Skip to content

Eigenvalue Helpers

1-D Laplacian eigenvalues for spectral elliptic solvers. See the theory page for derivations.

Same-BC Eigenvalues

Dirichlet

dst1_eigenvalues(N, dx)

1-D Laplacian eigenvalues for the DST-I basis (homogeneous Dirichlet BCs).

Returns the eigenvalues of the second-order finite-difference Laplacian

(L·v)_i = (v_{i-1} − 2v_i + v_{i+1}) / dx²

on N interior points with v_0 = v_{N+1} = 0. The DST-I diagonalises this operator, yielding:

λ_k = −4/dx² · sin²(π(k+1) / (2(N+1))),   k = 0, …, N−1

All eigenvalues are strictly negative (the discrete Dirichlet Laplacian is negative definite).

Parameters

N : int Number of interior grid points (excluding the two zero-boundary points). dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. All < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
def dst1_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for the DST-I basis (homogeneous Dirichlet BCs).

    Returns the eigenvalues of the second-order finite-difference Laplacian

        (L·v)_i = (v_{i-1} − 2v_i + v_{i+1}) / dx²

    on *N* interior points with v_0 = v_{N+1} = 0.  The DST-I diagonalises
    this operator, yielding:

        λ_k = −4/dx² · sin²(π(k+1) / (2(N+1))),   k = 0, …, N−1

    All eigenvalues are strictly negative (the discrete Dirichlet Laplacian
    is negative definite).

    Parameters
    ----------
    N : int
        Number of interior grid points (excluding the two zero-boundary points).
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.  All < 0.
    """
    k = jnp.arange(N)
    return -4.0 / dx**2 * jnp.sin(jnp.pi * (k + 1) / (2 * (N + 1))) ** 2

dst2_eigenvalues(N, dx)

1-D Laplacian eigenvalues for the DST-II basis (Dirichlet BCs, staggered grid).

Returns the eigenvalues of the second-order finite-difference Laplacian on N cell-centred points with homogeneous Dirichlet BCs at both boundaries (ψ = 0 at the cell edges, half a grid spacing outside the first and last points).

The DST-II diagonalises this operator, yielding:

λ_k = −4/dx² · sin²(π(k+1) / (2N)),   k = 0, …, N−1

All eigenvalues are strictly negative. The forward transform is DST-II; the inverse is DST-III (SciPy convention: idstn(type=2)).

Parameters

N : int Number of cell-centred grid points. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. All < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
def dst2_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for the DST-II basis (Dirichlet BCs, staggered grid).

    Returns the eigenvalues of the second-order finite-difference Laplacian
    on *N* cell-centred points with homogeneous Dirichlet BCs at both
    boundaries (ψ = 0 at the cell edges, half a grid spacing outside the
    first and last points).

    The DST-II diagonalises this operator, yielding:

        λ_k = −4/dx² · sin²(π(k+1) / (2N)),   k = 0, …, N−1

    All eigenvalues are strictly negative.  The forward transform is DST-II;
    the inverse is DST-III (SciPy convention: ``idstn(type=2)``).

    Parameters
    ----------
    N : int
        Number of cell-centred grid points.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.  All < 0.
    """
    k = jnp.arange(N)
    return -4.0 / dx**2 * jnp.sin(jnp.pi * (k + 1) / (2 * N)) ** 2

Neumann

dct1_eigenvalues(N, dx)

1-D Laplacian eigenvalues for the DCT-I basis (Neumann BCs, regular grid).

Returns the eigenvalues of the second-order finite-difference Laplacian on N vertex-centred points with homogeneous Neumann BCs (∂ψ/∂n = 0 at both boundaries, which coincide with the first and last grid points).

The DCT-I diagonalises this operator, yielding:

λ_k = −4/dx² · sin²(πk / (2(N−1))),   k = 0, …, N−1

The k = 0 eigenvalue is exactly zero (constant null mode). Requires N ≥ 2. The DCT-I is self-inverse (up to normalisation).

Parameters

N : int Number of grid points (including both boundary points). dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. λ_0 = 0; all others < 0.

Raises

ValueError If N < 2 (DCT-I requires at least two grid points).

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
def dct1_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for the DCT-I basis (Neumann BCs, regular grid).

    Returns the eigenvalues of the second-order finite-difference Laplacian
    on *N* vertex-centred points with homogeneous Neumann BCs
    (∂ψ/∂n = 0 at both boundaries, which coincide with the first and last
    grid points).

    The DCT-I diagonalises this operator, yielding:

        λ_k = −4/dx² · sin²(πk / (2(N−1))),   k = 0, …, N−1

    The k = 0 eigenvalue is exactly zero (constant null mode).  Requires
    N ≥ 2.  The DCT-I is self-inverse (up to normalisation).

    Parameters
    ----------
    N : int
        Number of grid points (including both boundary points).
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.
        λ_0 = 0; all others < 0.

    Raises
    ------
    ValueError
        If N < 2 (DCT-I requires at least two grid points).
    """
    if N < 2:
        raise ValueError(f"dct1_eigenvalues requires N >= 2, got N={N}")
    k = jnp.arange(N)
    return -4.0 / dx**2 * jnp.sin(jnp.pi * k / (2 * (N - 1))) ** 2

dct2_eigenvalues(N, dx)

1-D Laplacian eigenvalues for the DCT-II basis (homogeneous Neumann BCs).

Returns the eigenvalues of the second-order finite-difference Laplacian

(L·v)_i = (v_{i-1} − 2v_i + v_{i+1}) / dx²

on N points with ∂v/∂n = 0 at both boundaries (ghost-point Neumann). The DCT-II diagonalises this operator, yielding:

λ_k = −4/dx² · sin²(πk / (2N)),   k = 0, …, N−1

The k=0 eigenvalue is exactly zero (the constant mode), corresponding to the one-dimensional null space of the Neumann Laplacian.

Parameters

N : int Number of grid points. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. λ_0 = 0; all others < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
def dct2_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for the DCT-II basis (homogeneous Neumann BCs).

    Returns the eigenvalues of the second-order finite-difference Laplacian

        (L·v)_i = (v_{i-1} − 2v_i + v_{i+1}) / dx²

    on *N* points with ∂v/∂n = 0 at both boundaries (ghost-point Neumann).
    The DCT-II diagonalises this operator, yielding:

        λ_k = −4/dx² · sin²(πk / (2N)),   k = 0, …, N−1

    The k=0 eigenvalue is exactly zero (the constant mode), corresponding
    to the one-dimensional null space of the Neumann Laplacian.

    Parameters
    ----------
    N : int
        Number of grid points.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.
        λ_0 = 0; all others < 0.
    """
    k = jnp.arange(N)
    return -4.0 / dx**2 * jnp.sin(jnp.pi * k / (2 * N)) ** 2

Periodic

fft_eigenvalues(N, dx)

1-D Laplacian eigenvalues for the FFT basis (periodic BCs).

Returns the eigenvalues of the second-order finite-difference Laplacian

(L·v)_i = (v_{i-1} − 2v_i + v_{i+1}) / dx²

on N points with periodic boundary conditions v_{N} = v_0. The DFT diagonalises this operator, yielding:

λ_k = −4/dx² · sin²(πk / N),   k = 0, …, N−1

The k=0 eigenvalue is exactly zero (the constant / zero-wavenumber mode), corresponding to the one-dimensional null space of the periodic Laplacian.

Parameters

N : int Number of grid points in one period. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues in FFT order (k = 0, …, N−1). λ_0 = 0; all others ≤ 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
def fft_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for the FFT basis (periodic BCs).

    Returns the eigenvalues of the second-order finite-difference Laplacian

        (L·v)_i = (v_{i-1} − 2v_i + v_{i+1}) / dx²

    on *N* points with periodic boundary conditions v_{N} = v_0.  The DFT
    diagonalises this operator, yielding:

        λ_k = −4/dx² · sin²(πk / N),   k = 0, …, N−1

    The k=0 eigenvalue is exactly zero (the constant / zero-wavenumber mode),
    corresponding to the one-dimensional null space of the periodic Laplacian.

    Parameters
    ----------
    N : int
        Number of grid points in one period.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues in FFT order (k = 0, …, N−1).
        λ_0 = 0; all others ≤ 0.
    """
    k = jnp.arange(N)
    return -4.0 / dx**2 * jnp.sin(jnp.pi * k / N) ** 2

Mixed-BC Eigenvalues

dst3_eigenvalues(N, dx)

1-D Laplacian eigenvalues for Dirichlet-left + Neumann-right on a regular grid.

Pairs with DST-III forward / DST-II backward.

λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

All eigenvalues are strictly negative.

Parameters

N : int Number of grid points. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. All < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
def dst3_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for Dirichlet-left + Neumann-right on a regular grid.

    Pairs with DST-III forward / DST-II backward.

        λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

    All eigenvalues are strictly negative.

    Parameters
    ----------
    N : int
        Number of grid points.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.  All < 0.
    """
    return _mixed_bc_eigenvalues(N, dx)

dct3_eigenvalues(N, dx)

1-D Laplacian eigenvalues for Neumann-left + Dirichlet-right on a regular grid.

Pairs with DCT-III forward / DCT-II backward.

λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

All eigenvalues are strictly negative.

Parameters

N : int Number of grid points. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. All < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
def dct3_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for Neumann-left + Dirichlet-right on a regular grid.

    Pairs with DCT-III forward / DCT-II backward.

        λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

    All eigenvalues are strictly negative.

    Parameters
    ----------
    N : int
        Number of grid points.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.  All < 0.
    """
    return _mixed_bc_eigenvalues(N, dx)

dst4_eigenvalues(N, dx)

1-D Laplacian eigenvalues for Dirichlet-left + Neumann-right on a staggered grid.

Pairs with DST-IV (self-inverse).

λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

All eigenvalues are strictly negative.

Parameters

N : int Number of cell-centred grid points. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. All < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
def dst4_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for Dirichlet-left + Neumann-right on a staggered grid.

    Pairs with DST-IV (self-inverse).

        λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

    All eigenvalues are strictly negative.

    Parameters
    ----------
    N : int
        Number of cell-centred grid points.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.  All < 0.
    """
    return _mixed_bc_eigenvalues(N, dx)

dct4_eigenvalues(N, dx)

1-D Laplacian eigenvalues for Neumann-left + Dirichlet-right on a staggered grid.

Pairs with DCT-IV (self-inverse).

λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

All eigenvalues are strictly negative.

Parameters

N : int Number of cell-centred grid points. dx : float Uniform grid spacing.

Returns

Float[Array, " N"] 1-D array of N eigenvalues, ordered k = 0, …, N−1. All < 0.

Source code in spectraldiffx/_src/fourier/eigenvalues.py 
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
def dct4_eigenvalues(N: int, dx: float) -> Float[Array, " N"]:
    """1-D Laplacian eigenvalues for Neumann-left + Dirichlet-right on a staggered grid.

    Pairs with DCT-IV (self-inverse).

        λ_k = −4/dx² · sin²(π(2k+1) / (4N)),   k = 0, …, N−1

    All eigenvalues are strictly negative.

    Parameters
    ----------
    N : int
        Number of cell-centred grid points.
    dx : float
        Uniform grid spacing.

    Returns
    -------
    Float[Array, " N"]
        1-D array of *N* eigenvalues, ordered k = 0, …, N−1.  All < 0.
    """
    return _mixed_bc_eigenvalues(N, dx)