JESÚS AVALOS RODRÍGUEZ

Docente del Departamento Académico de Matemáticas

Facultad de Ciencias Físicas y Matemáticas

Universidad Nacional de Trujillo

E-mail: javalos@unitru.edu.pe

NUMERICAL METHODS

EJEMPLO DE APLICACIÓN DE LA SECANTE

EJEMPLO DE APLICACIÓN DEL MÉTODO DE NEWTON

SECANT METHOD - WOLFRAM MATHEMATICA V10

ROOTFINDING

Secant Method

www.jesus-avalos.ucoz.com

ALGORITHM CODE:

SecantMethod[x0_,x1_,max_]:=Module[{},k=1;p0=N[x0];p1=N[x1];

Print["p0=",PaddedForm[p0,{16,16}]", f[p0]=",NumberForm[f[p0],16]];

Print["p1=", PaddedForm[p1,{16,16}],", f[p1]=", NumberForm[f[p1],16]];

p2=p1;

p1=p0;

While[k<max, p0=p1;p1=p2; p2=p1-(f[p1](p1-p0))/(f[p1]-f[p0]);

k=k+1;

Print["p "k, "=", PaddedForm[p2,{16,16}],", f[","p"k,"]=",NumberForm[f[p2],16]];];

Print["p =",NumberForm[p2,16]];

Print[" Δp=±"Abs[p2-p1]];

Print["f[p]=",NumberForm[f[p2],16]];]

EXAMPLE:

f[x_]=x^3+x-1

-1+x+x3

SecantMethod[0,1,9]

p0= , f[p0]= 0.0000000000000000 -1.

p1= 1.0000000000000000 , f[p1]= 1.

2 p = 0.5000000000000000 , f[ 2 p ]= -0.375

3 p = 0.6363636363636363 , f[ 3 p ]= -0.1059353869271225

4 p = 0.6900523560209423 , f[ 4 p ]= 0.01863614179998446

5 p = 0.6820204196481856 , f[ 5 p ]= -0.000736518493373528

6 p = 0.6823257814098928 , f[ 6 p ]= -4.847148849740357*10-6

7 p = 0.6823278043590257 , f[ 7 p ]= 1.272670357987948*10-9

8 p = 0.6823278038280184 , f[ 8 p ]= -2.275957200481571*10-15

9 p = 0.6823278038280193 , f[ 9 p ]= 1.665334536937735*10-16

p= 0.6823278038280193

9.99201*10-16 Δp=±

f[p]= 1.665334536937735*10-16

Plot[-1+x+x3,{x,0,1}]

REGULA FALSI - WOLFRAM MATHEMATICA V10

ROOTFINDING

Regula Falsi Method

www.jesus-avalos.ucoz.com

ALGORITHM CODE:

RegulaFalsi[a0_,b0_,m_]:=Module[{},a=N[a0];b=N[b0]; c=(a*f[b]-b*f[a])/(f[b]-f[a]);k=0;

output={{k,a,c,b,f[c]}};

While[k<m, If[Sign[f[b]]==Sign[f[c]],b=c, a=c;];

c=(a*f[b]-b*f[a])/(f[b]-f[a]);

k=k+1;

output=Append[output,{k,a,c,b,f[c]}];];

Print[NumberForm[TableForm[output,TableHeadings->{None,{"k","ak","ck","bk","f[ck]"}}],16]];

Print[" c=",NumberForm[c,16]];

Print[" f[c]=",NumberForm[f[c],16]];]

EXAMPLE:

f[x_]=x^3-2x^2+3x/2

(3 x)/2-2 x2+x3

RegulaFalsi[-1,1,12]

{

{k, ak, ck, bk, f[ck]},

{0, -1., 0.8, 1., 0.432},

{1, -1., 0.6423357664233577, 0.8, 0.4033378536513656},

{2, -1., 0.5072408162536918, 0.6423357664233577, 0.3767843689852098},

{3, -1., 0.3907901518625014, 0.5072408162536918, 0.340431619474261},

{4, -1., 0.2929747128750934, 0.3907901518625014, 0.2929409494830454},

{5, -1., 0.2139490699474353, 0.2929747128750934, 0.2391685443454786},

{6, -1., 0.1526854897957456, 0.2139490699474353, 0.1859620523688113},

{7, -1., 0.1069412526417537, 0.1526854897957456, 0.1387620422415913},

{8, -1., 0.07382866194205702, 0.1069412526417537, 0.1002440660353247},

{9, -1., 0.0504288271173981, 0.07382866194205702, 0.07068535133336745},

{10, -1., 0.03418401370580981, 0.0504288271173981, 0.04897887259222585},

{11, -1., 0.02304895054924056, 0.03418401370580981, 0.03352316243099189},

{12, -1., 0.0154840093511178, 0.02304895054924056, 0.02275021729713151}

}

c= 0.0154840093511178

f[c]= 0.02275021729713151

Concise Program for the Regula Falsi

RegulaFalsi2[a0_,b0_,m_]:=Module[{},a=N[a0];b=N[b0];Ya=f[a];Yb=f[b];

c=(a*Yb-b*Ya)/(Yb-Ya); Yc=f[c]; k=0;While[k<m,

If[Sign[Yb]==Sign[Yc],

b=c;

Yb=Yc,

a=c;

Ya=Yc;];

c=(a*Yb-b*Ya)/(Yb-Ya);

Yc=f[c];

k=k+1;];

Print[" c=", NumberForm[c,11]];

Print["f[c]=", NumberForm[Yc,11]];]

EXAMPLE:

f[x_]=x^3-2x^2+3x/2

(3 x)/2-2 x2+x3

RegulaFalsi2[-1,1,12]

c= 0.015484009351

f[c]= 0.022750217297

Plot[(3 x)/2-2 x2+x3,{x,-1.,1.}]

FIXED POINT METHOD - WOLFRAM MATHEMATICA V10

ROOTFINDING

Fixed Point Method

www.jesus-avalos.ucoz.com

ALGORITHM CODE:

FixedPointIteration[x0_,max_]:=Module[{},p0=N[x0];k=0;

Print["0 p=",PaddedForm[p0,{15,15}]];

While[k<max,Module[{},p1=g[p0];k=k+1;

Print[" p"k, "=", PaddedForm[p1,{15,15}]];

p0=p1;];];

p=p0;

Print[" "];

Print["The function is g[x]=", g[x]];

Print[" p=",PaddedForm[p,{15,15}]];

Print["g[p]=",PaddedForm[g[p],{15,15}]];];

EXAMPLE: x^3+x-1=0

g[x_]=(1-x)1/3

FixedPointIteration[0.5,12]

0 p= 0.500000000000000

p = 0.793700525984100

2 p = 0.590880113275177

3 p = 0.742363932168006

4 p = 0.636310203481661

5 p = 0.713800814144207

6 p = 0.659006145622400

7 p = 0.698632605730219

8 p = 0.670448496228072

9 p = 0.690729120589141

10 p = 0.676258924926827

11 p = 0.686645536864490

12 p = 0.679222339897004

13 p = 0.684544005469716

The function is g[x]= (1-x)1/3

p= 0.684544005469716

g[p]= 0.680737373803562

Plot[{x^3+x-1,(1-x)1/3, x},{x,0,1}]

BISECTION METHOD - WOLFRAM MATHEMATICA V10

ROOTFINDING

Bisection Method

www.jesus-avalos.ucoz.com

ALGORITHM CODE:

Bisection[a0_,b0_,m_]:=Module[{},a=N[a0];b=N[b0]; c=(a+b)/2; k=0;

output={{k,a,c,b,f[c]}};

While[k<m,

If[Sign[f[b]]==Sign[f[c]],

b=c, a=c;];

c=(a+b)/2;

k=k+1;

output=Append[output,{k,a,c,b,f[c]}];];

Print[NumberForm[TableForm[output,

TableHeadings->{None,{"k","a_k","c_k", "b_k","f[c_k]"}}],16]];

Print[" c=",NumberForm[c,16]];

Print[" Δc=±",(b-a)/2];

Print["f[c]=", NumberForm[f[c],16]];]

EXAMPLE:

f[x_]=x^3+x-1

-1+x+x3

Bisection[0,1,10]

{

{k, a_k, c_k, b_k, f[c_k]},

{0, 0., 0.5, 1., -0.375},

{1, 0.5, 0.75, 1., 0.171875},

{2, 0.5, 0.625, 0.75, -0.130859375},

{3, 0.625, 0.6875, 0.75, 0.012451171875},

{4, 0.625, 0.65625, 0.6875, -0.061126708984375},

{5, 0.65625, 0.671875, 0.6875, -0.02482986450195312},

{6, 0.671875, 0.6796875, 0.6875, -0.006313800811767578},

{7, 0.6796875, 0.68359375, 0.6875, 0.003037393093109131},

{8, 0.6796875, 0.681640625, 0.68359375, -0.001646004617214203},

{9, 0.681640625, 0.6826171875, 0.68359375, 0.0006937412545084953},

{10, 0.681640625, 0.68212890625, 0.6826171875, -0.0004766195779666305}

}

c= 0.68212890625

Δc=± 0.000488281

f[c]= -0.0004766195779666305

Plot[{x^3 + x - 1, (1 - x)^(1/3), x}, {x, 0, 1}]

NEWTON RAPHSON METHOD

CURVAS FRACTALES

Las curvas fractales se generan a partir de una curva inicial (a menudo un polígono regular) y una o más curvas de recambio. Repetidamente, cada línea se sustituye por una copia correctamente a escala de una de las curvas de recambio. Los botones muestran la primera iteración de la curva de sustitución utilizado. Para ver la curva inicial, establecer el número de pasos a 0.

CÁLCULO DE ÍNDICE DE MASA CORPORAL (IMC): PROGRAMA HECHO EN OCTAVE GNU

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% ÍNDICE DE MASA CORPORAL - IMC

% P=PESO DEL PACIENTE (EN kg)

% E=ESTATURA DEL PACIENTE (EN m)

% Resultado= ÍnDICE DE MASA CORPORAL

% AUTOR:JAR

%https://sites.google.com/unitru.edu.pe/jar

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clc, clear

disp(" TABLA DE VALORES IMC\n");

disp(" =================================================\n");

disp("\t IMC :\t CONDICIÓN\t\n");

disp("\t menor a 18.5 :\t bajo peso\n");

disp("\t 18.5 a 24.9 :\t peso normal\n");

disp("\t 25 a 26.9 :\t sobrepeso grado I\n");

disp("\t 27 a 29.9 :\t sobrepeso grado II\n");

disp("\t 30 a 34.9 :\t obesidad tipo I\n");

disp("\t 35 a 39.9 :\t obesidad tipo II\n");

disp("\t 40 a 49.9 :\t obesidad tipo III\n");

disp("\t mayor a 50 :\t obesidad tipo IV\n");

disp(" =================================================\n");

disp("Se necesita tu peso P (en kg) y estatura E (en metros)");

P=input("\n\nIngresa tu peso P:");

E=input("Ingresa tu estatura E:");

if(P<0||E<0)

printf("Dato ingresado inválido, vuelva a ejecutar");

return

endif

Resultado=P/(E^2);

if(Resultado<18.5)

C="Estás bajo de peso.";

elseif(Resultado>=18.5&&Resultado<24.9)

C="Estás con un peso normal.";

elseif(Resultado>25&&Resultado<26.9)

C="Estás con un sobrepeso grado I.";

elseif(Resultado>=27&&Resultado<29.9)

C="Estás con un sobrepeso grado II.";

elseif(Resultado>=30&&Resultado<34.9)

C="Estás con obesidad tipo I.";

elseif(Resultado>=35&&Resultado<39.9)

C="Estás con obesidad tipo II.";

elseif(Resultado>=40&&Resultado<49.9)

C="Estás con obesidad tipo III.";

elseif(Resultado>=50)

C="Estás con obesidad tipo IV.";

endif

fprintf("\n Tienes un índice de masa coroporal es: %f. \n %s",Resultado, C);

%%%%%%%%%%%

PUEDE EJECUTAR DESDE:

https://octave-online.net/bucket~FV2pdyyLb3SGHcxj7ptyvo

https://www.12000.org/index.htm