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m (→‎Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1)
m (→‎Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1)
Linha 151: Linha 151:
 
O espectro tem autovalores próximos do eixo real (talvez devido a erros de ponto flutuante) com σ⊂ 1.6537 ∪ (-1,1) incluindo 0.
 
O espectro tem autovalores próximos do eixo real (talvez devido a erros de ponto flutuante) com σ⊂ 1.6537 ∪ (-1,1) incluindo 0.
  
 +
Incluindo CC na matriz                    Excluindo CC na matriz
 +
lambda(1)=  1.72605097442453            lambda(1)=  0.955274744954830     
 +
lambda(n)=  0.000000000000000E+000        lambda(n)=  4.218228659674831E-003
 +
( 0.173E+01, 0.000E+00)    (-0.955E+00, 0.000E+00)
 +
( 0.975E+00, 0.000E+00)    ( 0.955E+00, 0.000E+00)
 +
( 0.969E+00, 0.000E+00)    (-0.896E+00, 0.000E+00)
 +
(-0.945E+00, 0.000E+00)    ( 0.896E+00, 0.000E+00)
 +
( 0.942E+00, 0.589E-02)    ( 0.884E+00, 0.000E+00)
 +
( 0.942E+00,-0.589E-02)    (-0.884E+00, 0.000E+00)
 +
( 0.893E+00, 0.000E+00)    ( 0.825E+00, 0.000E+00)
 +
(-0.875E+00, 0.000E+00)    (-0.825E+00, 0.000E+00)
 +
( 0.874E+00, 0.000E+00)    (-0.803E+00, 0.000E+00)
 +
(-0.859E+00, 0.000E+00)    ( 0.803E+00, 0.000E+00)
 +
( 0.857E+00, 0.000E+00)    ( 0.774E+00, 0.000E+00)
 +
( 0.795E+00, 0.244E-02)    (-0.774E+00, 0.000E+00)
 +
( 0.795E+00,-0.244E-02)    ( 0.732E+00, 0.000E+00)
 +
(-0.788E+00, 0.000E+00)    (-0.732E+00, 0.000E+00)
 +
(-0.764E+00, 0.000E+00)    ( 0.715E+00, 0.000E+00)
 +
( 0.762E+00, 0.000E+00)    (-0.715E+00, 0.000E+00)
 +
( 0.748E+00, 0.000E+00)    ( 0.683E+00, 0.000E+00)
 +
(-0.726E+00, 0.000E+00)    (-0.683E+00, 0.000E+00)
 +
( 0.715E+00, 0.774E-02)    (-0.634E+00, 0.000E+00)
 +
( 0.715E+00,-0.774E-02)    ( 0.634E+00, 0.000E+00)
 +
( 0.679E+00, 0.000E+00)    ( 0.621E+00, 0.000E+00)
 +
(-0.677E+00, 0.000E+00)    (-0.621E+00, 0.000E+00)
 +
(-0.656E+00, 0.000E+00)    ( 0.612E+00, 0.000E+00)
 +
( 0.655E+00, 0.000E+00)    (-0.612E+00, 0.000E+00)
 +
(-0.625E+00, 0.000E+00)    ( 0.575E+00, 0.000E+00)
 +
( 0.620E+00, 0.362E-02)    (-0.575E+00, 0.000E+00)
 +
( 0.620E+00,-0.362E-02)    ( 0.547E+00, 0.000E+00)
 +
(-0.562E+00, 0.000E+00)    (-0.547E+00, 0.000E+00)
 +
( 0.561E+00, 0.000E+00)    ( 0.502E+00, 0.000E+00)
 +
(-0.545E+00, 0.000E+00)    (-0.502E+00, 0.000E+00)
 +
( 0.544E+00, 0.277E-02)    ( 0.482E+00, 0.000E+00)
 +
( 0.544E+00,-0.277E-02)    (-0.482E+00, 0.000E+00)
 +
(-0.538E+00, 0.000E+00)    ( 0.480E+00, 0.000E+00)
 +
( 0.500E+00, 0.219E-02)    (-0.480E+00, 0.000E+00)
 +
( 0.500E+00,-0.219E-02)    ( 0.476E+00, 0.000E+00)
 +
(-0.492E+00, 0.000E+00)    (-0.476E+00, 0.000E+00)
 +
(-0.470E+00, 0.000E+00)    ( 0.421E+00, 0.000E+00)
 +
( 0.469E+00, 0.000E+00)    (-0.421E+00, 0.000E+00)
 +
(-0.418E+00, 0.000E+00)    ( 0.404E+00, 0.000E+00)
 +
( 0.412E+00, 0.000E+00)    (-0.404E+00, 0.000E+00)
 +
( 0.401E+00, 0.000E+00)    ( 0.365E+00, 0.000E+00)
 +
(-0.389E+00, 0.000E+00)    (-0.365E+00, 0.000E+00)
 +
( 0.387E+00, 0.137E-01)    ( 0.362E+00, 0.000E+00)
 +
( 0.387E+00,-0.137E-01)    (-0.362E+00, 0.000E+00)
 +
( 0.386E+00, 0.000E+00)    (-0.333E+00, 0.000E+00)
 +
(-0.384E+00, 0.000E+00)    ( 0.333E+00, 0.000E+00)
 +
( 0.382E+00, 0.000E+00)    ( 0.327E+00, 0.000E+00)
 +
(-0.381E+00, 0.000E+00)    (-0.327E+00, 0.000E+00)
 +
(-0.332E+00, 0.000E+00)    (-0.325E+00, 0.000E+00)
 +
( 0.317E+00, 0.000E+00)    ( 0.325E+00, 0.000E+00)
 +
(-0.316E+00, 0.000E+00)    (-0.268E+00, 0.000E+00)
 +
( 0.314E+00, 0.000E+00)    ( 0.268E+00, 0.000E+00)
 +
( 0.290E+00, 0.000E+00)    (-0.266E+00, 0.000E+00)
 +
(-0.256E+00, 0.000E+00)    ( 0.266E+00, 0.000E+00)
 +
( 0.250E+00, 0.000E+00)    ( 0.226E+00, 0.000E+00)
 +
(-0.245E+00, 0.000E+00)    (-0.226E+00, 0.000E+00)
 +
( 0.244E+00, 0.000E+00)    ( 0.223E+00, 0.000E+00)
 +
(-0.233E+00, 0.000E+00)    (-0.223E+00, 0.000E+00)
 +
( 0.232E+00, 0.000E+00)    (-0.208E+00, 0.000E+00)
 +
(-0.226E+00, 0.000E+00)    ( 0.208E+00, 0.000E+00)
 +
( 0.225E+00, 0.000E+00)    ( 0.197E+00, 0.000E+00)
 +
(-0.221E+00, 0.000E+00)    (-0.197E+00, 0.000E+00)
 +
( 0.207E+00, 0.000E+00)    ( 0.186E+00, 0.000E+00)
 +
( 0.206E+00, 0.000E+00)    (-0.186E+00, 0.000E+00)
 +
(-0.204E+00, 0.000E+00)    ( 0.173E+00, 0.000E+00)
 +
(-0.176E+00, 0.000E+00)    (-0.173E+00, 0.000E+00)
 +
( 0.176E+00, 0.000E+00)    ( 0.148E+00, 0.000E+00)
 +
( 0.163E+00, 0.000E+00)    (-0.148E+00, 0.000E+00)
 +
(-0.154E+00, 0.000E+00)    ( 0.127E+00, 0.000E+00)
 +
( 0.117E+00, 0.000E+00)    (-0.127E+00, 0.000E+00)
 +
( 0.107E+00, 0.000E+00)    (-0.862E-01, 0.000E+00)
 +
( 0.955E-01, 0.000E+00)    ( 0.862E-01, 0.000E+00)
 +
(-0.925E-01, 0.000E+00)    (-0.834E-01, 0.000E+00)
 +
( 0.925E-01, 0.000E+00)    ( 0.834E-01, 0.000E+00)
 +
(-0.894E-01, 0.000E+00)    (-0.771E-01, 0.000E+00)
 +
( 0.891E-01, 0.000E+00)    ( 0.771E-01, 0.000E+00)
 +
(-0.878E-01, 0.000E+00)    (-0.752E-01, 0.000E+00)
 +
( 0.867E-01, 0.000E+00)    ( 0.752E-01, 0.000E+00)
 +
(-0.855E-01, 0.000E+00)    (-0.712E-01, 0.000E+00)
 +
( 0.675E-01, 0.000E+00)    ( 0.712E-01, 0.000E+00)
 +
(-0.656E-01, 0.000E+00)    ( 0.549E-01, 0.000E+00)
 +
( 0.652E-01, 0.000E+00)    (-0.549E-01, 0.000E+00)
 +
(-0.457E-01, 0.000E+00)    (-0.532E-01, 0.000E+00)
 +
( 0.436E-01, 0.000E+00)    ( 0.532E-01, 0.000E+00)
 +
(-0.427E-01, 0.000E+00)    ( 0.335E-01, 0.000E+00)
 +
( 0.372E-01, 0.000E+00)    (-0.335E-01, 0.000E+00)
 +
( 0.245E-01, 0.000E+00)    ( 0.161E-01, 0.000E+00)
 +
( 0.215E-01, 0.000E+00)    (-0.161E-01, 0.000E+00)
 +
(-0.107E-01, 0.000E+00)    ( 0.422E-02, 0.000E+00)
 +
( 0.568E-02, 0.000E+00)    (-0.422E-02, 0.000E+00)
 +
(-0.567E-02, 0.000E+00)   
 +
( 0.701E-14, 0.566E-15)
 +
( 0.701E-14,-0.566E-15)
 +
(-0.579E-14, 0.000E+00)
 +
(-0.219E-14, 0.000E+00)
 +
(-0.161E-14, 0.000E+00)
 +
( 0.497E-15, 0.120E-14)
 +
( 0.497E-15,-0.120E-14)
 +
(-0.683E-16, 0.975E-15)
 +
(-0.683E-16,-0.975E-15)
 +
(-0.478E-15, 0.405E-15)
 +
(-0.478E-15,-0.405E-15)
 +
( 0.624E-15, 0.000E+00)
 +
(-0.431E-15, 0.000E+00)
 +
( 0.147E-15, 0.360E-15)
 +
( 0.147E-15,-0.360E-15)
 +
(-0.183E-15, 0.295E-15)
 +
(-0.183E-15,-0.295E-15)
 +
( 0.317E-15, 0.000E+00)
 +
( 0.124E-15, 0.232E-15)
 +
( 0.124E-15,-0.232E-15)
 +
( 0.197E-15, 0.000E+00)
 +
( 0.165E-16, 0.190E-15)
 +
( 0.165E-16,-0.190E-15)
 +
(-0.188E-15, 0.000E+00)
 +
( 0.104E-15, 0.000E+00)
 +
( 0.322E-16, 0.971E-16)
 +
( 0.322E-16,-0.971E-16)
 +
(-0.187E-16, 0.857E-16)
 +
(-0.187E-16,-0.857E-16)
 +
(-0.229E-16, 0.000E+00)
 +
(-0.904E-17, 0.887E-17)
 +
(-0.904E-17,-0.887E-17)
 +
( 0.651E-28, 0.000E+00)
 +
( 0.264E-29, 0.000E+00)
 +
( 0.522E-30, 0.000E+00)
 +
(-0.923E-31, 0.000E+00)
 +
( 0.121E-31, 0.000E+00)
 +
( 0.000E+00, 0.000E+00)
 +
( 0.000E+00, 0.000E+00)
 +
( 0.000E+00, 0.000E+00)
 +
( 0.000E+00, 0.000E+00)
  
 
===Condição de Neumann em todos os lados e subtraindo P(1,2) ===
 
===Condição de Neumann em todos os lados e subtraindo P(1,2) ===

Edição das 10h43min de 30 de julho de 2009

A análise dos autovalores de uma matriz de iteração pode ser usada para estudar a estabilidade de um método iterativo.

Vamos relatar um estudo para um problema específico.

O problema

Queremos aproximar a solução da equação de Navier Stokes em um duto. Para isso devemos resolver a cada passo de tempo uma equação de Poisson como <math>\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}= f(u)\,\!</math>

Condição de Neumann em todos os lados

O espectro é real com espectro σ⊂(-1,1). 

Incluindo CC na matriz        Eliminando CC na matriz

lambda(1) = 1.0000E+00        lambda(1)=  0.955274744954830
lambda(n) = 0.0000E+00        lambda(n)=  4.218228659674831E-003

( 0.100E+01, 0.000E+00)      (-0.955E+00, 0.000E+00)
( 0.976E+00, 0.000E+00)      ( 0.955E+00, 0.000E+00)           
( 0.970E+00, 0.000E+00)      (-0.896E+00, 0.000E+00)           
(-0.945E+00, 0.000E+00)      ( 0.896E+00, 0.000E+00)           
( 0.945E+00, 0.000E+00)      ( 0.884E+00, 0.000E+00)           
( 0.905E+00, 0.000E+00)      (-0.884E+00, 0.000E+00)           
( 0.883E+00, 0.000E+00)      ( 0.825E+00, 0.000E+00)           
(-0.874E+00, 0.000E+00)      (-0.825E+00, 0.000E+00)           
( 0.874E+00, 0.000E+00)      (-0.803E+00, 0.000E+00)           
( 0.859E+00, 0.000E+00)      ( 0.803E+00, 0.000E+00)           
(-0.859E+00, 0.000E+00)      ( 0.774E+00, 0.000E+00)           
( 0.794E+00, 0.000E+00)      (-0.774E+00, 0.000E+00)           
(-0.788E+00, 0.000E+00)      ( 0.732E+00, 0.000E+00)           
( 0.788E+00, 0.000E+00)      (-0.732E+00, 0.000E+00)           
( 0.764E+00, 0.000E+00)      ( 0.715E+00, 0.000E+00)           
(-0.764E+00, 0.000E+00)      (-0.715E+00, 0.000E+00)           
( 0.750E+00, 0.000E+00)      ( 0.683E+00, 0.000E+00)           
(-0.726E+00, 0.000E+00)      (-0.683E+00, 0.000E+00)           
( 0.726E+00, 0.000E+00)      (-0.634E+00, 0.000E+00)           
( 0.677E+00, 0.000E+00)      ( 0.634E+00, 0.000E+00)           
(-0.677E+00, 0.000E+00)      ( 0.621E+00, 0.000E+00)           
(-0.655E+00, 0.000E+00)      (-0.621E+00, 0.000E+00)           
( 0.655E+00, 0.000E+00)      ( 0.612E+00, 0.000E+00)           
( 0.655E+00, 0.000E+00)      (-0.612E+00, 0.000E+00)           
( 0.624E+00, 0.000E+00)      ( 0.575E+00, 0.000E+00)           
(-0.624E+00, 0.000E+00)      (-0.575E+00, 0.000E+00)           
( 0.587E+00, 0.000E+00)      ( 0.547E+00, 0.000E+00)           
(-0.562E+00, 0.000E+00)      (-0.547E+00, 0.000E+00)           
( 0.562E+00, 0.000E+00)      ( 0.502E+00, 0.000E+00)           
(-0.544E+00, 0.000E+00)      (-0.502E+00, 0.000E+00)           
( 0.544E+00, 0.000E+00)      ( 0.482E+00, 0.000E+00)           
( 0.538E+00, 0.000E+00)      (-0.482E+00, 0.000E+00)           
(-0.538E+00, 0.000E+00)      ( 0.480E+00, 0.000E+00)           
( 0.500E+00, 0.000E+00)      (-0.480E+00, 0.000E+00)           
( 0.491E+00, 0.000E+00)      ( 0.476E+00, 0.000E+00)           
(-0.491E+00, 0.000E+00)      (-0.476E+00, 0.000E+00)           
(-0.470E+00, 0.000E+00)      ( 0.421E+00, 0.000E+00)           
( 0.470E+00, 0.000E+00)      (-0.421E+00, 0.000E+00)           
( 0.413E+00, 0.000E+00)      ( 0.404E+00, 0.000E+00)           
(-0.405E+00, 0.000E+00)      (-0.404E+00, 0.000E+00)           
( 0.405E+00, 0.000E+00)      ( 0.365E+00, 0.000E+00)           
(-0.389E+00, 0.000E+00)      (-0.365E+00, 0.000E+00)           
( 0.389E+00, 0.000E+00)      ( 0.362E+00, 0.000E+00)           
(-0.383E+00, 0.000E+00)      (-0.362E+00, 0.000E+00)           
( 0.383E+00, 0.000E+00)      (-0.333E+00, 0.000E+00)           
(-0.381E+00, 0.000E+00)      ( 0.333E+00, 0.000E+00)           
( 0.381E+00, 0.000E+00)      ( 0.327E+00, 0.000E+00)           
( 0.345E+00, 0.000E+00)      (-0.327E+00, 0.000E+00)           
(-0.318E+00, 0.000E+00)      (-0.325E+00, 0.000E+00)           
( 0.318E+00, 0.000E+00)      ( 0.325E+00, 0.000E+00)           
( 0.315E+00, 0.000E+00)      (-0.268E+00, 0.000E+00)           
(-0.315E+00, 0.000E+00)      ( 0.268E+00, 0.000E+00)           
(-0.250E+00, 0.000E+00)      (-0.266E+00, 0.000E+00)           
( 0.250E+00, 0.000E+00)      ( 0.266E+00, 0.000E+00)           
( 0.250E+00, 0.000E+00)      ( 0.226E+00, 0.000E+00)           
(-0.241E+00, 0.000E+00)      (-0.226E+00, 0.000E+00)           
( 0.241E+00, 0.000E+00)      ( 0.223E+00, 0.000E+00)           
(-0.229E+00, 0.000E+00)      (-0.223E+00, 0.000E+00)           
( 0.229E+00, 0.000E+00)      (-0.208E+00, 0.000E+00)           
(-0.226E+00, 0.000E+00)      ( 0.208E+00, 0.000E+00)           
( 0.226E+00, 0.000E+00)      ( 0.197E+00, 0.000E+00)           
( 0.207E+00, 0.000E+00)      (-0.197E+00, 0.000E+00)           
(-0.207E+00, 0.000E+00)      ( 0.186E+00, 0.000E+00)           
( 0.206E+00, 0.000E+00)      (-0.186E+00, 0.000E+00)           
( 0.176E+00, 0.000E+00)      ( 0.173E+00, 0.000E+00)           
(-0.176E+00, 0.000E+00)      (-0.173E+00, 0.000E+00)           
(-0.155E+00, 0.000E+00)      ( 0.148E+00, 0.000E+00)           
( 0.155E+00, 0.000E+00)      (-0.148E+00, 0.000E+00)           
( 0.117E+00, 0.000E+00)      ( 0.127E+00, 0.000E+00)           
( 0.955E-01, 0.000E+00)      (-0.127E+00, 0.000E+00)           
( 0.955E-01, 0.000E+00)      (-0.862E-01, 0.000E+00)           
(-0.955E-01, 0.000E+00)      ( 0.862E-01, 0.000E+00)           
(-0.925E-01, 0.000E+00)      (-0.834E-01, 0.000E+00)           
( 0.925E-01, 0.000E+00)      ( 0.834E-01, 0.000E+00)           
(-0.891E-01, 0.000E+00)      (-0.771E-01, 0.000E+00)           
( 0.891E-01, 0.000E+00)      ( 0.771E-01, 0.000E+00)           
(-0.868E-01, 0.000E+00)      (-0.752E-01, 0.000E+00)           
( 0.868E-01, 0.000E+00)      ( 0.752E-01, 0.000E+00)           
( 0.677E-01, 0.000E+00)      (-0.712E-01, 0.000E+00)           
(-0.677E-01, 0.000E+00)      ( 0.712E-01, 0.000E+00)           
(-0.653E-01, 0.000E+00)      ( 0.549E-01, 0.000E+00)           
( 0.653E-01, 0.000E+00)      (-0.549E-01, 0.000E+00)           
( 0.439E-01, 0.000E+00)      (-0.532E-01, 0.000E+00)           
(-0.439E-01, 0.000E+00)      ( 0.532E-01, 0.000E+00)           
( 0.302E-01, 0.000E+00)      ( 0.335E-01, 0.000E+00)           
( 0.245E-01, 0.000E+00)      (-0.335E-01, 0.000E+00)           
( 0.215E-01, 0.000E+00)      ( 0.161E-01, 0.000E+00)           
(-0.215E-01, 0.000E+00)      (-0.161E-01, 0.000E+00)           
(-0.568E-02, 0.000E+00)      ( 0.422E-02, 0.000E+00)           
( 0.568E-02, 0.000E+00)      (-0.422E-02, 0.000E+00)           
( 0.251E-14, 0.857E-15)      
( 0.251E-14,-0.857E-15)      
(-0.255E-14, 0.000E+00)
( 0.233E-15, 0.235E-14)
( 0.233E-15,-0.235E-14)
(-0.124E-14, 0.565E-15)
(-0.124E-14,-0.565E-15)
( 0.113E-14, 0.000E+00)
(-0.772E-15, 0.000E+00)
( 0.632E-15, 0.207E-15)
( 0.632E-15,-0.207E-15)
(-0.490E-15, 0.439E-15)
(-0.490E-15,-0.439E-15)
( 0.165E-15, 0.488E-15)
( 0.165E-15,-0.488E-15)
( 0.458E-15, 0.000E+00)
(-0.433E-15, 0.000E+00)
(-0.330E-15, 0.101E-15)
(-0.330E-15,-0.101E-15)
( 0.212E-15, 0.141E-15)
( 0.212E-15,-0.141E-15)
(-0.123E-15, 0.217E-15)
(-0.123E-15,-0.217E-15)
( 0.750E-17, 0.128E-15)
( 0.750E-17,-0.128E-15)
( 0.277E-16, 0.412E-16)
( 0.277E-16,-0.412E-16)
(-0.217E-16, 0.252E-16)
(-0.217E-16,-0.252E-16)
( 0.327E-16, 0.000E+00)
(-0.230E-16, 0.000E+00)
(-0.824E-30, 0.885E-30)
(-0.824E-30,-0.885E-30)
( 0.680E-30, 0.000E+00)
(-0.684E-31, 0.000E+00)
( 0.317E-31, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)

Condição de Neumann em todos os lados fixando um ponto, P(1,2)=1

O espectro tem autovalores próximos do eixo real (talvez devido a erros de ponto flutuante) com σ⊂ 1.6537 ∪ (-1,1) incluindo 0. 

Incluindo CC na matriz                    Excluindo CC na matriz
lambda(1)=   1.72605097442453             lambda(1)=  0.955274744954830      
lambda(n)=  0.000000000000000E+000        lambda(n)=  4.218228659674831E-003
( 0.173E+01, 0.000E+00)     (-0.955E+00, 0.000E+00)
( 0.975E+00, 0.000E+00)     ( 0.955E+00, 0.000E+00)
( 0.969E+00, 0.000E+00)     (-0.896E+00, 0.000E+00)
(-0.945E+00, 0.000E+00)     ( 0.896E+00, 0.000E+00)
( 0.942E+00, 0.589E-02)     ( 0.884E+00, 0.000E+00)
( 0.942E+00,-0.589E-02)     (-0.884E+00, 0.000E+00)
( 0.893E+00, 0.000E+00)     ( 0.825E+00, 0.000E+00)
(-0.875E+00, 0.000E+00)     (-0.825E+00, 0.000E+00)
( 0.874E+00, 0.000E+00)     (-0.803E+00, 0.000E+00)
(-0.859E+00, 0.000E+00)     ( 0.803E+00, 0.000E+00)
( 0.857E+00, 0.000E+00)     ( 0.774E+00, 0.000E+00)
( 0.795E+00, 0.244E-02)     (-0.774E+00, 0.000E+00)
( 0.795E+00,-0.244E-02)     ( 0.732E+00, 0.000E+00)
(-0.788E+00, 0.000E+00)     (-0.732E+00, 0.000E+00)
(-0.764E+00, 0.000E+00)     ( 0.715E+00, 0.000E+00)
( 0.762E+00, 0.000E+00)     (-0.715E+00, 0.000E+00)
( 0.748E+00, 0.000E+00)     ( 0.683E+00, 0.000E+00)
(-0.726E+00, 0.000E+00)     (-0.683E+00, 0.000E+00)
( 0.715E+00, 0.774E-02)     (-0.634E+00, 0.000E+00)
( 0.715E+00,-0.774E-02)     ( 0.634E+00, 0.000E+00)
( 0.679E+00, 0.000E+00)     ( 0.621E+00, 0.000E+00)
(-0.677E+00, 0.000E+00)     (-0.621E+00, 0.000E+00)
(-0.656E+00, 0.000E+00)     ( 0.612E+00, 0.000E+00)
( 0.655E+00, 0.000E+00)     (-0.612E+00, 0.000E+00)
(-0.625E+00, 0.000E+00)     ( 0.575E+00, 0.000E+00)
( 0.620E+00, 0.362E-02)     (-0.575E+00, 0.000E+00)
( 0.620E+00,-0.362E-02)     ( 0.547E+00, 0.000E+00)
(-0.562E+00, 0.000E+00)     (-0.547E+00, 0.000E+00)
( 0.561E+00, 0.000E+00)     ( 0.502E+00, 0.000E+00)
(-0.545E+00, 0.000E+00)     (-0.502E+00, 0.000E+00)
( 0.544E+00, 0.277E-02)     ( 0.482E+00, 0.000E+00)
( 0.544E+00,-0.277E-02)     (-0.482E+00, 0.000E+00)
(-0.538E+00, 0.000E+00)     ( 0.480E+00, 0.000E+00)
( 0.500E+00, 0.219E-02)     (-0.480E+00, 0.000E+00)
( 0.500E+00,-0.219E-02)     ( 0.476E+00, 0.000E+00)
(-0.492E+00, 0.000E+00)     (-0.476E+00, 0.000E+00)
(-0.470E+00, 0.000E+00)     ( 0.421E+00, 0.000E+00)
( 0.469E+00, 0.000E+00)     (-0.421E+00, 0.000E+00)
(-0.418E+00, 0.000E+00)     ( 0.404E+00, 0.000E+00)
( 0.412E+00, 0.000E+00)     (-0.404E+00, 0.000E+00)
( 0.401E+00, 0.000E+00)     ( 0.365E+00, 0.000E+00)
(-0.389E+00, 0.000E+00)     (-0.365E+00, 0.000E+00)
( 0.387E+00, 0.137E-01)     ( 0.362E+00, 0.000E+00)
( 0.387E+00,-0.137E-01)     (-0.362E+00, 0.000E+00)
( 0.386E+00, 0.000E+00)     (-0.333E+00, 0.000E+00)
(-0.384E+00, 0.000E+00)     ( 0.333E+00, 0.000E+00)
( 0.382E+00, 0.000E+00)     ( 0.327E+00, 0.000E+00)
(-0.381E+00, 0.000E+00)     (-0.327E+00, 0.000E+00)
(-0.332E+00, 0.000E+00)     (-0.325E+00, 0.000E+00)
( 0.317E+00, 0.000E+00)     ( 0.325E+00, 0.000E+00)
(-0.316E+00, 0.000E+00)     (-0.268E+00, 0.000E+00)
( 0.314E+00, 0.000E+00)     ( 0.268E+00, 0.000E+00)
( 0.290E+00, 0.000E+00)     (-0.266E+00, 0.000E+00)
(-0.256E+00, 0.000E+00)     ( 0.266E+00, 0.000E+00)
( 0.250E+00, 0.000E+00)     ( 0.226E+00, 0.000E+00)
(-0.245E+00, 0.000E+00)     (-0.226E+00, 0.000E+00)
( 0.244E+00, 0.000E+00)     ( 0.223E+00, 0.000E+00)
(-0.233E+00, 0.000E+00)     (-0.223E+00, 0.000E+00)
( 0.232E+00, 0.000E+00)     (-0.208E+00, 0.000E+00)
(-0.226E+00, 0.000E+00)     ( 0.208E+00, 0.000E+00)
( 0.225E+00, 0.000E+00)     ( 0.197E+00, 0.000E+00)
(-0.221E+00, 0.000E+00)     (-0.197E+00, 0.000E+00)
( 0.207E+00, 0.000E+00)     ( 0.186E+00, 0.000E+00)
( 0.206E+00, 0.000E+00)     (-0.186E+00, 0.000E+00)
(-0.204E+00, 0.000E+00)     ( 0.173E+00, 0.000E+00)
(-0.176E+00, 0.000E+00)     (-0.173E+00, 0.000E+00)
( 0.176E+00, 0.000E+00)     ( 0.148E+00, 0.000E+00)
( 0.163E+00, 0.000E+00)     (-0.148E+00, 0.000E+00)
(-0.154E+00, 0.000E+00)     ( 0.127E+00, 0.000E+00)
( 0.117E+00, 0.000E+00)     (-0.127E+00, 0.000E+00)
( 0.107E+00, 0.000E+00)     (-0.862E-01, 0.000E+00)
( 0.955E-01, 0.000E+00)     ( 0.862E-01, 0.000E+00)
(-0.925E-01, 0.000E+00)     (-0.834E-01, 0.000E+00)
( 0.925E-01, 0.000E+00)     ( 0.834E-01, 0.000E+00)
(-0.894E-01, 0.000E+00)     (-0.771E-01, 0.000E+00)
( 0.891E-01, 0.000E+00)     ( 0.771E-01, 0.000E+00)
(-0.878E-01, 0.000E+00)     (-0.752E-01, 0.000E+00)
( 0.867E-01, 0.000E+00)     ( 0.752E-01, 0.000E+00)
(-0.855E-01, 0.000E+00)     (-0.712E-01, 0.000E+00)
( 0.675E-01, 0.000E+00)     ( 0.712E-01, 0.000E+00)
(-0.656E-01, 0.000E+00)     ( 0.549E-01, 0.000E+00)
( 0.652E-01, 0.000E+00)     (-0.549E-01, 0.000E+00)
(-0.457E-01, 0.000E+00)     (-0.532E-01, 0.000E+00)
( 0.436E-01, 0.000E+00)     ( 0.532E-01, 0.000E+00)
(-0.427E-01, 0.000E+00)     ( 0.335E-01, 0.000E+00)
( 0.372E-01, 0.000E+00)     (-0.335E-01, 0.000E+00)
( 0.245E-01, 0.000E+00)     ( 0.161E-01, 0.000E+00)
( 0.215E-01, 0.000E+00)     (-0.161E-01, 0.000E+00)
(-0.107E-01, 0.000E+00)     ( 0.422E-02, 0.000E+00)
( 0.568E-02, 0.000E+00)     (-0.422E-02, 0.000E+00)
(-0.567E-02, 0.000E+00)     
( 0.701E-14, 0.566E-15)
( 0.701E-14,-0.566E-15)
(-0.579E-14, 0.000E+00)
(-0.219E-14, 0.000E+00)
(-0.161E-14, 0.000E+00)
( 0.497E-15, 0.120E-14)
( 0.497E-15,-0.120E-14)
(-0.683E-16, 0.975E-15)
(-0.683E-16,-0.975E-15)
(-0.478E-15, 0.405E-15)
(-0.478E-15,-0.405E-15)
( 0.624E-15, 0.000E+00)
(-0.431E-15, 0.000E+00)
( 0.147E-15, 0.360E-15)
( 0.147E-15,-0.360E-15)
(-0.183E-15, 0.295E-15)
(-0.183E-15,-0.295E-15)
( 0.317E-15, 0.000E+00)
( 0.124E-15, 0.232E-15)
( 0.124E-15,-0.232E-15)
( 0.197E-15, 0.000E+00)
( 0.165E-16, 0.190E-15)
( 0.165E-16,-0.190E-15)
(-0.188E-15, 0.000E+00)
( 0.104E-15, 0.000E+00)
( 0.322E-16, 0.971E-16)
( 0.322E-16,-0.971E-16)
(-0.187E-16, 0.857E-16)
(-0.187E-16,-0.857E-16)
(-0.229E-16, 0.000E+00)
(-0.904E-17, 0.887E-17)
(-0.904E-17,-0.887E-17)
( 0.651E-28, 0.000E+00)
( 0.264E-29, 0.000E+00)
( 0.522E-30, 0.000E+00)
(-0.923E-31, 0.000E+00)
( 0.121E-31, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)

Condição de Neumann em todos os lados e subtraindo P(1,2)

Usando Pnew=Pnew-P(1,2) temos o espectro 

 Incluindo CC na Matriz                 Excluindo CC na Matriz
 
lambda(1)=  0.975528258147575          lambda(1)=  0.955274744954828       
lambda(n)=  0.000000000000000E+000     lambda(n)=  0.000000000000000E+000
 
( 0.976E+00, 0.000E+00)     (-0.955E+00, 0.000E+00)
( 0.970E+00, 0.000E+00)     ( 0.911E+00, 0.000E+00)
( 0.945E+00, 0.000E+00)     ( 0.896E+00, 0.000E+00)
(-0.945E+00, 0.000E+00)     (-0.896E+00, 0.000E+00)
( 0.905E+00, 0.000E+00)     ( 0.884E+00, 0.000E+00)
( 0.883E+00, 0.000E+00)     (-0.884E+00, 0.000E+00)
(-0.874E+00, 0.000E+00)     (-0.825E+00, 0.000E+00)
( 0.874E+00, 0.000E+00)     ( 0.825E+00, 0.000E+00)
( 0.859E+00, 0.000E+00)     (-0.803E+00, 0.000E+00)
(-0.859E+00, 0.000E+00)     ( 0.789E+00, 0.000E+00)
( 0.794E+00, 0.000E+00)     (-0.774E+00, 0.000E+00)
( 0.788E+00, 0.000E+00)     (-0.732E+00, 0.000E+00)
(-0.788E+00, 0.000E+00)     ( 0.732E+00, 0.000E+00)
( 0.764E+00, 0.000E+00)     ( 0.715E+00, 0.000E+00)
(-0.764E+00, 0.000E+00)     (-0.712E+00, 0.000E+00)
( 0.750E+00, 0.000E+00)     ( 0.693E+00, 0.000E+00)
(-0.726E+00, 0.000E+00)     ( 0.683E+00, 0.000E+00)
( 0.726E+00, 0.000E+00)     (-0.682E+00, 0.000E+00)
(-0.677E+00, 0.000E+00)     (-0.634E+00, 0.000E+00)
( 0.677E+00, 0.000E+00)     ( 0.634E+00, 0.000E+00)
( 0.655E+00, 0.000E+00)     (-0.621E+00, 0.000E+00)
(-0.655E+00, 0.000E+00)     (-0.612E+00, 0.000E+00)
( 0.655E+00, 0.000E+00)     ( 0.612E+00, 0.000E+00)
(-0.624E+00, 0.000E+00)     ( 0.583E+00, 0.000E+00)
( 0.624E+00, 0.000E+00)     (-0.575E+00, 0.000E+00)
( 0.587E+00, 0.000E+00)     ( 0.575E+00, 0.000E+00)
( 0.562E+00, 0.000E+00)     (-0.547E+00, 0.000E+00)
(-0.562E+00, 0.000E+00)     ( 0.507E+00, 0.000E+00)
( 0.544E+00, 0.000E+00)     ( 0.502E+00, 0.000E+00)
(-0.544E+00, 0.000E+00)     (-0.494E+00, 0.000E+00)
(-0.538E+00, 0.000E+00)     ( 0.482E+00, 0.000E+00)
( 0.538E+00, 0.000E+00)     (-0.482E+00, 0.000E+00)
( 0.500E+00, 0.000E+00)     (-0.480E+00, 0.000E+00)
( 0.491E+00, 0.000E+00)     (-0.476E+00, 0.000E+00)
(-0.491E+00, 0.000E+00)     ( 0.476E+00, 0.000E+00)
(-0.470E+00, 0.000E+00)     ( 0.421E+00, 0.000E+00)
( 0.470E+00, 0.000E+00)     (-0.416E+00, 0.000E+00)
( 0.413E+00, 0.000E+00)     ( 0.412E+00, 0.000E+00)
(-0.405E+00, 0.000E+00)     ( 0.404E+00, 0.000E+00)
( 0.405E+00, 0.000E+00)     (-0.402E+00, 0.000E+00)
(-0.389E+00, 0.000E+00)     (-0.365E+00, 0.000E+00)
( 0.389E+00, 0.000E+00)     (-0.362E+00, 0.000E+00)
( 0.383E+00, 0.000E+00)     ( 0.362E+00, 0.000E+00)
(-0.383E+00, 0.000E+00)     ( 0.343E+00, 0.000E+00)
(-0.381E+00, 0.000E+00)     ( 0.333E+00, 0.000E+00)
( 0.381E+00, 0.000E+00)     (-0.333E+00, 0.000E+00)
( 0.345E+00, 0.000E+00)     (-0.327E+00, 0.000E+00)
( 0.318E+00, 0.000E+00)     (-0.325E+00, 0.000E+00)
(-0.318E+00, 0.000E+00)     ( 0.325E+00, 0.000E+00)
( 0.315E+00, 0.000E+00)     ( 0.283E+00, 0.000E+00)
(-0.315E+00, 0.000E+00)     (-0.268E+00, 0.000E+00)
( 0.250E+00, 0.000E+00)     ( 0.266E+00, 0.000E+00)
(-0.250E+00, 0.000E+00)     (-0.266E+00, 0.000E+00)
( 0.250E+00, 0.000E+00)     (-0.226E+00, 0.000E+00)
( 0.241E+00, 0.000E+00)     ( 0.226E+00, 0.000E+00)
(-0.241E+00, 0.000E+00)     ( 0.223E+00, 0.000E+00)
(-0.229E+00, 0.000E+00)     (-0.218E+00, 0.000E+00)
( 0.229E+00, 0.000E+00)     ( 0.208E+00, 0.000E+00)
(-0.226E+00, 0.000E+00)     ( 0.202E+00, 0.000E+00)
( 0.226E+00, 0.000E+00)     (-0.197E+00, 0.000E+00)
( 0.207E+00, 0.000E+00)     ( 0.197E+00, 0.000E+00)
(-0.207E+00, 0.000E+00)     (-0.190E+00, 0.000E+00)
( 0.206E+00, 0.000E+00)     (-0.186E+00, 0.000E+00)
( 0.176E+00, 0.000E+00)     (-0.173E+00, 0.000E+00)
(-0.176E+00, 0.000E+00)     ( 0.173E+00, 0.000E+00)
( 0.155E+00, 0.000E+00)     ( 0.148E+00, 0.000E+00)
(-0.155E+00, 0.000E+00)     (-0.146E+00, 0.000E+00)
( 0.117E+00, 0.000E+00)     ( 0.127E+00, 0.000E+00)
(-0.955E-01, 0.000E+00)     ( 0.122E+00, 0.000E+00)
( 0.955E-01, 0.411E-15)     (-0.120E+00, 0.000E+00)
( 0.955E-01,-0.411E-15)     (-0.862E-01, 0.000E+00)
( 0.925E-01, 0.000E+00)     (-0.834E-01, 0.000E+00)
(-0.925E-01, 0.000E+00)     ( 0.834E-01, 0.000E+00)
(-0.891E-01, 0.000E+00)     ( 0.789E-01, 0.000E+00)
( 0.891E-01, 0.000E+00)     (-0.771E-01, 0.000E+00)
( 0.868E-01, 0.000E+00)     ( 0.771E-01, 0.000E+00)
(-0.868E-01, 0.000E+00)     (-0.752E-01, 0.000E+00)
(-0.677E-01, 0.000E+00)     ( 0.752E-01, 0.000E+00)
( 0.677E-01, 0.000E+00)     (-0.712E-01, 0.000E+00)
( 0.653E-01, 0.000E+00)     ( 0.573E-01, 0.000E+00)
(-0.653E-01, 0.000E+00)     (-0.549E-01, 0.000E+00)
(-0.439E-01, 0.000E+00)     (-0.532E-01, 0.000E+00)
( 0.439E-01, 0.000E+00)     ( 0.532E-01, 0.000E+00)
( 0.302E-01, 0.000E+00)     ( 0.395E-01, 0.000E+00)
( 0.245E-01, 0.000E+00)     (-0.335E-01, 0.000E+00)
(-0.215E-01, 0.000E+00)     (-0.161E-01, 0.000E+00)
( 0.215E-01, 0.000E+00)     ( 0.161E-01, 0.000E+00)
( 0.568E-02, 0.000E+00)     ( 0.523E-02, 0.000E+00)
(-0.568E-02, 0.000E+00)     (-0.422E-02, 0.000E+00)
(-0.444E-14, 0.000E+00)     ( 0.000E+00, 0.000E+00)
( 0.214E-14, 0.128E-14)     
( 0.214E-14,-0.128E-14)
( 0.217E-14, 0.000E+00)
(-0.168E-14, 0.000E+00)
(-0.140E-14, 0.798E-15)
(-0.140E-14,-0.798E-15)
(-0.890E-15, 0.126E-14)
(-0.890E-15,-0.126E-14)
( 0.565E-15, 0.105E-14)
( 0.565E-15,-0.105E-14)
( 0.876E-15, 0.000E+00)
(-0.773E-15, 0.208E-15)
(-0.773E-15,-0.208E-15)
( 0.587E-15, 0.000E+00)
(-0.150E-16, 0.567E-15)
(-0.150E-16,-0.567E-15)
( 0.352E-15, 0.337E-15)
( 0.352E-15,-0.337E-15)
( 0.127E-15, 0.324E-15)
( 0.127E-15,-0.324E-15)
(-0.137E-16, 0.199E-15)
(-0.137E-16,-0.199E-15)
( 0.172E-15, 0.372E-16)
( 0.172E-15,-0.372E-16)
( 0.719E-16, 0.157E-15)
( 0.719E-16,-0.157E-15)
(-0.153E-15, 0.661E-16)
(-0.153E-15,-0.661E-16)
(-0.304E-16, 0.683E-16)
(-0.304E-16,-0.683E-16)
(-0.192E-16, 0.000E+00)
(-0.955E-28, 0.000E+00)
(-0.281E-30, 0.325E-30)
(-0.281E-30,-0.325E-30)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)
( 0.000E+00, 0.000E+00)

Condição de Neumann em 3 lados fixando P=1 na entrada (ou na saída)

O espectro tem a maioria dos autovalores próximos do eixo real com alguns autovalores complexos (um pequeno circulo de valores complexos dentro do círculo unitário) com σ ⊂ 11.23 ∪ (-1,1) incluindo 0.

Teste 4

  • O espectro do problema não depende de dt, Re ou U0. Depende apenas do coeficientes de P (que dependem de dx,dy
  • Aplicando as condições de contorno e também subtraindo uma constante de toda a solução, por exemplo, P=P-P(1,2)

Teste 5

Disponível em "http://hunter.mat.ufrgs.br/mediawiki/index.php?title=Análise_espectral_e_estabilidade&oldid=1647"