Information:
The Gauss-Seidel method updates variables sequentially, immediately using newly computed values:

A student implements Gauss-Seidel for a 3 3 system as follows:

def gauss_seidel(A, b, x, n_iter):
n = len(b)
for k in range(n_iter):
for i in range(n-1, -1, -1):
sigma = sum(A[i][j]*x[j] for j in range(n) if j != i)
x[i] = (b[i] - sigma) / A[i][i]
return x

The code updates variables in reverse order ( , then , then ) instead of forward order. How does this affect the method?

Question

Information: 
 The Gauss-Seidel method updates variables sequentially, immediately using newly computed values:
  
A student implements Gauss-Seidel for a 3  3 system as follows:

def gauss_seidel(A, b, x, n_iter):
    n = len(b)
    for k in range(n_iter):
        for i in range(n-1, -1, -1):
            sigma = sum(A[i][j]*x[j] for j in range(n) if j != i)
            x[i] = (b[i] - sigma) / A[i][i]
    return x

The code updates variables in reverse order (  , then   , then   ) instead of forward order. How does this affect the method?

Answer

The method becomes numerically unstable, accumulating round-off errors exponentially faster than forward-order Gauss-Seidel, leading to inaccurate solutions.

Answer

The method transforms into the Jacobi method because reverse ordering prevents the use of newly updated values within iterations.

Answer

The method achieves faster convergence than forward-order Gauss-Seidel by propagating corrections backward through the system more efficiently.

Answer

The method fails completely because Gauss-Seidel mathematically requires forward indexing; reverse order violates the algorithm's theoretical foundation and causes divergence.

Answer

The method produces identical iteration trajectories as forward-order Gauss-Seidel due to commutativity of sequential variable updates.

Answer

The method still converges to the same solution (if convergent), but convergence speed and trajectory differ due to altered information propagation order.

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