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kleinere Optimierungen

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Tammo.Weber
2026-02-06 14:10:48 +01:00
parent 9591045ee2
commit 7a2a843285
2 changed files with 203 additions and 210 deletions
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297
GHA_triaxial/gha2_num.py
View File

@@ -1,24 +1,19 @@
import numpy as np
from ellipsoide import EllipsoidTriaxial
import runge_kutta as rk
from runge_kutta import rk4, rk4_step, rk4_end, rk4_integral
import GHA_triaxial.numeric_examples_karney as ne_karney
import GHA_triaxial.numeric_examples_panou as ne_panou
import winkelumrechnungen as wu
from typing import Tuple
from numpy.typing import NDArray
import ausgaben as aus
from utils_angle import cot, wrap_to_pi
from utils_angle import cot, arccot, wrap_to_pi
def arccot(x):
x = np.asarray(x)
a = np.arctan2(1.0, x)
return np.where(x < 0.0, a - np.pi, a)
def normalize_alpha_0_pi(alpha):
if alpha < 0.0:
alpha += np.pi
return alpha
def norm_a(a):
if a < 0.0:
a += np.pi
return a
def sph_azimuth(beta1, lam1, beta2, lam2):
dlam = wrap_to_pi(lam2 - lam1)
@@ -32,17 +27,29 @@ def sph_azimuth(beta1, lam1, beta2, lam2):
# Panou 2013
def gha2_num(
ell: EllipsoidTriaxial,
beta_0: float,
lamb_0: float,
beta_1: float,
lamb_1: float,
beta_2: float,
lamb_2: float,
n: int = 16000,
epsilon: float = 10**-12,
iter_max: int = 30,
all_points: bool = False,
) -> Tuple[float, float, float] | Tuple[float, float, float, NDArray, NDArray]:
"""
:param ell: Ellipsoid
:param beta_0: Beta Punkt 0
:param lamb_0: Lambda Punkt 0
:param beta_1: Beta Punkt 1
:param lamb_1: Lambda Punkt 1
:param n: Anzahl Schritte
:param epsilon: Genauigkeit
:param iter_max: Maximale Iterationen
:param all_points: Ausgabe aller Punkte
:return: Azimut Startpunkt, Azumit Zielpunkt, Strecke
"""
# Berechnung Koeffizienten, Gaußschen Fundamentalgrößen 1. Ordnung sowie deren Ableitungen
def BETA_LAMBDA(beta, lamb):
BETA = (ell.ay ** 2 * np.sin(beta) ** 2 + ell.b ** 2 * np.cos(beta) ** 2) / (
ell.Ex ** 2 - ell.Ey ** 2 * np.sin(beta) ** 2
@@ -123,6 +130,7 @@ def gha2_num(
G_lamb_lamb,
)
# Berechnung der ODE Koeffizienten für Fall 1 (lambda_0 != lambda_1)
def p_coef(beta, lamb):
(
BETA,
@@ -161,6 +169,7 @@ def gha2_num(
return (BETA, LAMBDA, E, G, p_3, p_2, p_1, p_0, p_33, p_22, p_11, p_00)
# Berechnung der ODE Koeffizienten für Fall 2 (lambda_0 == lambda_1)
def q_coef(beta, lamb):
(
BETA,
@@ -197,69 +206,7 @@ def gha2_num(
)
q_00 = 0.5 * ((E_lamb_lamb * G - E_lamb * G_lamb) / (G**2))
return (BETA, LAMBDA, E, G, q_3, q_2, q_1, q_0, q_33, q_22, q_11, q_00)
def rk4_last(f, t0, y0, dt, N):
h = dt / N
t = t0
y = np.array(y0, dtype=float, copy=True)
for _ in range(N):
k1 = f(t, y)
k2 = f(t + 0.5 * h, y + 0.5 * h * k1)
k3 = f(t + 0.5 * h, y + 0.5 * h * k2)
k4 = f(t + h, y + h * k3)
y = y + (h / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4)
t = t + h
return t, y
def rk4_last_with_integral(f, t0, y0, dt, N, integrand_at):
h = dt / N
habs = abs(h)
t = t0
y = np.array(y0, dtype=float, copy=True)
if N % 2 == 0:
# Simpson streaming
f0 = integrand_at(t, y)
odd_sum = 0.0
even_sum = 0.0
for i in range(1, N + 1):
k1 = f(t, y)
k2 = f(t + 0.5 * h, y + 0.5 * h * k1)
k3 = f(t + 0.5 * h, y + 0.5 * h * k2)
k4 = f(t + h, y + h * k3)
y = y + (h / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4)
t = t + h
fi = integrand_at(t, y)
if i == N:
fN = fi
elif i % 2 == 1:
odd_sum += fi
else:
even_sum += fi
S = f0 + fN + 4.0 * odd_sum + 2.0 * even_sum
s = (habs / 3.0) * S
return t, y, s
# Trapez streaming
f_prev = integrand_at(t, y)
acc = 0.0
for _ in range(N):
k1 = f(t, y)
k2 = f(t + 0.5 * h, y + 0.5 * h * k1)
k3 = f(t + 0.5 * h, y + 0.5 * h * k2)
k4 = f(t + h, y + h * k3)
y = y + (h / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4)
t = t + h
f_cur = integrand_at(t, y)
acc += 0.5 * (f_prev + f_cur)
f_prev = f_cur
s = habs * acc
return t, y, s
return BETA, LAMBDA, E, G, q_3, q_2, q_1, q_0, q_33, q_22, q_11, q_00
def integrand_lambda(lamb, y):
beta = y[0]
@@ -273,42 +220,41 @@ def gha2_num(
(_, _, E, G, *_) = BETA_LAMBDA(beta, lamb)
return np.sqrt(E + G * lamb_p**2)
if lamb_1 != lamb_2:
N = n
dlamb = lamb_2 - lamb_1
# Fall 1 (lambda_0 != lambda_1)
if abs(lamb_1 - lamb_0) >= 1e-15:
N = int(n)
dlamb = float(lamb_1 - lamb_0)
def buildODElamb():
def ODE(lamb, v):
beta0 = float(beta_0)
lamb0 = float(lamb_0)
beta1 = float(beta_1)
lamb1 = float(lamb_1)
def ode_lamb(lamb, v):
beta, beta_p, X3, X4 = v
(_, _, _, _, p_3, p_2, p_1, p_0, p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
dbeta = beta_p
dbeta_p = p_3 * beta_p**3 + p_2 * beta_p**2 + p_1 * beta_p + p_0
dX3 = X4
dX4 = (p_33 * beta_p**3 + p_22 * beta_p**2 + p_11 * beta_p + p_00) * X3 + (
3 * p_3 * beta_p**2 + 2 * p_2 * beta_p + p_1
) * X4
dX4 = (p_33 * beta_p**3 + p_22 * beta_p**2 + p_11 * beta_p + p_00) * X3 + (3*p_3*beta_p**2 + 2*p_2*beta_p + p_1) * X4
return np.array([dbeta, dbeta_p, dX3, dX4], dtype=float)
return ODE
ode_lamb = buildODElamb()
alpha0_sph = sph_azimuth(beta_1, lamb_1, beta_2, lamb_2)
(_, _, E1, G1, *_) = BETA_LAMBDA(beta_1, lamb_1)
beta_p0_sph = np.sqrt(G1 / E1) * cot(alpha0_sph) if abs(dlamb) >= 1e-15 else 0.0
alpha0_sph = sph_azimuth(beta0, lamb0, beta1, lamb1)
(_, _, E0, G0, *_) = BETA_LAMBDA(beta0, lamb0)
beta_p0_sph = np.sqrt(G0 / E0) * cot(alpha0_sph)
N_newton = min(N, 4000)
def solve_newton(beta_p0_init: float):
beta_p0 = float(beta_p0_init)
for _ in range(iter_max):
startwerte = np.array([beta_1, beta_p0, 0.0, 1.0], dtype=float)
_, y_end = rk4_last(ode_lamb, lamb_1, startwerte, dlamb, N_newton)
beta_end, _, X3_end, _ = y_end
v0 = np.array([beta0, beta_p0, 0.0, 1.0], dtype=float)
_, y_end = rk4_end(ode_lamb, lamb0, v0, dlamb, N_newton)
beta_end, _, X3_end, _ = y_end
delta = beta_end - beta1
delta = beta_end - beta_2
if abs(delta) < epsilon:
return True, beta_p0
@@ -316,59 +262,45 @@ def gha2_num(
return False, None
step = delta / X3_end
max_step = 0.5
if abs(step) > max_step:
step = np.sign(step) * max_step
beta_p0 = beta_p0 - step
step = np.clip(step, -0.5, 0.5)
beta_p0 -= step
return False, None
ok, beta_p0_sol = solve_newton(beta_p0_sph)
if not ok:
candidates = [-beta_p0_sph, 0.5 * beta_p0_sph, 2.0 * beta_p0_sph]
N_quick = min(N, 2000)
best = None
for g in candidates:
ok_g, beta_p0_sol_g = solve_newton(g)
ok_g, sol = solve_newton(g)
if not ok_g:
continue
startwerte_g = np.array([beta_1, beta_p0_sol_g, 0.0, 1.0], dtype=float)
_, _, s_quick = rk4_last_with_integral(
ode_lamb, lamb_1, startwerte_g, dlamb, N_quick, integrand_lambda
)
v0_g = np.array([beta0, sol, 0.0, 1.0], dtype=float)
_, _, s_quick = rk4_integral(ode_lamb, lamb0, v0_g, dlamb, N_quick, integrand_lambda)
if (best is None) or (s_quick < best[0]):
best = (s_quick, beta_p0_sol_g)
best = (s_quick, sol)
if best is None:
raise RuntimeError("Keine Startwert-Variante konvergiert.")
raise RuntimeError("Keine Startwert-Variante konvergiert (lambda-Fall).")
beta_p0_sol = best[1]
beta_0 = beta_p0_sol
startwerte_final = np.array([beta_1, beta_0, 0.0, 1.0], dtype=float)
beta_p0 = float(beta_p0_sol)
v0_final = np.array([beta0, beta_p0, 0.0, 1.0], dtype=float)
if all_points:
lamb_list, states = rk.rk4(ode_lamb, lamb_1, startwerte_final, dlamb, N, False)
lamb_list, states = rk4(ode_lamb, lamb0, v0_final, dlamb, N, False)
lamb_arr = np.array(lamb_list, dtype=float)
beta_arr = np.array([st[0] for st in states], dtype=float)
beta_p_arr = np.array([st[1] for st in states], dtype=float)
(_, _, E1, G1, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
(_, _, E2, G2, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
(_, _, E_start, G_start, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
(_, _, E_end, G_end, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
alpha_1 = arccot(np.sqrt(E1 / G1) * beta_p_arr[0])
alpha_2 = arccot(np.sqrt(E2 / G2) * beta_p_arr[-1])
alpha_1 = norm_a(arccot(np.sqrt(E_start / G_start) * beta_p_arr[0]))
alpha_2 = norm_a(arccot(np.sqrt(E_end / G_end) * beta_p_arr[-1]))
alpha_1 = normalize_alpha_0_pi(float(alpha_1))
alpha_2 = normalize_alpha_0_pi(float(alpha_2))
# Distanz s aus Arrays (Simpson/Trapz)
# Distanz aus Arrays
integrand = np.zeros(N + 1, dtype=float)
for i in range(N + 1):
(_, _, Ei, Gi, *_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
@@ -381,91 +313,75 @@ def gha2_num(
else:
s = np.trapz(integrand, dx=h)
return alpha_1, alpha_2, s, beta_arr, lamb_arr
return float(alpha_1), float(alpha_2), float(s), beta_arr, lamb_arr
_, y_end, s = rk4_last_with_integral(ode_lamb, lamb_1, startwerte_final, dlamb, N, integrand_lambda)
_, y_end, s = rk4_integral(ode_lamb, lamb0, v0_final, dlamb, N, integrand_lambda)
beta_end, beta_p_end, _, _ = y_end
(_, _, E1, G1, *_) = BETA_LAMBDA(beta_1, lamb_1)
(_, _, E2, G2, *_) = BETA_LAMBDA(beta_2, lamb_2)
(_, _, E_start, G_start, *_) = BETA_LAMBDA(beta0, lamb0)
(_, _, E_end, G_end, *_) = BETA_LAMBDA(beta1, lamb1)
alpha_1 = arccot(np.sqrt(E1 / G1) * beta_0)
alpha_2 = arccot(np.sqrt(E2 / G2) * beta_p_end)
alpha_1 = norm_a(arccot(np.sqrt(E_start / G_start) * beta_p0))
alpha_2 = norm_a(arccot(np.sqrt(E_end / G_end) * beta_p_end))
alpha_1 = normalize_alpha_0_pi(float(alpha_1))
alpha_2 = normalize_alpha_0_pi(float(alpha_2))
return float(alpha_1), float(alpha_2), float(s)
return alpha_1, alpha_2, s
N = n
dbeta = beta_2 - beta_1
# Fall 2 (lambda_0 == lambda_1)
N = int(n)
dbeta = float(beta_1 - beta_0)
if abs(dbeta) < 1e-15:
if all_points:
return 0.0, 0.0, 0.0, np.array([]), np.array([])
return 0.0, 0.0, 0.0
# ODE-System (lambda, lambda', Y3, Y4) in Abhängigkeit von beta
def buildODEbeta():
def ODE(beta, v):
beta0 = float(beta_0)
lamb0 = float(lamb_0)
beta1 = float(beta_1)
lamb1 = float(lamb_1)
def ode_beta(beta, v):
lamb, lamb_p, Y3, Y4 = v
(_, _, _, _, q_3, q_2, q_1, q_0, q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
dlamb = lamb_p
dlamb_p = q_3*lamb_p**3 + q_2*lamb_p**2 + q_1*lamb_p + q_0
dY3 = Y4
dY4 = (q_33 * lamb_p**3 + q_22 * lamb_p**2 + q_11 * lamb_p + q_00) * Y3 + (
3 * q_3 * lamb_p**2 + 2 * q_2 * lamb_p + q_1
) * Y4
dY4 = (q_33*lamb_p**3 + q_22*lamb_p**2 + q_11*lamb_p + q_00)*Y3 + (3*q_3*lamb_p**2 + 2*q_2*lamb_p + q_1)*Y4
return np.array([dlamb, dlamb_p, dY3, dY4], dtype=float)
return ODE
ode_beta = buildODEbeta()
# Newton auf lambda'_0
lamb_0 = 0.0
lamb_p0 = 0.0
for _ in range(iter_max):
startwerte = np.array([lamb_1, lamb_0, 0.0, 1.0], dtype=float)
beta_list, states = rk.rk4(ode_beta, beta_1, startwerte, dbeta, N, False)
v0 = np.array([lamb0, lamb_p0, 0.0, 1.0], dtype=float)
_, y_end = rk4_end(ode_beta, beta0, v0, dbeta, N)
lamb_end, lamb_p_end, Y3_end, _ = states[-1]
delta = lamb_end - lamb_2
lamb_end, _, Y3_end, _ = y_end
delta = lamb_end - lamb1
if abs(delta) < epsilon:
break
if abs(Y3_end) < 1e-20:
raise RuntimeError("Abbruch (Ableitung ~ 0).")
raise RuntimeError("Abbruch (Ableitung ~ 0) im beta-Fall.")
step = delta / Y3_end
max_step = 1.0
if abs(step) > max_step:
step = np.sign(step) * max_step
lamb_0 = lamb_0 - step
step = np.clip(step, -1.0, 1.0)
lamb_p0 -= step
startwerte_final = np.array([lamb_1, lamb_0, 0.0, 1.0], dtype=float)
v0_final = np.array([lamb0, lamb_p0, 0.0, 1.0], dtype=float)
if all_points:
beta_list, states = rk.rk4(ode_beta, beta_1, startwerte_final, dbeta, N, False)
beta_list, states = rk4(ode_beta, beta0, v0_final, dbeta, N, False)
beta_arr = np.array(beta_list, dtype=float)
lamb_arr = np.array([st[0] for st in states], dtype=float)
lamb_p_arr = np.array([st[1] for st in states], dtype=float)
# Azimute
(BETA1, LAMBDA1, _, _, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
(BETA2, LAMBDA2, _, _, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
(BETA_s, LAMBDA_s, _, _, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
(BETA_e, LAMBDA_e, _, _, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
alpha_1 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA1 / BETA1) * lamb_p_arr[0])
alpha_2 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA2 / BETA2) * lamb_p_arr[-1])
alpha_1 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_s / BETA_s) * lamb_p_arr[0]))
alpha_2 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_e / BETA_e) * lamb_p_arr[-1]))
# optionaler Quadrantenfix (robust)
alpha_1 = normalize_alpha_0_pi(float(alpha_1))
alpha_2 = normalize_alpha_0_pi(float(alpha_2))
# Distanz
integrand = np.zeros(N + 1, dtype=float)
for i in range(N + 1):
(_, _, Ei, Gi, *_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
@@ -478,32 +394,29 @@ def gha2_num(
else:
s = np.trapz(integrand, dx=h)
return alpha_1, alpha_2, s, beta_arr, lamb_arr
return float(alpha_1), float(alpha_2), float(s), beta_arr, lamb_arr
# all_points == False: streaming Integral
_, y_end, s = rk4_last_with_integral(ode_beta, beta_1, startwerte_final, dbeta, N, integrand_beta)
_, y_end, s = rk4_integral(ode_beta, beta0, v0_final, dbeta, N, integrand_beta)
lamb_end, lamb_p_end, _, _ = y_end
(BETA1, LAMBDA1, _, _, *_) = BETA_LAMBDA(beta_1, lamb_1)
(BETA2, LAMBDA2, _, _, *_) = BETA_LAMBDA(beta_2, lamb_2)
(BETA_s, LAMBDA_s, _, _, *_) = BETA_LAMBDA(beta0, lamb0)
(BETA_e, LAMBDA_e, _, _, *_) = BETA_LAMBDA(beta1, lamb1)
alpha_1 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA1 / BETA1) * lamb_0)
alpha_2 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA2 / BETA2) * lamb_p_end)
alpha_1 = normalize_alpha_0_pi(float(alpha_1))
alpha_2 = normalize_alpha_0_pi(float(alpha_2))
return alpha_1, alpha_2, s
alpha_1 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_s / BETA_s) * lamb_p0))
alpha_2 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_e / BETA_e) * lamb_p_end))
return float(alpha_1), float(alpha_2), float(s)
if __name__ == "__main__":
ell = EllipsoidTriaxial.init_name("BursaSima1980round")
beta1 = np.deg2rad(75)
lamb1 = np.deg2rad(-90)
beta2 = np.deg2rad(75)
lamb2 = np.deg2rad(66)
a1, a2, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=5000)
print(aus.gms("a1", a1, 4))
# ell = EllipsoidTriaxial.init_name("BursaSima1980round")
# beta1 = np.deg2rad(75)
# lamb1 = np.deg2rad(-90)
# beta2 = np.deg2rad(75)
# lamb2 = np.deg2rad(66)
# a0, a1, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=5000)
# print(aus.gms("a0", a0, 4))
# print(aus.gms("a1", a1, 4))
# print("s: ", s)
# # print(aus.gms("a2", a2, 4))
# # print(s)
# cart1 = ell.para2cart(0, 0)

80
runge_kutta.py
View File

@@ -41,3 +41,83 @@ def rk4_step(ode, t: float, v: np.ndarray, h: float) -> np.ndarray:
k3 = ode(t + 0.5 * h, v + 0.5 * h * k2)
k4 = ode(t + h, v + h * k3)
return v + (h / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4)
def rk4_end(ode, t0: float, v0: np.ndarray, weite: float, schritte: int, fein: bool = False):
h = weite / schritte
t = float(t0)
v = np.array(v0, dtype=float, copy=True)
for _ in range(schritte):
if not fein:
v_next = rk4_step(ode, t, v, h)
else:
v_grob = rk4_step(ode, t, v, h)
v_half = rk4_step(ode, t, v, 0.5 * h)
v_fein = rk4_step(ode, t + 0.5 * h, v_half, 0.5 * h)
v_next = v_fein + (v_fein - v_grob) / 15.0
t += h
v = v_next
return t, v
# RK4 mit Simpson bzw. Trapez
def rk4_integral( ode, t0: float, v0: np.ndarray, weite: float, schritte: int, integrand_at, fein: bool = False, simpson: bool = True, ):
h = weite / schritte
habs = abs(h)
t = float(t0)
v = np.array(v0, dtype=float, copy=True)
if simpson and (schritte % 2 == 0):
f0 = float(integrand_at(t, v))
odd_sum = 0.0
even_sum = 0.0
fN = None
for i in range(1, schritte + 1):
if not fein:
v_next = rk4_step(ode, t, v, h)
else:
v_grob = rk4_step(ode, t, v, h)
v_half = rk4_step(ode, t, v, 0.5 * h)
v_fein = rk4_step(ode, t + 0.5 * h, v_half, 0.5 * h)
v_next = v_fein + (v_fein - v_grob) / 15.0
t += h
v = v_next
fi = float(integrand_at(t, v))
if i == schritte:
fN = fi
elif i % 2 == 1:
odd_sum += fi
else:
even_sum += fi
S = f0 + fN + 4.0 * odd_sum + 2.0 * even_sum
s = (habs / 3.0) * S
return t, v, s
f_prev = float(integrand_at(t, v))
acc = 0.0
for _ in range(schritte):
if not fein:
v_next = rk4_step(ode, t, v, h)
else:
v_grob = rk4_step(ode, t, v, h)
v_half = rk4_step(ode, t, v, 0.5 * h)
v_fein = rk4_step(ode, t + 0.5 * h, v_half, 0.5 * h)
v_next = v_fein + (v_fein - v_grob) / 15.0
t += h
v = v_next
f_cur = float(integrand_at(t, v))
acc += 0.5 * (f_prev + f_cur)
f_prev = f_cur
s = habs * acc
return t, v, s