Journey 05 — Banded + DAE

solve_banded on tridiagonal MOL operators and reduce_dae for linear descriptor systems.

Prerequisites: Journey 04 works. Scenario: Exploit sparsity, and reduce a linear DAE to canonical form

Key APIs

solvlrde.banded_from_dense()
solvlrde.solve_banded()
solvlrde.reduce_dae()

Concepts to internalize

Banded layout — (kl+ku+1, n) LAPACK band storage.
MOL heat/diffusion — typical thermal and PDE workflows land here.
reduce_dae — returns reduced (A, b) for standard solve().

Important code

The guided tour script wraps this pattern. Copy the production shape into your pipeline:

ab = lrde.banded_from_dense(A, kl=1, ku=1)
result = lrde.solve_banded(ab, 1, 1, y0, T=T)
 
# DAE: M v' = -G v + f
A_red, b_red = lrde.reduce_dae(M, G, f)

Act-by-act walkthrough

Act 1 — Banded is for structured operators

Many linear operators - a 1-D heat equation discretized by method of lines, a chain of coupled masses - are banded. Passing the dense matrix wastes O(n^2) memory; the banded path takes the compact band layout and the sub/super bandwidths (kl, ku).

Act 2 — Dense vs banded agree

Solve the same heat problem two ways. banded_from_dense() builds the (kl+ku+1, n) layout; solve_banded() consumes it. The exact answer is exp(-pi^2 T) sin(pi x), so we can score both.

Field	Value	Notes
`dense_survived`	True
`banded_survived`	True
`dense_vs_banded_maxdiff`	0.000e+00
`banded_vs_exact_maxerr`	3.580e-05
`band_shape`	(3, 60)

Act 3 — DAE reduction

A linear descriptor system M v' = -G v (+ f) is not yet in y' = A y + b form. reduce_dae(M, G, f) returns (A, b) with A = -M^-1 G. M may be a 1-D diagonal or a full matrix. Here M = diag(2, 4), G = diag(2, 8) gives v' = diag(-1, -2) v.

Field	Value	Notes
`A_from_dae`	[[-1.0, -0.0], [-0.0, -2.0]]
`b_from_dae`	None
`survived`	True
`status`	ok
`y`	[0.367879, 0.135335]
`n_talbot_used`	32
`n_solves`	32
`nu_used`	0.6
`contour_scale`	12.8
`wall_s`	0
`expected`	[0.367879, 0.135335]
`max_abs_error`	9.492e-11

Guided tour output

Verified console capture from 05_banded_dae.py:

Exploit sparsity, and reduce a linear DAE to canonical form

-- Act 1 - Banded is for structured operators --
  -> Many linear operators - a 1-D heat equation discretized by method of lines, a chain of coupled masses - are banded. Passing the dense matrix wastes O(n^2) memory; the banded path takes the compact band layout and the sub/super bandwidths (kl, ku).

-- Act 2 - Dense vs banded agree --
  -> Solve the same heat problem two ways. banded_from_dense() builds the (kl+ku+1, n) layout; solve_banded() consumes it. The exact answer is exp(-pi^2 T) sin(pi x), so we can score both.
  dense_survived: True
  banded_survived: True
  dense_vs_banded_maxdiff: 0.000e+00
  banded_vs_exact_maxerr: 3.580e-05
  band_shape: (3, 60)

-- Act 3 - DAE reduction --
  -> A linear descriptor system M v' = -G v (+ f) is not yet in y' = A y + b form. reduce_dae(M, G, f) returns (A, b) with A = -M^-1 G. M may be a 1-D diagonal or a full matrix. Here M = diag(2, 4), G = diag(2, 8) gives v' = diag(-1, -2) v.
  A_from_dae: [[-1.0, -0.0], [-0.0, -2.0]]
  b_from_dae: None
  survived: True
  status: ok
  y: [0.367879, 0.135335]
  n_talbot_used: 32
  n_solves: 32
  nu_used: 0.6
  contour_scale: 12.8
  wall_s: 0
  expected: [0.367879, 0.135335]
  max_abs_error: 9.492e-11

-- Run complete --
  next_journey: 06_limits.py

Run it

cd python
# Guided tour — narrated scenario walkthrough (recommended first run):
python examples/journeys/05_banded_dae.py
 
# Production one-liner — same API call you ship:
python examples/journeys/05_banded_dae.py

What comes next

Journey 06 states routing to SolvSRK/SolvJump and honest refusals.

Navigation

← Journey 04 — Time-independent cost · Journey 06 — Limits →

04 Time-independent cost 06 Limits