AceleracaoCong-02-03-21, R

AceleracaoCong-02-03-21

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# Resolvendo o sistema linear Au=v pelo método dos gradientes.

c=c(8, 1, 1, 0, 2, 20, 9, 2, 1, 1, 2, 1, 11,2, 0, 3, 2, 1, -8, 1, 2, 1, 2, 3, 10)
A=matrix(c,5,5,,byrow=TRUE); A   # Entrada da matriz A
b=c(1,2,3,4,5); b                # Entrada do vetor v

f<-function(u){A%*%u-b}          # Au=v quando f(u)=0.

B=t(A)%*%A; bb=t(A)%*%b           # Adaptando o sistema para A*Au=A*v, ou Bu=b, em que B=A*A é positiva definida e b=A*v.

gradg1<-function(w){u}
gradg2<-function(u){B%*%u-bb}

n=5
u=c(1,1,1,1,1)
                                 # Início do método
w=B%*%u-bb                        # w0
v=w                              # d1
alpha= t(w)%*%w /(t(B%*%w) %*%w); alpha=alpha[1]     # Escolha de h0
u=u-alpha*w                       # u1

for ( j in 1:n){
w=(B%*%u-bb)
beta=t(B%*%w) %*%v/(t(B%*%v) %*%v); beta=beta[1]
v=w-beta*v                   # Método de Euler
alpha= t(v) %*%w/(t(B%*%v) %*%v);alpha=alpha[1]
u=u-alpha*v                      # Método quase Euler (integral)
}
print(" Aproximação para u");u;v                               # Aproximação para u(20).
print(" Aproximação para f(u)") ;f(u)                         # Verificação do erro de aproximação.
print(" Número de iterações"); n
eigen(B)$values

Bg=matrix(0,10,10); Bg[6:10,1:5]=-alpha*B; Bg[1:5,6:10]=diag(c(1,1,1,1,1));Bg
lambda=(eigen(Bg)$values); (lambda)

Bg=matrix(0,10,10); Bg[6:10,1:5]=-alpha*B; Bg[6:10,6:10]=-beta*diag(c(1,1,1,1,1));Bg[1:5,6:10]=diag(c(1,1,1,1,1));Bg
lambda=(eigen(Bg)$values);Re(lambda); Im(lambda)

(beta*lambda+lambda^2)/alpha

[ + ] Show input
Absolute running time: 0.64 sec, cpu time: 1.03 sec, memory peak: 44 Mb, absolute service time: 0,87 sec

[,1] [,2] [,3] [,4] [,5] [1,] 8 1 1 0 2 [2,] 20 9 2 1 1 [3,] 2 1 11 2 0 [4,] 3 2 1 -8 1 [5,] 2 1 2 3 10 [1] 1 2 3 4 5 [1] " Aproximação para u" [,1] [1,] -0.08679601 [2,] 0.32372124 [3,] 0.32195742 [4,] -0.34583043 [5,] 0.52434472 [,1] [1,] 1.375917e-10 [2,] 5.730710e-11 [3,] 2.907565e-11 [4,] 3.801069e-12 [5,] 2.145112e-11 [1] " Aproximação para f(u)" [,1] [1,] -6.661338e-16 [2,] -6.217249e-15 [3,] 2.087219e-14 [4,] -8.881784e-15 [5,] 1.953993e-14 [1] " Número de iterações" [1] 5 [1] 588.796475 135.948589 92.608724 61.425217 5.220995 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 1 0 [2,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 0 1 [3,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 0 0 [4,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 0 0 [5,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 0 0 [6,] -0.81692648 -0.336281586 -0.13077617 -0.010190351 -0.10020512 0 0 [7,] -0.33628159 -0.149458483 -0.05774532 0.003396784 -0.03906301 0 0 [8,] -0.13077617 -0.057745323 -0.22248933 -0.037364621 -0.04245980 0 0 [9,] -0.01019035 0.003396784 -0.03736462 -0.132474564 -0.03906301 0 0 [10,] -0.10020512 -0.039063013 -0.04245980 -0.039063013 -0.18002954 0 0 [,8] [,9] [,10] [1,] 0 0 0 [2,] 0 0 0 [3,] 1 0 0 [4,] 0 1 0 [5,] 0 0 1 [6,] 0 0 0 [7,] 0 0 0 [8,] 0 0 0 [9,] 0 0 0 [10,] 0 0 0 [1] 0+1.00000357i 0-1.00000357i 0+0.48051428i 0-0.48051428i 0+0.39659287i [6] 0-0.39659287i 0+0.32299240i 0-0.32299240i 0+0.09416632i 0-0.09416632i [,1] [,2] [,3] [,4] [,5] [1,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 [2,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 [3,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 [4,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 [5,] 0.00000000 0.000000000 0.00000000 0.000000000 0.00000000 [6,] -0.81692648 -0.336281586 -0.13077617 -0.010190351 -0.10020512 [7,] -0.33628159 -0.149458483 -0.05774532 0.003396784 -0.03906301 [8,] -0.13077617 -0.057745323 -0.22248933 -0.037364621 -0.04245980 [9,] -0.01019035 0.003396784 -0.03736462 -0.132474564 -0.03906301 [10,] -0.10020512 -0.039063013 -0.04245980 -0.039063013 -0.18002954 [,6] [,7] [,8] [,9] [,10] [1,] 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [2,] 0.000000e+00 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [3,] 0.000000e+00 0.000000e+00 1.000000e+00 0.000000e+00 0.000000e+00 [4,] 0.000000e+00 0.000000e+00 0.000000e+00 1.000000e+00 0.000000e+00 [5,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 1.000000e+00 [6,] -4.048618e-15 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [7,] 0.000000e+00 -4.048618e-15 0.000000e+00 0.000000e+00 0.000000e+00 [8,] 0.000000e+00 0.000000e+00 -4.048618e-15 0.000000e+00 0.000000e+00 [9,] 0.000000e+00 0.000000e+00 0.000000e+00 -4.048618e-15 0.000000e+00 [10,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 -4.048618e-15 [1] -2.017235e-15 -2.017235e-15 -2.053913e-15 -2.053913e-15 -1.998401e-15 [6] -1.998401e-15 -2.030935e-15 -2.030935e-15 -2.085027e-15 -2.085027e-15 [1] 1.00000357 -1.00000357 0.48051428 -0.48051428 0.39659287 -0.39659287 [7] 0.32299240 -0.32299240 0.09416632 -0.09416632 [1] -588.796475+0i -588.796475-0i -135.948589-0i -135.948589+0i -92.608724+0i [6] -92.608724-0i -61.425217-0i -61.425217+0i -5.220995-0i -5.220995+0i