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 lu_decomp(a: Dmatrix, N: int=N) -> Tuple[Dmatrix, List[int]]:
eps = 2.0 ** -50.0 # eps = 2^{-50}とする
p = [0] * (a.row_last_idx - a.row_head_idx + 1) # p[1...N-1] を利用, p[0] は未使用
a_lu = a.copy() # 値渡し
for k in range(1, N):
# ピボットの選択
amax = abs(a_lu[k][k])
ip = k
for i in range(k+1, N+1):
if abs(a_lu[i][k]) > amax:
amax = abs(a_lu[i][k])
ip = i
# 正則性の判定
if amax < eps:
print("入力した行列は正則ではない!!")
# ipを配列pに保存
p[k] = ip
# 行交換
if ip != k:
for j in range(k, N+1):
a_lu[k][j], a_lu[ip][j] = a_lu[ip][j], a_lu[k][j]
# 前進消去
for i in range(k+1, N+1):
alpha = - a_lu[i][k] / a_lu[k][k]
a_lu[i][k] = alpha
for j in range(k+1, N+1):
a_lu[i][j] += alpha * a_lu[k][j]
return (a_lu, p)
def qr(a: Dmatrix, eps: float, n: int) -> Dmatrix:
# 領域の確保
q = Dmatrix(1, n, 1, n)
work = Dvector(1, n)
a_qr = a.copy()
m = n
while m > 1:
# 収束判定
if abs(a_qr[m][m-1]) < eps:
m = m - 1
continue
# 原点移動
s = a_qr[n][n]
if m == n: # m=n のときは原点移動なし
s = 0.0
for i in range(1, m+1):
a_qr[i][i] -= s
# QR法
for i in range(1, m+1):
for j in range(1, m+1):
q[i][j] = 0.0 # Q を m x m 単位行列で初期化
q[i][i] = 1.0
# R と Q の計算
for i in range(1, m):
r = sqrt( a_qr[i][i]*a_qr[i][i] + a_qr[i+1][i]*a_qr[i+1][i] )
if r == 0.0:
sint = 0.0
cost = 0.0
else:
sint = a_qr[i+1][i] / r
cost = a_qr[i][i] / r
for j in range(i+1, m+1):
tmp = a_qr[i][j]*cost + a_qr[i+1][j]*sint
a_qr[i+1][j] = -a_qr[i][j]*sint + a_qr[i+1][j]*cost
a_qr[i][j] = tmp
a_qr[i+1][i] = 0.0
a_qr[i][i] = r
for j in range(1, m+1): # Q は P の転置
tmp = q[j][i]*cost + q[j][i+1]*sint
q[j][i+1] = -q[j][i]*sint + q[j][i+1]*cost
q[j][i] = tmp
# RQ の計算
for i in range(1, m+1):
for j in range(i, m+1):
work[j] = a_qr[i][j]
for j in range(1, m+1):
a_qr[i][j] = sum( ( work[k] * q[k][j] for k in range(i, m+1) ) )
# 原点移動後の処理
for i in range(1, m+1):
a_qr[i][i] += s
return a_qr
def cholesky_decomp(a: Dmatrix, N: int = N):
a_cd = a.copy()
for i in range(2, N + 1):
for j in range(1, i):
a_cd[i][j] = (a_cd[i][j] - sum(
(a_cd[i][k] * a_cd[k][k] * a_cd[j][k]
for k in range(1, j)))) / a_cd[j][j]
a_cd[i][i] -= sum(
(a_cd[i][k] * a_cd[i][k] * a_cd[k][k] for k in range(1, j + 1)))
return a_cd
def householder(a: Dmatrix, n: int) -> Dmatrix:
u = Dvector(1, n)
f = Dvector(1, n)
g = Dvector(1, n)
a_hh = a.copy()
for k in range(1, n - 1):
# v の計算
for i in range(1, k + 1):
u[i] = 0.0
for i in range(k + 1, n + 1):
u[i] = a_hh[i][k]
# s の計算
ss = 0.0
for i in range(k + 2, n + 1):
ss += u[i] * u[i]
if abs(ss) <= 0.0: # 消去が必要ない場合の処理
continue
s = sqrt(ss + u[k + 1] * u[k + 1])
if u[k + 1] > 0.0:
s = -s
# uの計算
u[k + 1] -= s
uu = sqrt(ss + u[k + 1] * u[k + 1])
for i in range(k + 1, n + 1):
u[i] /= uu
# f, gの計算
for i in range(1, n + 1):
f[i], g[i] = 0.0, 0.0
for j in range(k + 1, n + 1):
f[i] += a_hh[i][j] * u[j]
g[i] += a_hh[j][i] * u[j]
# gammaの計算
gamma = 0.0
for j in range(1, n + 1):
gamma += u[j] * g[j]
# f, gの計算
for i in range(1, n + 1):
f[i] -= gamma * u[i]
g[i] -= gamma * u[i]
# A の計算
for i in range(1, n + 1):
for j in range(1, n + 1):
a_hh[i][j] -= 2.0 * u[i] * g[j] + 2.0 * f[i] * u[j]
return a_hh