review using Claude Code and update for consistency and correctness

.gitignore

@@ -39,4 +39,6 @@ test_util
 test_util.dat
 
 # Jupyter Notebook checkpoints
-*.ipynb_checkpoints
+*.ipynb_checkpoints
+
+arXiv-0708.4399v2

CLAUDE.md

@@ -0,0 +1,54 @@
+# CLAUDE.md
+
+This file provides guidance to Claude Code (claude.ai/code) when working with code in this repository.
+
+## Project purpose
+
+A teaching collection of FFTW 3.3.9 examples in C. Each example computes a 1D or 2D DFT both via FFTW and via a hand-rolled naive reference, then compares the two — the goal is to make explicit the symmetries, scaling factors, and array layouts that FFTW assumes. The README is the long-form tutorial; the `src/` programs are the runnable companions to it.
+
+## Build / run / test
+
+Top-level Makefile builds with `gcc` and links `-lfftw3 -lm`. Each target is a single-file executable that `#include`s `src/util.h`.
+
+```
+make            # build all test_* and app_* binaries (compiled into repo root, not src/)
+make test       # build all targets, then run each one via the %_test pattern rule
+make clean      # remove binaries and generated *.dat files
+make <name>     # build a single target, e.g. `make test_1d_redft00` or `make app_flux_surface`
+./<name>        # run a single test/app (binaries land in the repo root)
+```
+
+There is no separate test framework — each `test_*.c` has its own `main` that returns non-zero on mismatch, and `make test` simply runs them in sequence.
+
+The `eqn/` subdirectory has its own Makefile that renders `.tex` snippets to `.png` via `./pnglatex` for the README; ignore it unless touching the rendered equations.
+
+## Architecture
+
+**Header-only utility layer (`src/util.h`).** All shared helpers — random/zero array fills, element-wise comparison with tolerance, file dumpers in NumPy-loadable formats, a generalized-DFT reference (`dft_1d_cplx`), and the LCFS file readers — live as `static`-style definitions directly in the header. Each `.c` file is compiled into its own executable, so there is no link-time conflict. Adding a new helper means editing `util.h` directly; there is no `util.c`.
+
+**Per-example pattern.** Every `test_*.c` follows the same skeleton:
+1. Allocate input/output arrays with `fftw_alloc_real` / `fftw_alloc_complex`.
+2. Create the FFTW plan **before** filling input (FFTW_ESTIMATE; some plans destroy input on execute).
+3. Fill input with `fill_random_*`.
+4. Compute a reference output by a direct, naive transform formula.
+5. `fftw_execute`, then `compare_*` with `eps = 1e-12`.
+6. Free plan and arrays.
+For r2r tests (REDFT/RODFT), the reference is built by constructing the "logically equivalent" full DFT input (mirrored/anti-mirrored according to the transform's implicit symmetry) and comparing FFTW's output to a slice of that full DFT.
+
+**Application layer (`src/app_*.c`).** Three real-world stellarator-physics examples that consume the same primitives: `app_magn_axis` (1D c2r), `app_magn_axis_stellsym` (1D REDFT01, exploits stellarator symmetry), `app_flux_surface` (2D c2r, reads `lcfs.dat`). They write `axis_R*.dat` / `lcfs_R.dat` / `lcfs_Z.dat` to the repo root for plotting; `clean` removes these.
+
+**Python companions.** `src/plot_lcfs_realspace.py` (Mayavi) reads the `app_flux_surface` output. `src/redft_symmetries.py` and `src/redft_symmetries.ipynb` are exploratory NumPy notebooks on even/odd-harmonic DCT decomposition.
+
+## Non-obvious conventions
+
+- **2D arrays are row-major flat `double*` / `fftw_complex*`**, indexed as `arr[i*cols + j]`. The dumpers and comparators all assume this.
+- **VMEC vs FFTW sign convention** (relevant in `app_flux_surface.c` and the corresponding README section): VMEC uses angle argument `(m θ − n ζ)` while FFTW's 2D DFT uses `(m θ + n ζ)`. Coefficients from VMEC must be placed at index `-n` in the FFTW input — see the `idx_in` computation. `m=0` row is special-cased because VMEC keeps the positive-`n` sign there.
+- **Hermitian-symmetry scaling for c2r/r2c.** For real-input DFTs the complex array has `n/2+1` entries; all but DC (and Nyquist when `n` even) need a factor-of-2 weighting in the manual reference. Both array-layout figure (`img/array_structures.png`) and `test_1d_c2r.c` document this.
+- **R\*DFT00 caveat.** Per the FFTW manual, `REDFT00`/`RODFT00` are numerically less stable; `app_magn_axis_stellsym` deliberately uses `REDFT01` instead. Don't "fix" a stellsym example to use 00.
+- **Output `.dat` files are gitignored** but consumed by Python plotters and by some tests' own assertions — don't assume `make clean` is safe to run mid-debugging session if a Python plot is open.
+
+## When extending
+
+- New transform example → add `src/test_<name>.c`, add the bare name to `TARGETS` in the Makefile. The pattern rule `$(TARGETS): % : src/%.c src/util.h` handles compilation; no other wiring needed.
+- New utility → add it to `src/util.h` as a definition (not just a declaration). Keep it dependency-free beyond `<fftw3.h>` and libc.
+- The repo also carries Eclipse CDT (`.cproject`, `.project`) and PyDev (`.pydevproject`) project files; leave them alone unless asked.