Journey 02 — Core path

import numpy as np
import solvlrde as lrde
 
A = np.diag([-0.5, -1.0, -2.0])
result = lrde.solve(A, [1.0, 1.0, 1.0], T=2.0)
 
# Constant source: y' = -y + 1
forced = lrde.solve(np.array([[-1.0]]), [0.0], T=3.0, b=[1.0])

Solve y' = A y directly at a target time T

-- Act 1 - The workhorse call --
  -> solvlrde.solve(A, y0, T) is the one call you will use most. Give it a dense system matrix A, an initial state y0, and a target time T; it returns y(T) plus a receipt describing the contour it used.

-- Act 2 - A decoupled 3-mode system --
  -> A = diag(-0.5, -1, -2) makes three independent decay modes. Because the modes are decoupled, y_i(T) = exp(lambda_i * T) * y0_i - a closed form we can check against.
  A_diag: [-0.5, -1.0, -2.0]
  y0: [1.0, 1.0, 1.0]
  T: 2.0

-- Act 3 - Solve and read the receipt --
  -> The result dict carries y plus n_talbot_used and n_solves: how many contour nodes (each a complex linear solve) the auto-tuned contour needed. That count - not T - is what drives the cost.
  survived: True
  status: ok
  y: [0.367879, 0.135335, 0.0183156]
  n_talbot_used: 32
  n_solves: 32
  nu_used: 0.6
  contour_scale: 6.4
  wall_s: 0
  expected: [0.367879, 0.135335, 0.0183156]
  max_abs_error: 4.347e-10

-- Act 4 - Constant forcing --
  -> Add a constant source with b=. For y' = -y + 1, y(0)=0 the exact answer is 1 - exp(-T). Same call, one extra keyword.
  survived: True
  status: ok
  y: [0.950213]
  n_talbot_used: 32
  n_solves: 32
  nu_used: 0.6
  contour_scale: 4.267
  wall_s: 0
  expected: [0.950213]
  max_abs_error: 2.688e-10

-- Run complete --
  next_journey: 03_configuration.py

cd python
# Guided tour — narrated scenario walkthrough (recommended first run):
python examples/journeys/02_core_path.py
 
# Production one-liner — same API call you ship:
python -c "import solvlrde,numpy as np; print(solvlrde.solve(np.diag([-0.5,-1.,-2.]),[1,1,1],T=2.0)['y'])"