Mudanças entre as edições de "Análise espectral e estabilidade"
De WikiLICC
m (→Condição de Neumann em todos os lados) |
m (→Condição de Dirichlet em todos os lados) |
||
| Linha 17: | Linha 17: | ||
==Condição de Dirichlet em todos os lados== | ==Condição de Dirichlet em todos os lados== | ||
| − | O espectro é real com 0<|λ|<1. A condição de contorno contribui com autovalores &lambda=1. | + | O espectro é real com 0<|λ|<1. A condição de contorno contribui com autovalores λ=1. |
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830 | lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830 | ||
| Linha 153: | Linha 153: | ||
( 0.161E-01, 0.000E+00) (-0.161E-01, 0.000E+00) | ( 0.161E-01, 0.000E+00) (-0.161E-01, 0.000E+00) | ||
( 0.422E-02, 0.000E+00) ( 0.422E-02, 0.000E+00) | ( 0.422E-02, 0.000E+00) ( 0.422E-02, 0.000E+00) | ||
| − | (-0.422E-02, 0.000E+00) (-0.422E-02, 0.000E+00) | + | (-0.422E-02, 0.000E+00) (-0.422E-02, 0.000E+00) |
| − | |||
| − | |||
==Condição de Neumann em todos os lados== | ==Condição de Neumann em todos os lados== | ||
Edição das 14h40min 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.
Índice
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>
que é discretizada como
<math>A p_{i+1,j}+A p_{i-1,j}+C p_{i,j} +B p_{i,j+1}+B p_{i,j-1} = f_{i,j} \,\!</math>
- A matriz possui o estêncil [...B,0...0,A,C,A,0...0,B,0...] onde
<math>A=\frac{-1}{dx^2}, B=\frac{-1}{dy^2}, C=-2A-2B \,\!</math>
- O espectro do problema não depende de dt, Re ou U0. Depende apenas de A, B e C
Condição de Dirichlet em todos os lados
O espectro é real com 0<|λ|<1. A condição de contorno contribui com autovalores λ=1.
lambda(1)= 1.00000000000000 lambda(1)= 0.955274744954830
lambda(n)= 4.218228659668808E-003 lambda(n)= 4.218228659674831E-003
Eigenvalues Eigenvalues
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
( 0.100E+01, 0.000E+00)
(-0.955E+00, 0.000E+00) (-0.955E+00, 0.000E+00)
( 0.955E+00, 0.000E+00) ( 0.955E+00, 0.000E+00)
( 0.896E+00, 0.000E+00) (-0.896E+00, 0.000E+00)
(-0.896E+00, 0.000E+00) ( 0.896E+00, 0.000E+00)
(-0.884E+00, 0.000E+00) ( 0.884E+00, 0.000E+00)
( 0.884E+00, 0.000E+00) (-0.884E+00, 0.000E+00)
( 0.825E+00, 0.000E+00) ( 0.825E+00, 0.000E+00)
(-0.825E+00, 0.000E+00) (-0.825E+00, 0.000E+00)
( 0.803E+00, 0.000E+00) (-0.803E+00, 0.000E+00)
(-0.803E+00, 0.000E+00) ( 0.803E+00, 0.000E+00)
(-0.774E+00, 0.000E+00) ( 0.774E+00, 0.000E+00)
( 0.774E+00, 0.000E+00) (-0.774E+00, 0.000E+00)
( 0.732E+00, 0.000E+00) ( 0.732E+00, 0.000E+00)
(-0.732E+00, 0.000E+00) (-0.732E+00, 0.000E+00)
(-0.715E+00, 0.000E+00) ( 0.715E+00, 0.000E+00)
( 0.715E+00, 0.000E+00) (-0.715E+00, 0.000E+00)
( 0.683E+00, 0.000E+00) ( 0.683E+00, 0.000E+00)
(-0.683E+00, 0.000E+00) (-0.683E+00, 0.000E+00)
(-0.634E+00, 0.000E+00) (-0.634E+00, 0.000E+00)
( 0.634E+00, 0.000E+00) ( 0.634E+00, 0.000E+00)
( 0.621E+00, 0.000E+00) ( 0.621E+00, 0.000E+00)
(-0.621E+00, 0.000E+00) (-0.621E+00, 0.000E+00)
(-0.612E+00, 0.000E+00) ( 0.612E+00, 0.000E+00)
( 0.612E+00, 0.000E+00) (-0.612E+00, 0.000E+00)
( 0.575E+00, 0.000E+00) ( 0.575E+00, 0.000E+00)
(-0.575E+00, 0.000E+00) (-0.575E+00, 0.000E+00)
(-0.547E+00, 0.000E+00) ( 0.547E+00, 0.000E+00)
( 0.547E+00, 0.000E+00) (-0.547E+00, 0.000E+00)
( 0.502E+00, 0.000E+00) ( 0.502E+00, 0.000E+00)
(-0.502E+00, 0.000E+00) (-0.502E+00, 0.000E+00)
(-0.482E+00, 0.000E+00) ( 0.482E+00, 0.000E+00)
( 0.482E+00, 0.000E+00) (-0.482E+00, 0.000E+00)
(-0.480E+00, 0.000E+00) ( 0.480E+00, 0.000E+00)
( 0.480E+00, 0.000E+00) (-0.480E+00, 0.000E+00)
(-0.476E+00, 0.000E+00) ( 0.476E+00, 0.000E+00)
( 0.476E+00, 0.000E+00) (-0.476E+00, 0.000E+00)
( 0.421E+00, 0.000E+00) ( 0.421E+00, 0.000E+00)
(-0.421E+00, 0.000E+00) (-0.421E+00, 0.000E+00)
(-0.404E+00, 0.000E+00) ( 0.404E+00, 0.000E+00)
( 0.404E+00, 0.000E+00) (-0.404E+00, 0.000E+00)
(-0.365E+00, 0.000E+00) ( 0.365E+00, 0.000E+00)
( 0.365E+00, 0.000E+00) (-0.365E+00, 0.000E+00)
( 0.362E+00, 0.000E+00) ( 0.362E+00, 0.000E+00)
(-0.362E+00, 0.000E+00) (-0.362E+00, 0.000E+00)
( 0.333E+00, 0.000E+00) (-0.333E+00, 0.000E+00)
(-0.333E+00, 0.000E+00) ( 0.333E+00, 0.000E+00)
(-0.327E+00, 0.000E+00) ( 0.327E+00, 0.000E+00)
( 0.327E+00, 0.000E+00) (-0.327E+00, 0.000E+00)
( 0.325E+00, 0.000E+00) (-0.325E+00, 0.000E+00)
(-0.325E+00, 0.000E+00) ( 0.325E+00, 0.000E+00)
( 0.268E+00, 0.000E+00) (-0.268E+00, 0.000E+00)
(-0.268E+00, 0.000E+00) ( 0.268E+00, 0.000E+00)
( 0.266E+00, 0.000E+00) (-0.266E+00, 0.000E+00)
(-0.266E+00, 0.000E+00) ( 0.266E+00, 0.000E+00)
( 0.226E+00, 0.000E+00) ( 0.226E+00, 0.000E+00)
(-0.226E+00, 0.000E+00) (-0.226E+00, 0.000E+00)
(-0.223E+00, 0.000E+00) ( 0.223E+00, 0.000E+00)
( 0.223E+00, 0.000E+00) (-0.223E+00, 0.000E+00)
(-0.208E+00, 0.000E+00) (-0.208E+00, 0.000E+00)
( 0.208E+00, 0.000E+00) ( 0.208E+00, 0.000E+00)
(-0.197E+00, 0.000E+00) ( 0.197E+00, 0.000E+00)
( 0.197E+00, 0.000E+00) (-0.197E+00, 0.000E+00)
(-0.186E+00, 0.000E+00) ( 0.186E+00, 0.000E+00)
( 0.186E+00, 0.000E+00) (-0.186E+00, 0.000E+00)
( 0.173E+00, 0.000E+00) ( 0.173E+00, 0.000E+00)
(-0.173E+00, 0.000E+00) (-0.173E+00, 0.000E+00)
( 0.148E+00, 0.000E+00) ( 0.148E+00, 0.000E+00)
(-0.148E+00, 0.000E+00) (-0.148E+00, 0.000E+00)
(-0.127E+00, 0.000E+00) ( 0.127E+00, 0.000E+00)
( 0.127E+00, 0.000E+00) (-0.127E+00, 0.000E+00)
(-0.862E-01, 0.000E+00) (-0.862E-01, 0.000E+00)
( 0.862E-01, 0.000E+00) ( 0.862E-01, 0.000E+00)
(-0.834E-01, 0.000E+00) (-0.834E-01, 0.000E+00)
( 0.834E-01, 0.000E+00) ( 0.834E-01, 0.000E+00)
( 0.771E-01, 0.000E+00) (-0.771E-01, 0.000E+00)
(-0.771E-01, 0.000E+00) ( 0.771E-01, 0.000E+00)
( 0.752E-01, 0.000E+00) (-0.752E-01, 0.000E+00)
(-0.752E-01, 0.000E+00) ( 0.752E-01, 0.000E+00)
(-0.712E-01, 0.000E+00) (-0.712E-01, 0.000E+00)
( 0.712E-01, 0.000E+00) ( 0.712E-01, 0.000E+00)
(-0.549E-01, 0.000E+00) ( 0.549E-01, 0.000E+00)
( 0.549E-01, 0.000E+00) (-0.549E-01, 0.000E+00)
( 0.532E-01, 0.000E+00) (-0.532E-01, 0.000E+00)
(-0.532E-01, 0.000E+00) ( 0.532E-01, 0.000E+00)
(-0.335E-01, 0.000E+00) ( 0.335E-01, 0.000E+00)
( 0.335E-01, 0.000E+00) (-0.335E-01, 0.000E+00)
(-0.161E-01, 0.000E+00) ( 0.161E-01, 0.000E+00)
( 0.161E-01, 0.000E+00) (-0.161E-01, 0.000E+00)
( 0.422E-02, 0.000E+00) ( 0.422E-02, 0.000E+00)
(-0.422E-02, 0.000E+00) (-0.422E-02, 0.000E+00)
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, 0.0 ) (-0.955E+00, 0.0 ) ( 0.976E+00, 0.0 ) ( 0.955E+00, 0.0 ) ( 0.970E+00, 0.0 ) (-0.896E+00, 0.0 ) (-0.945E+00, 0.0 ) ( 0.896E+00, 0.0 ) ( 0.945E+00, 0.0 ) ( 0.884E+00, 0.0 ) ( 0.905E+00, 0.0 ) (-0.884E+00, 0.0 ) ( 0.883E+00, 0.0 ) ( 0.825E+00, 0.0 ) (-0.874E+00, 0.0 ) (-0.825E+00, 0.0 ) ( 0.874E+00, 0.0 ) (-0.803E+00, 0.0 ) ( 0.859E+00, 0.0 ) ( 0.803E+00, 0.0 ) (-0.859E+00, 0.0 ) ( 0.774E+00, 0.0 ) ( 0.794E+00, 0.0 ) (-0.774E+00, 0.0 ) (-0.788E+00, 0.0 ) ( 0.732E+00, 0.0 ) ( 0.788E+00, 0.0 ) (-0.732E+00, 0.0 ) ( 0.764E+00, 0.0 ) ( 0.715E+00, 0.0 ) (-0.764E+00, 0.0 ) (-0.715E+00, 0.0 ) ( 0.750E+00, 0.0 ) ( 0.683E+00, 0.0 ) (-0.726E+00, 0.0 ) (-0.683E+00, 0.0 ) ( 0.726E+00, 0.0 ) (-0.634E+00, 0.0 ) ( 0.677E+00, 0.0 ) ( 0.634E+00, 0.0 ) (-0.677E+00, 0.0 ) ( 0.621E+00, 0.0 ) (-0.655E+00, 0.0 ) (-0.621E+00, 0.0 ) ( 0.655E+00, 0.0 ) ( 0.612E+00, 0.0 ) ( 0.655E+00, 0.0 ) (-0.612E+00, 0.0 ) ( 0.624E+00, 0.0 ) ( 0.575E+00, 0.0 ) (-0.624E+00, 0.0 ) (-0.575E+00, 0.0 ) ( 0.587E+00, 0.0 ) ( 0.547E+00, 0.0 ) (-0.562E+00, 0.0 ) (-0.547E+00, 0.0 ) ( 0.562E+00, 0.0 ) ( 0.502E+00, 0.0 ) (-0.544E+00, 0.0 ) (-0.502E+00, 0.0 ) ( 0.544E+00, 0.0 ) ( 0.482E+00, 0.0 ) ( 0.538E+00, 0.0 ) (-0.482E+00, 0.0 ) (-0.538E+00, 0.0 ) ( 0.480E+00, 0.0 ) ( 0.500E+00, 0.0 ) (-0.480E+00, 0.0 ) ( 0.491E+00, 0.0 ) ( 0.476E+00, 0.0 ) (-0.491E+00, 0.0 ) (-0.476E+00, 0.0 ) (-0.470E+00, 0.0 ) ( 0.421E+00, 0.0 ) ( 0.470E+00, 0.0 ) (-0.421E+00, 0.0 ) ( 0.413E+00, 0.0 ) ( 0.404E+00, 0.0 ) (-0.405E+00, 0.0 ) (-0.404E+00, 0.0 ) ( 0.405E+00, 0.0 ) ( 0.365E+00, 0.0 ) (-0.389E+00, 0.0 ) (-0.365E+00, 0.0 ) ( 0.389E+00, 0.0 ) ( 0.362E+00, 0.0 ) (-0.383E+00, 0.0 ) (-0.362E+00, 0.0 ) ( 0.383E+00, 0.0 ) (-0.333E+00, 0.0 ) (-0.381E+00, 0.0 ) ( 0.333E+00, 0.0 ) ( 0.381E+00, 0.0 ) ( 0.327E+00, 0.0 ) ( 0.345E+00, 0.0 ) (-0.327E+00, 0.0 ) (-0.318E+00, 0.0 ) (-0.325E+00, 0.0 ) ( 0.318E+00, 0.0 ) ( 0.325E+00, 0.0 ) ( 0.315E+00, 0.0 ) (-0.268E+00, 0.0 ) (-0.315E+00, 0.0 ) ( 0.268E+00, 0.0 ) (-0.250E+00, 0.0 ) (-0.266E+00, 0.0 ) ( 0.250E+00, 0.0 ) ( 0.266E+00, 0.0 ) ( 0.250E+00, 0.0 ) ( 0.226E+00, 0.0 ) (-0.241E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.241E+00, 0.0 ) ( 0.223E+00, 0.0 ) (-0.229E+00, 0.0 ) (-0.223E+00, 0.0 ) ( 0.229E+00, 0.0 ) (-0.208E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.208E+00, 0.0 ) ( 0.226E+00, 0.0 ) ( 0.197E+00, 0.0 ) ( 0.207E+00, 0.0 ) (-0.197E+00, 0.0 ) (-0.207E+00, 0.0 ) ( 0.186E+00, 0.0 ) ( 0.206E+00, 0.0 ) (-0.186E+00, 0.0 ) ( 0.176E+00, 0.0 ) ( 0.173E+00, 0.0 ) (-0.176E+00, 0.0 ) (-0.173E+00, 0.0 ) (-0.155E+00, 0.0 ) ( 0.148E+00, 0.0 ) ( 0.155E+00, 0.0 ) (-0.148E+00, 0.0 ) ( 0.117E+00, 0.0 ) ( 0.127E+00, 0.0 ) ( 0.955E-01, 0.0 ) (-0.127E+00, 0.0 ) ( 0.955E-01, 0.0 ) (-0.862E-01, 0.0 ) (-0.955E-01, 0.0 ) ( 0.862E-01, 0.0 ) (-0.925E-01, 0.0 ) (-0.834E-01, 0.0 ) ( 0.925E-01, 0.0 ) ( 0.834E-01, 0.0 ) (-0.891E-01, 0.0 ) (-0.771E-01, 0.0 ) ( 0.891E-01, 0.0 ) ( 0.771E-01, 0.0 ) (-0.868E-01, 0.0 ) (-0.752E-01, 0.0 ) ( 0.868E-01, 0.0 ) ( 0.752E-01, 0.0 ) ( 0.677E-01, 0.0 ) (-0.712E-01, 0.0 ) (-0.677E-01, 0.0 ) ( 0.712E-01, 0.0 ) (-0.653E-01, 0.0 ) ( 0.549E-01, 0.0 ) ( 0.653E-01, 0.0 ) (-0.549E-01, 0.0 ) ( 0.439E-01, 0.0 ) (-0.532E-01, 0.0 ) (-0.439E-01, 0.0 ) ( 0.532E-01, 0.0 ) ( 0.302E-01, 0.0 ) ( 0.335E-01, 0.0 ) ( 0.245E-01, 0.0 ) (-0.335E-01, 0.0 ) ( 0.215E-01, 0.0 ) ( 0.161E-01, 0.0 ) (-0.215E-01, 0.0 ) (-0.161E-01, 0.0 ) (-0.568E-02, 0.0 ) ( 0.422E-02, 0.0 ) ( 0.568E-02, 0.0 ) (-0.422E-02, 0.0 ) ( 0.251E-14, 0.857E-15) ( 0.251E-14,-0.857E-15) (-0.255E-14, 0.0 ) ( 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, 0.0 ) (-0.772E-15, 0.0 ) ( 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, 0.0 ) (-0.433E-15, 0.0 ) (-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, 0.0 ) (-0.230E-16, 0.0 ) (-0.824E-30, 0.885E-30) (-0.824E-30,-0.885E-30) ( 0.680E-30, 0.0 ) (-0.684E-31, 0.0 ) ( 0.317E-31, 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 )
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, 0.0 ) (-0.955E+00, 0.0 ) ( 0.970E+00, 0.0 ) ( 0.911E+00, 0.0 ) ( 0.945E+00, 0.0 ) ( 0.896E+00, 0.0 ) (-0.945E+00, 0.0 ) (-0.896E+00, 0.0 ) ( 0.905E+00, 0.0 ) ( 0.884E+00, 0.0 ) ( 0.883E+00, 0.0 ) (-0.884E+00, 0.0 ) (-0.874E+00, 0.0 ) (-0.825E+00, 0.0 ) ( 0.874E+00, 0.0 ) ( 0.825E+00, 0.0 ) ( 0.859E+00, 0.0 ) (-0.803E+00, 0.0 ) (-0.859E+00, 0.0 ) ( 0.789E+00, 0.0 ) ( 0.794E+00, 0.0 ) (-0.774E+00, 0.0 ) ( 0.788E+00, 0.0 ) (-0.732E+00, 0.0 ) (-0.788E+00, 0.0 ) ( 0.732E+00, 0.0 ) ( 0.764E+00, 0.0 ) ( 0.715E+00, 0.0 ) (-0.764E+00, 0.0 ) (-0.712E+00, 0.0 ) ( 0.750E+00, 0.0 ) ( 0.693E+00, 0.0 ) (-0.726E+00, 0.0 ) ( 0.683E+00, 0.0 ) ( 0.726E+00, 0.0 ) (-0.682E+00, 0.0 ) (-0.677E+00, 0.0 ) (-0.634E+00, 0.0 ) ( 0.677E+00, 0.0 ) ( 0.634E+00, 0.0 ) ( 0.655E+00, 0.0 ) (-0.621E+00, 0.0 ) (-0.655E+00, 0.0 ) (-0.612E+00, 0.0 ) ( 0.655E+00, 0.0 ) ( 0.612E+00, 0.0 ) (-0.624E+00, 0.0 ) ( 0.583E+00, 0.0 ) ( 0.624E+00, 0.0 ) (-0.575E+00, 0.0 ) ( 0.587E+00, 0.0 ) ( 0.575E+00, 0.0 ) ( 0.562E+00, 0.0 ) (-0.547E+00, 0.0 ) (-0.562E+00, 0.0 ) ( 0.507E+00, 0.0 ) ( 0.544E+00, 0.0 ) ( 0.502E+00, 0.0 ) (-0.544E+00, 0.0 ) (-0.494E+00, 0.0 ) (-0.538E+00, 0.0 ) ( 0.482E+00, 0.0 ) ( 0.538E+00, 0.0 ) (-0.482E+00, 0.0 ) ( 0.500E+00, 0.0 ) (-0.480E+00, 0.0 ) ( 0.491E+00, 0.0 ) (-0.476E+00, 0.0 ) (-0.491E+00, 0.0 ) ( 0.476E+00, 0.0 ) (-0.470E+00, 0.0 ) ( 0.421E+00, 0.0 ) ( 0.470E+00, 0.0 ) (-0.416E+00, 0.0 ) ( 0.413E+00, 0.0 ) ( 0.412E+00, 0.0 ) (-0.405E+00, 0.0 ) ( 0.404E+00, 0.0 ) ( 0.405E+00, 0.0 ) (-0.402E+00, 0.0 ) (-0.389E+00, 0.0 ) (-0.365E+00, 0.0 ) ( 0.389E+00, 0.0 ) (-0.362E+00, 0.0 ) ( 0.383E+00, 0.0 ) ( 0.362E+00, 0.0 ) (-0.383E+00, 0.0 ) ( 0.343E+00, 0.0 ) (-0.381E+00, 0.0 ) ( 0.333E+00, 0.0 ) ( 0.381E+00, 0.0 ) (-0.333E+00, 0.0 ) ( 0.345E+00, 0.0 ) (-0.327E+00, 0.0 ) ( 0.318E+00, 0.0 ) (-0.325E+00, 0.0 ) (-0.318E+00, 0.0 ) ( 0.325E+00, 0.0 ) ( 0.315E+00, 0.0 ) ( 0.283E+00, 0.0 ) (-0.315E+00, 0.0 ) (-0.268E+00, 0.0 ) ( 0.250E+00, 0.0 ) ( 0.266E+00, 0.0 ) (-0.250E+00, 0.0 ) (-0.266E+00, 0.0 ) ( 0.250E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.241E+00, 0.0 ) ( 0.226E+00, 0.0 ) (-0.241E+00, 0.0 ) ( 0.223E+00, 0.0 ) (-0.229E+00, 0.0 ) (-0.218E+00, 0.0 ) ( 0.229E+00, 0.0 ) ( 0.208E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.202E+00, 0.0 ) ( 0.226E+00, 0.0 ) (-0.197E+00, 0.0 ) ( 0.207E+00, 0.0 ) ( 0.197E+00, 0.0 ) (-0.207E+00, 0.0 ) (-0.190E+00, 0.0 ) ( 0.206E+00, 0.0 ) (-0.186E+00, 0.0 ) ( 0.176E+00, 0.0 ) (-0.173E+00, 0.0 ) (-0.176E+00, 0.0 ) ( 0.173E+00, 0.0 ) ( 0.155E+00, 0.0 ) ( 0.148E+00, 0.0 ) (-0.155E+00, 0.0 ) (-0.146E+00, 0.0 ) ( 0.117E+00, 0.0 ) ( 0.127E+00, 0.0 ) (-0.955E-01, 0.0 ) ( 0.122E+00, 0.0 ) ( 0.955E-01, 0.411E-15) (-0.120E+00, 0.0 ) ( 0.955E-01,-0.411E-15) (-0.862E-01, 0.0 ) ( 0.925E-01, 0.0 ) (-0.834E-01, 0.0 ) (-0.925E-01, 0.0 ) ( 0.834E-01, 0.0 ) (-0.891E-01, 0.0 ) ( 0.789E-01, 0.0 ) ( 0.891E-01, 0.0 ) (-0.771E-01, 0.0 ) ( 0.868E-01, 0.0 ) ( 0.771E-01, 0.0 ) (-0.868E-01, 0.0 ) (-0.752E-01, 0.0 ) (-0.677E-01, 0.0 ) ( 0.752E-01, 0.0 ) ( 0.677E-01, 0.0 ) (-0.712E-01, 0.0 ) ( 0.653E-01, 0.0 ) ( 0.573E-01, 0.0 ) (-0.653E-01, 0.0 ) (-0.549E-01, 0.0 ) (-0.439E-01, 0.0 ) (-0.532E-01, 0.0 ) ( 0.439E-01, 0.0 ) ( 0.532E-01, 0.0 ) ( 0.302E-01, 0.0 ) ( 0.395E-01, 0.0 ) ( 0.245E-01, 0.0 ) (-0.335E-01, 0.0 ) (-0.215E-01, 0.0 ) (-0.161E-01, 0.0 ) ( 0.215E-01, 0.0 ) ( 0.161E-01, 0.0 ) ( 0.568E-02, 0.0 ) ( 0.523E-02, 0.0 ) (-0.568E-02, 0.0 ) (-0.422E-02, 0.0 ) (-0.444E-14, 0.0 ) ( 0.0 , 0.0 ) ( 0.214E-14, 0.128E-14) ( 0.214E-14,-0.128E-14) ( 0.217E-14, 0.0 ) (-0.168E-14, 0.0 ) (-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, 0.0 ) (-0.773E-15, 0.208E-15) (-0.773E-15,-0.208E-15) ( 0.587E-15, 0.0 ) (-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, 0.0 ) (-0.955E-28, 0.0 ) (-0.281E-30, 0.325E-30) (-0.281E-30,-0.325E-30) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 )
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⊂ 1.726 ∪ (-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 ( 1.726E+00, 0.0 ) (-0.955E+00, 0.0 ) ( 0.975E+00, 0.0 ) ( 0.955E+00, 0.0 ) ( 0.969E+00, 0.0 ) (-0.896E+00, 0.0 ) (-0.945E+00, 0.0 ) ( 0.896E+00, 0.0 ) ( 0.942E+00, 0.589E-02) ( 0.884E+00, 0.0 ) ( 0.942E+00,-0.589E-02) (-0.884E+00, 0.0 ) ( 0.893E+00, 0.0 ) ( 0.825E+00, 0.0 ) (-0.875E+00, 0.0 ) (-0.825E+00, 0.0 ) ( 0.874E+00, 0.0 ) (-0.803E+00, 0.0 ) (-0.859E+00, 0.0 ) ( 0.803E+00, 0.0 ) ( 0.857E+00, 0.0 ) ( 0.774E+00, 0.0 ) ( 0.795E+00, 0.244E-02) (-0.774E+00, 0.0 ) ( 0.795E+00,-0.244E-02) ( 0.732E+00, 0.0 ) (-0.788E+00, 0.0 ) (-0.732E+00, 0.0 ) (-0.764E+00, 0.0 ) ( 0.715E+00, 0.0 ) ( 0.762E+00, 0.0 ) (-0.715E+00, 0.0 ) ( 0.748E+00, 0.0 ) ( 0.683E+00, 0.0 ) (-0.726E+00, 0.0 ) (-0.683E+00, 0.0 ) ( 0.715E+00, 0.774E-02) (-0.634E+00, 0.0 ) ( 0.715E+00,-0.774E-02) ( 0.634E+00, 0.0 ) ( 0.679E+00, 0.0 ) ( 0.621E+00, 0.0 ) (-0.677E+00, 0.0 ) (-0.621E+00, 0.0 ) (-0.656E+00, 0.0 ) ( 0.612E+00, 0.0 ) ( 0.655E+00, 0.0 ) (-0.612E+00, 0.0 ) (-0.625E+00, 0.0 ) ( 0.575E+00, 0.0 ) ( 0.620E+00, 0.362E-02) (-0.575E+00, 0.0 ) ( 0.620E+00,-0.362E-02) ( 0.547E+00, 0.0 ) (-0.562E+00, 0.0 ) (-0.547E+00, 0.0 ) ( 0.561E+00, 0.0 ) ( 0.502E+00, 0.0 ) (-0.545E+00, 0.0 ) (-0.502E+00, 0.0 ) ( 0.544E+00, 0.277E-02) ( 0.482E+00, 0.0 ) ( 0.544E+00,-0.277E-02) (-0.482E+00, 0.0 ) (-0.538E+00, 0.0 ) ( 0.480E+00, 0.0 ) ( 0.500E+00, 0.219E-02) (-0.480E+00, 0.0 ) ( 0.500E+00,-0.219E-02) ( 0.476E+00, 0.0 ) (-0.492E+00, 0.0 ) (-0.476E+00, 0.0 ) (-0.470E+00, 0.0 ) ( 0.421E+00, 0.0 ) ( 0.469E+00, 0.0 ) (-0.421E+00, 0.0 ) (-0.418E+00, 0.0 ) ( 0.404E+00, 0.0 ) ( 0.412E+00, 0.0 ) (-0.404E+00, 0.0 ) ( 0.401E+00, 0.0 ) ( 0.365E+00, 0.0 ) (-0.389E+00, 0.0 ) (-0.365E+00, 0.0 ) ( 0.387E+00, 0.137E-01) ( 0.362E+00, 0.0 ) ( 0.387E+00,-0.137E-01) (-0.362E+00, 0.0 ) ( 0.386E+00, 0.0 ) (-0.333E+00, 0.0 ) (-0.384E+00, 0.0 ) ( 0.333E+00, 0.0 ) ( 0.382E+00, 0.0 ) ( 0.327E+00, 0.0 ) (-0.381E+00, 0.0 ) (-0.327E+00, 0.0 ) (-0.332E+00, 0.0 ) (-0.325E+00, 0.0 ) ( 0.317E+00, 0.0 ) ( 0.325E+00, 0.0 ) (-0.316E+00, 0.0 ) (-0.268E+00, 0.0 ) ( 0.314E+00, 0.0 ) ( 0.268E+00, 0.0 ) ( 0.290E+00, 0.0 ) (-0.266E+00, 0.0 ) (-0.256E+00, 0.0 ) ( 0.266E+00, 0.0 ) ( 0.250E+00, 0.0 ) ( 0.226E+00, 0.0 ) (-0.245E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.244E+00, 0.0 ) ( 0.223E+00, 0.0 ) (-0.233E+00, 0.0 ) (-0.223E+00, 0.0 ) ( 0.232E+00, 0.0 ) (-0.208E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.208E+00, 0.0 ) ( 0.225E+00, 0.0 ) ( 0.197E+00, 0.0 ) (-0.221E+00, 0.0 ) (-0.197E+00, 0.0 ) ( 0.207E+00, 0.0 ) ( 0.186E+00, 0.0 ) ( 0.206E+00, 0.0 ) (-0.186E+00, 0.0 ) (-0.204E+00, 0.0 ) ( 0.173E+00, 0.0 ) (-0.176E+00, 0.0 ) (-0.173E+00, 0.0 ) ( 0.176E+00, 0.0 ) ( 0.148E+00, 0.0 ) ( 0.163E+00, 0.0 ) (-0.148E+00, 0.0 ) (-0.154E+00, 0.0 ) ( 0.127E+00, 0.0 ) ( 0.117E+00, 0.0 ) (-0.127E+00, 0.0 ) ( 0.107E+00, 0.0 ) (-0.862E-01, 0.0 ) ( 0.955E-01, 0.0 ) ( 0.862E-01, 0.0 ) (-0.925E-01, 0.0 ) (-0.834E-01, 0.0 ) ( 0.925E-01, 0.0 ) ( 0.834E-01, 0.0 ) (-0.894E-01, 0.0 ) (-0.771E-01, 0.0 ) ( 0.891E-01, 0.0 ) ( 0.771E-01, 0.0 ) (-0.878E-01, 0.0 ) (-0.752E-01, 0.0 ) ( 0.867E-01, 0.0 ) ( 0.752E-01, 0.0 ) (-0.855E-01, 0.0 ) (-0.712E-01, 0.0 ) ( 0.675E-01, 0.0 ) ( 0.712E-01, 0.0 ) (-0.656E-01, 0.0 ) ( 0.549E-01, 0.0 ) ( 0.652E-01, 0.0 ) (-0.549E-01, 0.0 ) (-0.457E-01, 0.0 ) (-0.532E-01, 0.0 ) ( 0.436E-01, 0.0 ) ( 0.532E-01, 0.0 ) (-0.427E-01, 0.0 ) ( 0.335E-01, 0.0 ) ( 0.372E-01, 0.0 ) (-0.335E-01, 0.0 ) ( 0.245E-01, 0.0 ) ( 0.161E-01, 0.0 ) ( 0.215E-01, 0.0 ) (-0.161E-01, 0.0 ) (-0.107E-01, 0.0 ) ( 0.422E-02, 0.0 ) ( 0.568E-02, 0.0 ) (-0.422E-02, 0.0 ) (-0.567E-02, 0.0 ) ( 0.701E-14, 0.566E-15) ( 0.701E-14,-0.566E-15) (-0.579E-14, 0.0 ) (-0.219E-14, 0.0 ) (-0.161E-14, 0.0 ) ( 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, 0.0 ) (-0.431E-15, 0.0 ) ( 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, 0.0 ) ( 0.124E-15, 0.232E-15) ( 0.124E-15,-0.232E-15) ( 0.197E-15, 0.0 ) ( 0.165E-16, 0.190E-15) ( 0.165E-16,-0.190E-15) (-0.188E-15, 0.0 ) ( 0.104E-15, 0.0 ) ( 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, 0.0 ) (-0.904E-17, 0.887E-17) (-0.904E-17,-0.887E-17) ( 0.651E-28, 0.0 ) ( 0.264E-29, 0.0 ) ( 0.522E-30, 0.0 ) (-0.923E-31, 0.0 ) ( 0.121E-31, 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 )
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 ⊂ 11.26 ∪ (-1,1) incluindo 0.
Incluindo CC na matriz Excluindo CC na matriz lambda(1)= 11.2617565379434 lambda(1)= 0.955274744954830 lambda(n)= 0.000000000000000E+000 lambda(n)= 4.218228659674831E-003 (11.262E+00, 0.0 ) (-0.955E+00, 0.0 ) ( 0.975E+00, 0.0 ) ( 0.955E+00, 0.0 ) ( 0.964E+00, 0.0 ) (-0.896E+00, 0.0 ) (-0.948E+00, 0.0 ) ( 0.896E+00, 0.0 ) ( 0.932E+00, 0.0 ) ( 0.884E+00, 0.0 ) ( 0.920E+00, 0.0 ) (-0.884E+00, 0.0 ) (-0.883E+00, 0.0 ) ( 0.825E+00, 0.0 ) ( 0.877E+00, 0.0 ) (-0.825E+00, 0.0 ) (-0.861E+00, 0.0 ) (-0.803E+00, 0.0 ) ( 0.836E+00, 0.0 ) ( 0.803E+00, 0.0 ) ( 0.834E+00, 0.0 ) ( 0.774E+00, 0.0 ) (-0.796E+00, 0.0 ) (-0.774E+00, 0.0 ) ( 0.790E+00, 0.0 ) ( 0.732E+00, 0.0 ) (-0.782E+00, 0.0 ) (-0.732E+00, 0.0 ) ( 0.750E+00, 0.0 ) ( 0.715E+00, 0.0 ) ( 0.750E+00, 0.0 ) (-0.715E+00, 0.0 ) ( 0.744E+00, 0.0 ) ( 0.683E+00, 0.0 ) (-0.728E+00, 0.0 ) (-0.683E+00, 0.0 ) ( 0.720E+00, 0.0 ) (-0.634E+00, 0.0 ) ( 0.700E+00, 0.0 ) ( 0.634E+00, 0.0 ) (-0.695E+00, 0.0 ) ( 0.621E+00, 0.0 ) (-0.663E+00, 0.0 ) (-0.621E+00, 0.0 ) (-0.653E+00, 0.0 ) ( 0.612E+00, 0.0 ) ( 0.633E+00, 0.0 ) (-0.612E+00, 0.0 ) ( 0.617E+00, 0.0 ) ( 0.575E+00, 0.0 ) ( 0.581E+00, 0.0 ) (-0.575E+00, 0.0 ) ( 0.581E+00, 0.0 ) ( 0.547E+00, 0.0 ) (-0.566E+00, 0.0 ) (-0.547E+00, 0.0 ) (-0.565E+00, 0.0 ) ( 0.502E+00, 0.0 ) (-0.562E+00, 0.0 ) (-0.502E+00, 0.0 ) ( 0.537E+00, 0.0 ) ( 0.482E+00, 0.0 ) (-0.507E+00, 0.0 ) (-0.482E+00, 0.0 ) ( 0.500E+00, 0.0 ) ( 0.480E+00, 0.0 ) (-0.500E+00, 0.0 ) (-0.480E+00, 0.0 ) ( 0.498E+00, 0.138E-01) ( 0.476E+00, 0.0 ) ( 0.498E+00,-0.138E-01) (-0.476E+00, 0.0 ) ( 0.494E+00, 0.0 ) ( 0.421E+00, 0.0 ) ( 0.453E+00, 0.0 ) (-0.421E+00, 0.0 ) (-0.433E+00, 0.0 ) ( 0.404E+00, 0.0 ) ( 0.432E+00, 0.0 ) (-0.404E+00, 0.0 ) (-0.420E+00, 0.0 ) ( 0.365E+00, 0.0 ) ( 0.408E+00, 0.0 ) (-0.365E+00, 0.0 ) (-0.399E+00, 0.0 ) ( 0.362E+00, 0.0 ) (-0.391E+00, 0.0 ) (-0.362E+00, 0.0 ) ( 0.364E+00, 0.0 ) (-0.333E+00, 0.0 ) ( 0.361E+00, 0.0 ) ( 0.333E+00, 0.0 ) (-0.359E+00, 0.0 ) ( 0.327E+00, 0.0 ) ( 0.346E+00, 0.0 ) (-0.327E+00, 0.0 ) ( 0.337E+00, 0.0 ) (-0.325E+00, 0.0 ) (-0.326E+00, 0.0 ) ( 0.325E+00, 0.0 ) (-0.287E+00, 0.0 ) (-0.268E+00, 0.0 ) ( 0.287E+00, 0.0 ) ( 0.268E+00, 0.0 ) ( 0.280E+00, 0.0 ) (-0.266E+00, 0.0 ) (-0.272E+00, 0.0 ) ( 0.266E+00, 0.0 ) (-0.269E+00, 0.0 ) ( 0.226E+00, 0.0 ) ( 0.247E+00, 0.0 ) (-0.226E+00, 0.0 ) ( 0.244E+00, 0.0 ) ( 0.223E+00, 0.0 ) (-0.228E+00, 0.0 ) (-0.223E+00, 0.0 ) (-0.225E+00, 0.0 ) (-0.208E+00, 0.0 ) (-0.220E+00, 0.0 ) ( 0.208E+00, 0.0 ) ( 0.213E+00, 0.0 ) ( 0.197E+00, 0.0 ) ( 0.207E+00, 0.0 ) (-0.197E+00, 0.0 ) ( 0.200E+00, 0.0 ) ( 0.186E+00, 0.0 ) ( 0.200E+00, 0.0 ) (-0.186E+00, 0.0 ) ( 0.198E+00, 0.0 ) ( 0.173E+00, 0.0 ) ( 0.163E+00, 0.0 ) (-0.173E+00, 0.0 ) (-0.163E+00, 0.0 ) ( 0.148E+00, 0.0 ) ( 0.158E+00, 0.0 ) (-0.148E+00, 0.0 ) (-0.139E+00, 0.0 ) ( 0.127E+00, 0.0 ) (-0.133E+00, 0.0 ) (-0.127E+00, 0.0 ) (-0.124E+00, 0.0 ) (-0.862E-01, 0.0 ) ( 0.117E+00, 0.0 ) ( 0.862E-01, 0.0 ) ( 0.111E+00, 0.0 ) (-0.834E-01, 0.0 ) (-0.103E+00, 0.0 ) ( 0.834E-01, 0.0 ) (-0.958E-01, 0.0 ) (-0.771E-01, 0.0 ) (-0.948E-01, 0.0 ) ( 0.771E-01, 0.0 ) ( 0.713E-01, 0.0 ) (-0.752E-01, 0.0 ) ( 0.675E-01, 0.0 ) ( 0.752E-01, 0.0 ) ( 0.673E-01, 0.0 ) (-0.712E-01, 0.0 ) ( 0.668E-01, 0.0 ) ( 0.712E-01, 0.0 ) (-0.617E-01, 0.0 ) ( 0.549E-01, 0.0 ) ( 0.567E-01, 0.0 ) (-0.549E-01, 0.0 ) ( 0.495E-01, 0.0 ) (-0.532E-01, 0.0 ) (-0.301E-01, 0.0 ) ( 0.532E-01, 0.0 ) ( 0.246E-01, 0.0 ) ( 0.335E-01, 0.0 ) ( 0.245E-01, 0.0 ) (-0.335E-01, 0.0 ) ( 0.244E-01, 0.0 ) ( 0.161E-01, 0.0 ) (-0.194E-01, 0.0 ) (-0.161E-01, 0.0 ) (-0.165E-01, 0.0 ) ( 0.422E-02, 0.0 ) (-0.794E-02, 0.0 ) (-0.422E-02, 0.0 ) (-0.568E-08, 0.0 ) ( 0.568E-08, 0.0 ) ( 0.308E-14, 0.0 ) (-0.178E-14, 0.136E-14) (-0.178E-14,-0.136E-14) (-0.126E-14, 0.0 ) ( 0.882E-15, 0.828E-15) ( 0.882E-15,-0.828E-15) ( 0.518E-15, 0.109E-14) ( 0.518E-15,-0.109E-14) (-0.466E-16, 0.552E-15) (-0.466E-16,-0.552E-15) ( 0.508E-15, 0.0 ) (-0.336E-15, 0.361E-15) (-0.336E-15,-0.361E-15) ( 0.154E-15, 0.417E-15) ( 0.154E-15,-0.417E-15) (-0.333E-15, 0.0 ) (-0.275E-15, 0.112E-15) (-0.275E-15,-0.112E-15) ( 0.156E-15, 0.208E-15) ( 0.156E-15,-0.208E-15) ( 0.253E-16, 0.658E-16) ( 0.253E-16,-0.658E-16) ( 0.666E-16, 0.0 ) (-0.536E-16, 0.0 ) (-0.205E-16, 0.0 ) (-0.119E-16, 0.0 ) ( 0.259E-18, 0.106E-16) ( 0.259E-18,-0.106E-16) (-0.216E-29, 0.0 ) ( 0.171E-29, 0.0 ) ( 0.357E-30, 0.914E-31) ( 0.357E-30,-0.914E-31) (-0.246E-30, 0.0 ) (-0.204E-31, 0.0 ) (-0.392E-32, 0.313E-32) (-0.392E-32,-0.313E-32) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 ) ( 0.0 , 0.0 )
