# Consider the linear system Ax = b with Analyze the convergence properties of the Jacobi and…

Consider the linear system Ax = b with

Analyze the convergence properties of the Jacobi and
Gauss-Seidel methods applied to the system above in their point and block forms
(for a 2 × 2 block partition of A).

Let A _
Rn×n be such that A = (1 + _)P _ (N + _P), with P_1N
nonsingular and with real eigenvalues 1 > _1 _ _2
_ … _ _n. Find the values of _ _ R for which the
following iterative method

converges _x(0)
to the solution of the linear
»

Consider the linear system Ax = b with

Analyze the convergence properties of the Jacobi and
Gauss-Seidel methods applied to the system above in their point and block forms
(for a 2 × 2 block partition of A).

Let A _
Rn×n be such that A = (1 + _)P _ (N + _P), with P_1N
nonsingular and with real eigenvalues 1 > _1 _ _2
_ … _ _n. Find the values of _ _ R for which the
following iterative method

converges _x(0)
to the solution of the linear system (3.2). Determine also the value of _
for which the convergence rate is maximum.

[Solution : _ > _(1 + _n)/2;
_opt = _(_1 + _n)/2.]

»

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