métodos-numéricos y topología-algebraica

Teorema:

Sea d_{x}[y(x)] = y+x+(k+(-1)) ==>

[Ej][ (1/h)·( y_{n+1}+(-1)·y_{n} ) = y_{n}+j ] es un método numérico convergente a y(x)

Demostración:

Sea h = 0a & j = (-1)·((k/a)+1) ==>

y_{n+1} = y_{n}+h·( y_{n}+j ) = y_{n}·(1+h)+hj

y_{n+1} = y_{0}·(1+h)^{n}+nhj

Sea y_{0} = 1 ==>

y(a) = y_{oo} = e^{a}+(-k)+(-a)

Teorema:

Sea d_{x}[y(x)] = y+x^{2}+(k+(-2)) ==>

[Ej][ (1/h)·( y_{n+1}+(-1)·y_{n} ) = y_{n}+j ] es un método numérico convergente a y(x)

Demostración:

Sea h = 0a & j = (-1)·((k/a)+2+a) ==>

y_{n+1} = y_{n}+h·( y_{n}+j ) = y_{n}·(1+h)+hj

y_{n+1} = y_{0}·(1+h)^{n}+nhj

Sea y_{0} = 1 ==>

y(a) = y_{oo} = e^{a}+(-k)+(-1)·2a+(-1)·a^{2}

Teorema: [ de sp-line cuadrática ]

P(x) = (x+(-1)·x_{j})·(x+(-1)·x_{k})·( (x_{i}+(-1)·x_{j})·(x_{i}+(-1)·x_{k}) )^{(-1)}·f(x_{i})

Teorema: [ de sp-line cúbica ]

Q(x) = ...

... (x+(-1)·x_{j})·x·(x+(-1)·x_{k})·( (x_{i}+(-1)·x_{j})·x_{i}·(x_{i}+(-1)·x_{k}) )^{(-1)}·f(x_{i})

Teorema:

Sea  a_{n+1} = (1/2)·( a_{n}+(a_{n})^{m}·y_{n} ) ==> a_{oo} = ( y_{n} )^{( 1/(1+(-m)) )}

Sea d_{x}[y(x)] = y^{m} ==>

[Ej][ (1/h)·( y_{n+1}+(-1)·( y_{n} )^{j} ) = ( y_{n} )^{m} ] es un método numérico convergente a y(x)

Demostración:

Sea h = 0 & j = ( 1/(1+(-m))^{0} ) ==>

y_{n+1} = ( y_{n} )^{j}+h·( y_{n} )^{m} = ( y_{n} )^{m+[j+(-m):h]}

y_{n+1} = ( y_{0} )^{( 1/(1+(-m)) )^{0n}}

Sea y_{0} = a ==>

y(a) = y_{oo} = a^{( 1/(1+(-m)) )}

Teorema:

Sea f_{n}(x): ( x+(-a) )^{n} ---> ( x+(-a) )^{n+1} ==>

[Ex][ f_{n}(x) está compactificada en 2 clases ]

Teorema:

Sea f_{n}(x): ( e^{x}+(-a) )^{n} ---> ( e^{x}+(-a) )^{n+1} ==>

[Ex][ f_{n}(x) está compactificada en 2 clases ]

Teorema:

Sea f_{n}(P(x)): d_{x...x}^{n}[P(x)]·h(x) ---> Q(x) [o(x)o] ( x /o(x)o/ H(x) ) ==>

[EP(x)][ f_{n}(P(x)) está compactificada en 2 clases ]

Demostración:

d_{x}[ sinh(x) [o(x)o] ( x /o(x)o/ H(x) ) ]·h(x) = cosh(x)

d_{x}[ cosh(x) [o(x)o] ( x /o(x)o/ H(x) ) ]·h(x) = sinh(x)