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thomas tridiagonal and spectral radius

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Giovanni Di Grezia
2014-11-02 16:48:39 +01:00
parent 0aff089106
commit fa0d6f3f4f
7 changed files with 123 additions and 3 deletions
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2
functions/convert_matrix_to_triangular_matrix_gauss_naif.m
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@@ -15,7 +15,7 @@ function [U,b] = convert_matrix_to_triangular_matrix_gauss_naif (U,b)
ok=1;
% convertiamo la matrice in una triangolare superiore
% convertiamo la matrice
for i=1:1:n-1
if abs(U(i,i)) <= eps * norma

2
functions/convert_matrix_to_triangular_matrix_gauss_pivoting.m
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@@ -15,7 +15,7 @@ function [U,b,pivot] = convert_matrix_to_triangular_matrix_gauss_pivoting (U,b)
pivot = 1:1:n;
pivot = pivot';
% convertiamo la matrice in una triangolare superiore
% convertiamo la matrice
for i=1:1:n-1
x_max = max ( abs(U(i:n,i)) );

23
functions/linear_system_resolver_lower_triangular_matrix.m Normal file
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@@ -0,0 +1,23 @@
%risoluzione matrice triangolare inferiore con algoritmo
% di sostituzione botton up (all'indietro)
%example input
% U=[1,0,0;1,1,0;1,5,1];
% b=[5;-8;-2];
function[x] = linear_system_resolver_lower_triangular_matrix(U,b)
x=[];
n= length(b);
x(1) = b(1)/U(1,1);
for i=2:1:n
somma = 0;
for k=i-1:-1:1
somma=somma+U(i,k) * x(k);
end
x(i) = (b(i) - somma)/ U(i,i);
end
end

2
functions/linear_system_resolver_triangular_matrix.m → functions/linear_system_resolver_upper_triangular_matrix.m
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@@ -6,7 +6,7 @@
% b=[5;-8;-2];
function[x] = linear_system_resolver_triangular_matrix(U,b)
function[x] = linear_system_resolver_upper_triangular_matrix(U,b)
x=[];
n= length(b);

54
functions/thomas_tridiagonal_matrix_no_pivoting.m Normal file
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@@ -0,0 +1,54 @@
% questo algoritmo prende in input una matrice tridiagonale
% e i termini noti : ax=b e ne calcola la soluzione x
% usando il metodo di thomas lineare
%U=[10,20,30;1,5,3;4,56,34]; matrice di input
%b=[1;2;17]; termini noti
function [x] = thomas_tridiagonal_matrix_no_pivoting (U,b)
n= length(b); % lunghezza del vettore dei termini noti o numero equazioni
norma = norm(U,inf);
ok=1;
% convertiamo la matrice
for i=1:1:n-1
if abs(U(i,i)) <= eps * norma
disp('WARNING: Elemento prossimo allo zero sulla diagonale')
end
if U(i,i) == 0
ok=0;
break
end
j=i+1;
m= U(j,i) / U(i,i);
U(j,i+1) = U(j,i+1) + ( U(i,i+1) * (- m ) ) ;
U(j,i)=m;
end
if ok == 0 || U(n,n) == 0
disp('impossibile convertire')
x=[];
else
if abs(U(n,n)) <= eps * norma
disp('WARNING: Elemento prossimo allo zero sulla diagonale')
end
Low=tril(U,-1) + eye (size(U));
Upp=triu(U);
%1 ly=b
%2 ux=y
y = linear_system_resolver_lower_triangular_matrix(Low,b);
x = linear_system_resolver_upper_triangular_matrix(Upp,y);
end
end

4
samples/test.m → samples/sample_matrices.m
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@@ -1 +1,5 @@
%band 2 2
A=[6,-1,-1,0,0,0,0,0,0;-1,6,-1,-2,0,0,0,0,0;-3,-1,6,0,-1,0,0,0,0;0,-1,0,6,-1,-1,0,0,0;0,0,-1,-1,6,-1,-1,0,0;0,0,0,-3,-1,6,0,-1,0;0,0,0,0,-1,0,6,-1,-2;0,0,0,0,0,-1,-1,6,-1;0,0,0,0,0,0,-1,-1,6]
%tridiagonal
A = [5,1,0,0;4,5,1,0;0,1,3,5;0,0,4,6]

39
samples/spettro.m Normal file
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@@ -0,0 +1,39 @@
%Ax=b
%A = X + Y
%x = inv(X) *b - (inv(X) * Y *x)
% se C = inv(X) * b e B = -(inv(X) * Y )
% allora x = Bx + C
%metodo iterativo converge se e solo se il raggio spettrale di B < 1
% raggio spettrale = max (abs(eig(B)))
%in jacobi X = diag(diag(A)) e Y = A - X
%in gauss-seidel X = tril(A) e Y = A - X
%quindi
%jacobi B = -(inv(diag(diag(A))) * (A - (diag(diag(A)))) )
%jacobi C = inv(diag(diag(A))) * b
%gauss B = -(inv(tril(A)) * (A - (tril(A))) )
%gauss C = inv(tril(A)) * b
%formula definitiva
%jacobi converge se
% max(abs(eig((-(inv(diag(diag(A))) * (A - (diag(diag(A)))) )))))
% <1
%gauss converge se
% max(abs(eig(-(inv(tril(A)) * (A - (tril(A))) ))))
% <1