LU ডিকম্পোজিশন

def lu_decompose(A):
    n = len(A)
    L = [[0.0] * n for _ in range(n)]
    U = [[0.0] * n for _ in range(n)]
    for i in range(n):
        L[i][i] = 1.0

    for k in range(n):
        for j in range(k, n):
            U[k][j] = A[k][j] - sum(L[k][m] * U[m][j] for m in range(k))
        for i in range(k + 1, n):
            L[i][k] = (A[i][k] - sum(L[i][m] * U[m][k] for m in range(k))) / U[k][k]
    return L, U

def mat_mult(X, Y):
    n = len(X)
    p = len(Y[0])
    m = len(Y)
    return [[sum(X[i][t] * Y[t][j] for t in range(m)) for j in range(p)] for i in range(n)]

def forward_sub(L, b):
    n = len(L)
    y = [0.0] * n
    for i in range(n):
        y[i] = b[i] - sum(L[i][j] * y[j] for j in range(i))
    return y

def back_sub(U, y):
    n = len(U)
    x = [0.0] * n
    for i in range(n - 1, -1, -1):
        x[i] = (y[i] - sum(U[i][j] * x[j] for j in range(i + 1, n))) / U[i][i]
    return x

A = [
    [4.0,  3.0,  2.0],
    [2.0, -1.0,  1.0],
    [6.0,  3.0,  5.0],
]
b = [21.0, 3.0, 26.0]

L, U = lu_decompose(A)
print("L =")
for row in L:
    print(["%.6f" % v for v in row])
print("U =")
for row in U:
    print(["%.6f" % v for v in row])

LU = mat_mult(L, U)
print()
print("যাচাই -- L*U পুনর্গঠন বনাম মূল A:")
max_diff = 0.0
for i in range(3):
    for j in range(3):
        diff = abs(LU[i][j] - A[i][j])
        max_diff = max(max_diff, diff)
print(f"সর্বোচ্চ পার্থক্য = {max_diff:.2e}")

y = forward_sub(L, b)
x = back_sub(U, y)
print()
print("Ly = b থেকে y =", [round(v, 6) for v in y])
print("Ux = y থেকে x =", [round(v, 6) for v in x])

print()
print("যাচাই -- মূল Ax = b-তে বসিয়ে অবশিষ্টাংশ:")
for i in range(3):
    lhs = sum(A[i][j] * x[j] for j in range(3))
    print(f"সমীকরণ {i+1}: LHS = {lhs:.10f}   b = {b[i]:.6f}   residual = {lhs - b[i]:.2e}")

L =
['1.000000', '0.000000', '0.000000']
['0.500000', '1.000000', '0.000000']
['1.500000', '0.600000', '1.000000']
U =
['4.000000', '3.000000', '2.000000']
['0.000000', '-2.500000', '0.000000']
['0.000000', '0.000000', '2.000000']

যাচাই -- L*U পুনর্গঠন বনাম মূল A:
সর্বোচ্চ পার্থক্য = 0.00e+00

Ly = b থেকে y = [21.0, -7.5, -1.0]
Ux = y থেকে x = [3.25, 3.0, -0.5]

যাচাই -- মূল Ax = b-তে বসিয়ে অবশিষ্টাংশ:
সমীকরণ 1: LHS = 21.0000000000   b = 21.000000   residual = 0.00e+00
সমীকরণ 2: LHS = 3.0000000000   b = 3.000000   residual = 0.00e+00
সমীকরণ 3: LHS = 26.0000000000   b = 26.000000   residual = 0.00e+00