Métodos Numéricos
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Reference

Formulario de métodos numéricos

Pensado para repasar antes de resolver problemas. Cada fórmula enlaza conceptualmente con guías largas y ejercicios resueltos del sitio.

Errores y Taylor

ε=∣y−y^∣,εr=∣y−y^∣∣y∣\varepsilon=|y-\hat y|,\qquad \varepsilon_r=\frac{|y-\hat y|}{|y|}
εk=∣y^k+1−y^k∣\varepsilon_k=|\hat y_{k+1}-\hat y_k|
f(x+h)=∑j=0mf(j)(x)j!hj+f(m+1)(ξ)(m+1)!hm+1f(x+h)=\sum_{j=0}^{m}\frac{f^{(j)}(x)}{j!}h^j+\frac{f^{(m+1)}(\xi)}{(m+1)!}h^{m+1}

Interpolación

Li(x)=∏j≠ix−xjxi−xj,pn(x)=∑i=0nLi(x)f(xi)L_i(x)=\prod_{j\ne i}\frac{x-x_j}{x_i-x_j},\qquad p_n(x)=\sum_{i=0}^n L_i(x)f(x_i)
f[x0,…,xk]=f[x1,…,xk]−f[x0,…,xk−1]xk−x0f[x_0,\dots,x_k]=\frac{f[x_1,\dots,x_k]-f[x_0,\dots,x_{k-1}]}{x_k-x_0}
f(x)−pn(x)=f(n+1)(ξ)(n+1)!∏i=0n(x−xi)f(x)-p_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^n(x-x_i)

Diferenciación

f′(x)≈f(x+h)−f(x)hf'(x)\approx\frac{f(x+h)-f(x)}{h}
f′(x)≈f(x+h)−f(x−h)2hf'(x)\approx\frac{f(x+h)-f(x-h)}{2h}
f′′(x)≈f(x+h)−2f(x)+f(x−h)h2f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}

Integración

M1=(b−a)f(a+b2),EM=(b−a)324f′′(ξ)M_1=(b-a)f\left(\frac{a+b}{2}\right),\qquad E_M=\frac{(b-a)^3}{24}f''(\xi)
Mn=h∑i=0n−1f(a+(i+12)h),EM=b−a24h2f′′(ξ)M_n=h\sum_{i=0}^{n-1}f\left(a+\left(i+\frac12\right)h\right),\qquad E_M=\frac{b-a}{24}h^2f''(\xi)
Tn=h2(f0+2∑i=1n−1fi+fn),ET=−b−a12h2f′′(ξ)T_n=\frac{h}{2}\left(f_0+2\sum_{i=1}^{n-1}f_i+f_n\right),\qquad E_T=-\frac{b-a}{12}h^2f''(\xi)
Sn=h3(f0+fn+4∑i oddfi+2∑i evenfi)S_n=\frac{h}{3}\left(f_0+f_n+4\sum_{i\ odd}f_i+2\sum_{i\ even}f_i\right)
∫abf(x)dx=b−a2∫−11f(a+b2+b−a2t)dt\int_a^b f(x)dx=\frac{b-a}{2}\int_{-1}^{1}f\left(\frac{a+b}{2}+\frac{b-a}{2}t\right)dt

EDO: métodos de un paso

yk+1=yk+hf(tk,yk)y_{k+1}=y_k+hf(t_k,y_k)
yk+1=yk+hf(tk+1,yk+1)y_{k+1}=y_k+hf(t_{k+1},y_{k+1})
yk+1=yk+12k1+12k2,k1=hf(tk,yk),  k2=hf(tk+1,yk+k1)y_{k+1}=y_k+\tfrac12 k_1+\tfrac12 k_2,\quad k_1=hf(t_k,y_k),\; k_2=hf(t_{k+1},y_k+k_1)
yk+1=yk+h6(k1+2k2+2k3+k4)y_{k+1}=y_k+\frac{h}{6}(k_1+2k_2+2k_3+k_4)

EDO: métodos multipaso

yk+1=yk+h2(3fk−fk−1)y_{k+1}=y_k+\frac{h}{2}(3f_k-f_{k-1})
yk+1=yk+h24(55fk−59fk−1+37fk−2−9fk−3)y_{k+1}=y_k+\frac{h}{24}(55f_k-59f_{k-1}+37f_{k-2}-9f_{k-3})
yk+1=yk+h2(fk+1+fk)y_{k+1}=y_k+\frac{h}{2}(f_{k+1}+f_k)

Sistemas lineales

x(k+1)=Bx(k)+c,ρ(B)<1x^{(k+1)}=Bx^{(k)}+c,\qquad \rho(B)<1
κ(A)=∥A∥ ∥A−1∥\kappa(A)=\|A\|\,\|A^{-1}\|

Ecuaciones no lineales

mk=ak+bk2,∣mk−α∣≤b−a2k+1m_k=\frac{a_k+b_k}{2},\qquad |m_k-\alpha|\le\frac{b-a}{2^{k+1}}
xk+1=xk−f(xk)f′(xk)x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}
xk+1=xk−f(xk)f[xk,xk−1]x_{k+1}=x_k-\frac{f(x_k)}{f[x_k,x_{k-1}]}

Sistemas no lineales

x(k+1)=x(k)−[F′(x(k))]−1F(x(k))x^{(k+1)}=x^{(k)}-[F'(x^{(k)})]^{-1}F(x^{(k)})
F′(x(k)) u=F(x(k)),x(k+1)=x(k)−uF'(x^{(k)})\,u=F(x^{(k)}),\qquad x^{(k+1)}=x^{(k)}-u