879 5.683866038 ¥ 10 -^7 4.053115846 ¥ 10 -
- 24 0.4126771069 0.4126771271 0.4126771474 2.65284019 ¥ 10 -^7 2.026557921 ¥ 10 -
- 25 0.4126771069 0.412677117 0.4126771271 1.137327292 ¥ 10 -^7 1.013278961 ¥ 10 -
- 26 0.4126771069 0.4126771119 0.412677117 3.795708126 ¥ 10 -^8 5.066394804 ¥ 10 -
