def inverse_iteration(a: Dmatrix, a_qr: Dmatrix, eps: float) -> Dmatrix:
mu = 0.0
y = Dvector(1, N)
a_ii = a_qr.copy()
for i in range(1, N + 1):
lambda_ = a_ii[i][i] # 近似固有値の代入
y[i] = 1.0 # 初期値設定
# 行列の作成およびLU分解
lu = a.copy()
for k in range(1, N + 1):
lu[k][k] -= lambda_
lu, p = lu_decomp(lu, N) # LU分解
# 逆反復法
while True:
muo = mu
v = y.copy()
v = lu_solve(lu, v, p, N) # 固有ベクトルの計算
mu = inner_product(v, y) # 補正
v2s = sqrt(inner_product(v, v))
for j in range(1, N + 1):
y[j] = v[j] / v2s
if abs((mu - muo) / mu) < eps:
break
# 結果の代入(固有ベクトルはaのi列に)
for j in range(1, N + 1):
a_ii[j][i] = y[j]
return a_ii
def gauss_seidel(a: Dmatrix, b: Dvector, x0: Dvector, N: int = N):
k = 0
x = x0.copy()
xo = Dvector(1, N) # xo[1...N]
while True:
# xo <- x_k, x <- x_{k+1}
for i in range(1, N + 1):
xo[i] = x[i] # x_k に x_(k+1) を代入
# i=1 の処理
x[1] = (b[1] - sum(
(a[1][j] * xo[j] for j in range(2, N + 1)))) / a[1][1]
# i=2,3,...N の処理
for i in range(2, N + 1):
s = sum((a[i][j] * x[j] for j in range(1, i))) # i-1列までの和
t = sum((a[i][j] * xo[j] for j in range(i + 1, N + 1))) # i+1列以降の和
x[i] = (b[i] - s - t) / a[i][i]
for i in range(1, N + 1):
xo[i] = xo[i] - x[i]
eps = vector_norm_max(xo)
k += 1
if eps <= EPS or k >= KMAX:
break
if k == KMAX:
print("答えが見つかりませんでした")
exit(1)
else:
print(f"反復回数は{k}回です")
return x
def cholesky_solve(a_cd: Dmatrix, b: Dvector, N: int = N):
b_cs = b.copy()
# LDy = b
b_cs[1] = b[1] / a_cd[1][1]
for i in range(2, N + 1):
b_cs[i] = (b_cs[i] - sum((a_cd[j][j] * a_cd[i][j] * b_cs[j]
for j in range(1, i)))) / a_cd[i][i]
# L^t x = y
for i in range(N - 1, 0, -1):
b_cs[i] -= sum((a_cd[j][i] * b_cs[j] for j in range(i + 1, N + 1)))
return b_cs
def lu_solve(a: Dmatrix, b: Dvector, p: List[int], N:int=N) -> Dvector:
b_lu = b.copy() # 値渡し
# 右辺の行交換
for k in range(1, N):
b_lu[k], b_lu[p[k]] = b_lu[p[k]], b_lu[k]
# 前進代入
for i in range(k+1, N+1):
b_lu[i] += a[i][k] * b_lu[k]
# 後退代入
b_lu[N] /= a[N][N]
for k in range(N-1, 0, -1):
b_lu[k] = ( b_lu[k] - sum( (a[k][j] * b_lu[j] for j in range(k+1, N+1)) ) ) / a[k][k]
return b_lu
def cg(a: Dmatrix, b: Dvector, x0: Dvector):
k = 0
r = Dvector(1, N) # r[1...N]
p = Dvector(1, N) # p[1...N]
x = x0.copy()
tmp = matrix_vector_product(a, x) # tmp <- A b
for i in range(1, N + 1):
p[i] = b[i] - tmp[i]
r[i] = p[i]
while True:
# alpha の計算
tmp = matrix_vector_product(a, p) # tmp <- A p_k
work = inner_product(p, tmp) # work <- (p, Ap_k)
alpha = inner_product(p, r) / work
# x_{k+1} と r_{k+1} の計算
for i in range(1, N + 1):
x[i] += alpha * p[i]
for i in range(1, N + 1):
r[i] -= alpha * tmp[i]
# 収束判定
eps = vector_norm1(r)
k += 1 # 反復回数の更新
if eps < EPS:
break
# beta と p_{k+1} の計算
beta = -inner_product(r, tmp) / work
for i in range(1, N + 1):
p[i] = r[i] + beta * p[i]
if k >= KMAX:
break
if k == KMAX:
print("答えが見つかりませんでした")
exit(1)
else:
print(f"反復回数は{k}回です") # 反復回数を画面に表示
return x
def sor(a: Dmatrix, b: Dvector, x0: Dvector, omega: float, N: int = N):
k = 0
x = x0.copy()
xo = Dvector(1, N) # xo[1...N]
while True:
# xo <- x_k, x <- x_{k+1}
for i in range(1, N + 1):
xo[i] = x[i] # x_k に x_(k+1) を代入
# i=1 の処理
x[1] = (b[1] - sum(
(a[1][j] * xo[j] for j in range(2, N + 1)))) / a[1][1]
# i=2,3,...N の処理
for i in range(2, N + 1):
s = sum((a[i][j] * x[j] for j in range(1, i))) # i-1列までの和
t = sum((a[i][j] * xo[j] for j in range(i + 1, N + 1))) # i+1列以降の和
x[i] = (b[i] - s - t) / a[i][i]
# ここまではガウス・ザイデル法と同じ
# SOR法
for i in range(1, N + 1):
x[i] = xo[i] + omega * (x[i] - xo[i]) # 補正
for i in range(1, N + 1):
xo[i] = xo[i] - x[i]
eps = vector_norm_max(xo)
k += 1
if eps <= EPS or k >= KMAX:
break
if k == KMAX:
print("答えが見つかりませんでした")
exit(1)
else:
print(f"反復回数は{k}回です") # 反復回数を画面に表示
return x