103 lines
3.3 KiB
Python
103 lines
3.3 KiB
Python
import numpy as np
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from core.orthonormal_basis import generate_muntz_laguerre_basis
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# ---------- 可选：离散再正交 (加权 QR) ----------
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# def trapezoid_weights(freqs: np.ndarray):
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# if len(freqs) == 1:
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# return np.ones(1)
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# df = np.diff(freqs)
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# w = np.zeros_like(freqs, dtype=float)
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# w[0] = 0.5 * df[0]
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# w[-1] = 0.5 * df[-1]
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# if len(freqs) > 2:
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# w[1:-1] = 0.5 * (df[:-1] + df[1:])
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# return w
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def weighted_qr_from_basis(basis_cols: list[np.ndarray], weights: np.ndarray | None = None):
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A = np.column_stack(basis_cols) # (Nf, M)
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if weights is None:
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sw = np.ones(A.shape[0])
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else:
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sw = np.sqrt(weights.real)
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Aw = sw[:, None] * A
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Qw, R = np.linalg.qr(Aw)
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Phi = Qw / (sw[:, None] + 1e-30)
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return Phi, R # Raw = Phi R
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def check_discrete_orthogonality(Phi: np.ndarray, w: np.ndarray):
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G = Phi.conj().T @ (w[:, None] * Phi)
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off = np.max(np.abs(G - np.eye(G.shape[0])))
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return G, off
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def verify_orthonormal(Phi: np.ndarray, w: np.ndarray, atol=1e-10, rtol=1e-8):
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"""
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返回:
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G : Gram 矩阵 (Φ^H W Φ)
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max_off : 最大非对角幅值
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diag_err : max |diag(G)-1|
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passed : 是否满足阈值
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"""
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G = Phi.conj().T @ (w[:, None] * Phi)
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I = np.eye(G.shape[0])
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diag_err = np.max(np.abs(np.diag(G) - 1.0))
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max_off = np.max(np.abs(G - I + np.diag(np.diag(G) - 1.0)))
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passed = (diag_err <= atol) and (max_off <= rtol)
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return G, max_off, diag_err, passed
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def omega_weights(freqs_hz: np.ndarray):
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"""
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基于 ω=2πf 的梯形法得到 w_ω = Δω/(2π)，使得
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(1/2π) ∫_{-∞}^{∞} → Σ w_ω,k
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"""
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f = freqs_hz
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if len(f) == 1:
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return np.ones(1)
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df = np.diff(f)
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w_f = np.zeros_like(f)
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w_f[0] = 0.5 * df[0]
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w_f[-1] = 0.5 * df[-1]
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if len(f) > 2:
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w_f[1:-1] = 0.5 * (df[:-1] + df[1:])
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# dω = 2π df, (1/2π) * dω = df ⇒ 直接 w_f 就是 w_ω
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return w_f # 已等价于 Δω/(2π)
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def evaluate(basis,freqs):
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w = omega_weights(freqs)
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Phi_num, R = weighted_qr_from_basis(basis, w)
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Gram, off = check_discrete_orthogonality(Phi_num, w)
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print("离散 Gram 最大非对角元素 =", off)
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print("R 形状:", R.shape)
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# 验证 Raw ≈ Phi R
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raw = np.column_stack(basis)
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err = np.max(np.abs(raw - Phi_num @ R))
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print("重构误差 ||Raw - Phi R||_∞ =", err)
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# 验证正交性
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print("离散 Gram 矩阵 (前5x5):")
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print(Gram[:5, :5])
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Gcheck, max_off, diag_err, ok = verify_orthonormal(Phi_num, w)
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print(f"Diag 误差={diag_err:.3e}, Max off={max_off:.3e}, Orthonormal={ok}")
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# 额外: 验证 R
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# raw = Φ R => R ≈ Φ^H W raw (因为 Φ^H W Φ = I)
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R_alt = Phi_num.conj().T @ (w[:,None] * raw)
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print("R 差异 ||R - R_alt||_max =", np.max(np.abs(R - R_alt)))
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# ------------------ 示例 ------------------
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if __name__ == "__main__":
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# 示例稳定极点 (复对正虚部在前)
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init_poles = [
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-1.0e3 + 2.5e9j,
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-1.0e3 - 2.5e9j,
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]
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freqs = np.linspace(1e8, 8e9, 40)
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s = 1j * 2 * np.pi * freqs
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basis = generate_muntz_laguerre_basis(s, init_poles)
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print("解析基函数数量 =", len(basis))
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print("基函数:")
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for i in range(len(basis)):
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print(f"φ_{i}:", basis[i][:])
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evaluate(basis, freqs)
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