Collaboration diagram for Fast nonlinear Fourier transforms:
Files | |
| file | fnft_kdv_discretization_t.h |
| Lists discretizations for the Korteweg-de Vries equation. | |
| file | fnft_kdvv.h |
| Fast nonlinear Fourier transform for the vanishing Korteweg-de Vries equation. | |
| file | fnft_nse_discretization_t.h |
| Lists discretizations for the nonlinear Schroedinger equation. | |
| file | fnft_nsep.h |
| Fast nonlinear Fourier transform for the periodic nonlinear Schroedinger equation. | |
| file | fnft_nsev.h |
| Fast nonlinear Fourier transform for the vanishing nonlinear Schroedinger equation. | |
| file | fnft_version.h |
| Provides a function to query the version of the FNFT library. | |
Classes | |
| struct | fnft_kdvv_opts_t |
| Stores additional options for the routine fnft_kdvv. More... | |
| struct | fnft_nsep_opts_t |
| Stores additional options for the routine fnft_nsep. More... | |
| struct | fnft_nsev_opts_t |
| Stores additional options for the routine fnft_nsev. More... | |
Functions | |
| fnft_kdvv_opts_t | fnft_kdvv_default_opts () |
| Creates a new options variable for fnft_kdvv with default settings. More... | |
| FNFT_INT | fnft_kdvv (const FNFT_UINT D, FNFT_COMPLEX *const q, FNFT_REAL const *const T, const FNFT_UINT M, FNFT_COMPLEX *const contspec, FNFT_REAL const *const XI, FNFT_UINT *const K_ptr, FNFT_COMPLEX *const bound_states, FNFT_COMPLEX *const normconsts_or_residues, fnft_kdvv_opts_t *opts_ptr) |
| Fast nonlinear Fourier transform for the Korteweg-de Vries equation with vanishing boundary conditions. More... | |
| fnft_nsep_opts_t | fnft_nsep_default_opts () |
| Creates a new options variable for fnft_nsep with default settings. More... | |
| FNFT_INT | fnft_nsep (const FNFT_UINT D, FNFT_COMPLEX const *const q, FNFT_REAL const *const T, FNFT_REAL const phase_shift, FNFT_UINT *const K_ptr, FNFT_COMPLEX *const main_spec, FNFT_UINT *const M_ptr, FNFT_COMPLEX *const aux_spec, FNFT_REAL *const sheet_indices, const FNFT_INT kappa, fnft_nsep_opts_t *opts) |
| Fast nonlinear Fourier transform for the nonlinear Schroedinger equation with (quasi-)periodic boundary conditions. More... | |
| fnft_nsev_opts_t | fnft_nsev_default_opts () |
| Creates a new options variable for fnft_nsev with default settings. More... | |
| FNFT_UINT | fnft_nsev_max_K (const FNFT_UINT D, fnft_nsev_opts_t const *const opts) |
| Returns the maximum number of bound states that can be detected by fnft_nsev. More... | |
| FNFT_INT | fnft_nsev (const FNFT_UINT D, FNFT_COMPLEX *const q, FNFT_REAL const *const T, const FNFT_UINT M, FNFT_COMPLEX *const contspec, FNFT_REAL const *const XI, FNFT_UINT *const K_ptr, FNFT_COMPLEX *const bound_states, FNFT_COMPLEX *const normconsts_or_residues, const FNFT_INT kappa, fnft_nsev_opts_t *opts) |
| Nonlinear Fourier transform for the nonlinear Schroedinger equation with vanishing boundary conditions. Fast algorithms are used if the discretization supports it. More... | |
| FNFT_INT | fnft_version (FNFT_UINT *const major, FNFT_UINT *const minor, FNFT_UINT *const patch, char suffix[FNFT_VERSION_SUFFIX_MAXLEN+1]) |
Detailed Description
Function Documentation
◆ fnft_kdvv()
| FNFT_INT fnft_kdvv | ( | const FNFT_UINT | D, |
| FNFT_COMPLEX *const | q, | ||
| FNFT_REAL const *const | T, | ||
| const FNFT_UINT | M, | ||
| FNFT_COMPLEX *const | contspec, | ||
| FNFT_REAL const *const | XI, | ||
| FNFT_UINT *const | K_ptr, | ||
| FNFT_COMPLEX *const | bound_states, | ||
| FNFT_COMPLEX *const | normconsts_or_residues, | ||
| fnft_kdvv_opts_t * | opts_ptr | ||
| ) |
Fast nonlinear Fourier transform for the Korteweg-de Vries equation with vanishing boundary conditions.
This routine computes the nonlinear Fourier transform for the Korteweg-de Vries equation
\[ q_x + 6qq_{t} + q_{ttt}=0, \quad q=q(x,t), \]
of Gardner et al. (Phys. Rev. Lett., 1967) for initial conditions with vanishing boundaries
\[ \lim_{t\to \pm \infty }q(x_0,t) = 0 \text{ sufficiently rapidly.} \]
- Parameters
-
[in] D Number of samples. [in] q Array of length D, contains samples \( q(t_n)=q(x_0, t_n) \), where \( t_n = T[0] + n(T[1]-T[0])/(D-1) \) and \(n=0,1,\dots,D-1\), of the to-be-transformed signal in ascending order (i.e., \( q(t_0), q(t_1), \dots, q(t_{D-1}) \)). [in] T Array of length 2, contains the position in time of the first and of the last sample. It should be T[0]<T[1]. [in] M Number of points at which the continuous spectrum (aka reflection coefficient) should be computed. [out] contspec Array of length M in which the routine will store the desired samples \( R(\xi_m) \) of the continuous spectrum (aka reflection coefficient) in ascending order, where \( \xi_m = XI[0]+m(XI[1]-XI[0])/(M-1) \) and \(m=0,1,\dots,M-1\). Has to be preallocated by the user. [in] XI Array of length 2, contains the position of the first and the last sample of the continuous spectrum. It should be XI[0]<XI[1]. Can also be NULL if contspec==NULL. [in,out] K_ptr Not yet implemented. Pass NULL. [out] bound_states Not yet implemented. Pass NULL. [out] normconsts_or_residues Not yet implemented. Pass NULL. [in] opts_ptr Pointer to a fnft_kdvv_opts_t object. The object can be used to modify the behavior of the routine. Use the routine fnft_kdvv_default_opts to generate such an object and modify as desired. It is also possible to pass NULL, in which case the routine will use the default options. The user is reponsible to freeing the object after the routine has returned.
- Returns
- FNFT_SUCCESS or one of the FNFT_EC_... error codes defined in fnft_errwarn.h.
◆ fnft_kdvv_default_opts()
| fnft_kdvv_opts_t fnft_kdvv_default_opts | ( | ) |
Creates a new options variable for fnft_kdvv with default settings.
- Returns
- A fnft_kdvv_opts_t object with default options.
◆ fnft_nsep()
| FNFT_INT fnft_nsep | ( | const FNFT_UINT | D, |
| FNFT_COMPLEX const *const | q, | ||
| FNFT_REAL const *const | T, | ||
| FNFT_REAL const | phase_shift, | ||
| FNFT_UINT *const | K_ptr, | ||
| FNFT_COMPLEX *const | main_spec, | ||
| FNFT_UINT *const | M_ptr, | ||
| FNFT_COMPLEX *const | aux_spec, | ||
| FNFT_REAL *const | sheet_indices, | ||
| const FNFT_INT | kappa, | ||
| fnft_nsep_opts_t * | opts | ||
| ) |
Fast nonlinear Fourier transform for the nonlinear Schroedinger equation with (quasi-)periodic boundary conditions.
This routine computes the nonlinear Fourier transform for the nonlinear Schroedinger equation
\[ iq_x + q_{tt} \pm 2q|q|^2=0, \quad q=q(x,t), \]
of Kotlyarov and Its (Problems of Mathemetical Physics and Functional Analysis, 1976) for initial conditions with (quasi-)periodic boundary conditions,
\[ q(x_0,t + L) = mq(x_0,t),|m|=1. \]
The main references for the numerical algorithm are:
- Wahls and Poor, "Fast numerical nonlinear Fourier transforms," IEEE Trans. Inform. Theor. 61(12), 2015.
- Prins and Wahls, "Higher order exponential splittings for the fast non-linear Fourier transform of the KdV equation,"in Proc.ICASSP 2018, Calgary, AB, 2018, pp. 4524-4528.
- Mertsching, " Quasiperiodie Solutions of the Nonlinear Schrödinger Equation," Fortschr. Phys. 35:519-536, 1987.
The routine supports the following discretizations of type fnft_nse_discretization_t:
- fnft_nse_discretization_2SPLIT1A
- fnft_nse_discretization_2SPLIT1B
- fnft_nse_discretization_2SPLIT2A
- fnft_nse_discretization_2SPLIT2B
- fnft_nse_discretization_2SPLIT2S
- fnft_nse_discretization_2SPLIT2_MODAL
- fnft_nse_discretization_2SPLIT3A
- fnft_nse_discretization_2SPLIT3B
- fnft_nse_discretization_2SPLIT3S
- fnft_nse_discretization_2SPLIT4A
- fnft_nse_discretization_2SPLIT4B
- fnft_nse_discretization_2SPLIT5A
- fnft_nse_discretization_2SPLIT5B
- fnft_nse_discretization_2SPLIT6A
- fnft_nse_discretization_2SPLIT6B
- fnft_nse_discretization_2SPLIT7A
- fnft_nse_discretization_2SPLIT7B
- fnft_nse_discretization_2SPLIT8A
- fnft_nse_discretization_2SPLIT8B
- fnft_nse_discretization_4SPLIT4A
- Parameters
-
[in] D Number of samples. Should be power of two and \( D\geq 2\). [in] q Array of length D, contains samples \( q(t_n)=q(x_0, t_n) \), where \( t_n = T[0] + n*L/D \), where \(L=T[1]-T[0]\) is the period and \(n=0,1,\dots,D-1\), of the to-be-transformed signal in ascending order (i.e., \( q(t_0), q(t_1), \dots, q(t_{D-1}) \)) [in] T Array of length 2. T[0] is the position in time of the first sample. T[1] is the beginning of the next period. (The location of the last sample is thus \(t_{D-1}=T[1]-L/D\).) It should be \(T[0]<T[1]\). [in] phase_shift Real scalar constant. It is the change in the phase over one quasi-period, \( arg(q(t+L)/q(t))\). For periodic signals it will be 0. [in,out] K_ptr Upon entry, *K_ptr should contain the length of the array main_spec. Upon return, *K_ptr contains the number of actually detected points in the main spectrum. If the length of the array main_spec was not sufficient to store all of the detected points in the main spectrum, a warning is printed and as many points as possible are returned instead. Note that in order to skip the computation of the main spectrum completely, it is not sufficient to pass *K_ptr==NULL. Instead, pass main_spec==NULL. [out] main_spec Array. Upon return, the routine has stored the detected main specrum points (i.e., the points for which the trace of the monodromy matrix is either +2 or -2) in the first *K_ptr entries of this array. If NULL is passed instead, the main spectrum will not be computed. Has to be preallocated by the user. The user can choose an arbitrary length. Typically, D is a good choice. [in,out] M_ptr Upon entry, *M_ptr should contain the length of the array aux_spec. Upon return, *M_ptr contains the number of actually detected points in the auxiliary spectrum. If the length of the array aux_spec was not sufficient to store all of the detected points in the auxiliary spectrum, a warning is printed and as many points as possible are returned instead. Note that in order to skip the computation of the auxiliary spectrum completely, it is not sufficient to pass *M_ptr==NULL. Instead, pass aux_spec==NULL. [out] aux_spec Array. Upon return, the routine has stored the detected auxiliary specrum points (i.e., the points for which the upper right element of the monodromy matrix is zero) in the first *M_ptr entries of this array. If NULL is passed instead, the auxiliary spectrum will not be computed. Has to be preallocated by the user. The user can choose an arbitrary length. Typically, D is a good choice. [in] sheet_indices Not yet implemented. Pass NULL. [in] kappa =+1 for the focusing nonlinear Schroedinger equation, =-1 for the defocusing one. [in] opts Pointer to a fnft_nsep_opts_t object. The object can be used to modify the behavior of the routine. Use the routine fnft_nsep_default_opts to generate such an object and modify as desired. It is also possible to pass NULL, in which case the routine will use the default options. The user is reponsible to freeing the object after the routine has returned.
- Returns
- FNFT_SUCCESS or one of the FNFT_EC_... error codes defined in fnft_errwarn.h.
◆ fnft_nsep_default_opts()
| fnft_nsep_opts_t fnft_nsep_default_opts | ( | ) |
Creates a new options variable for fnft_nsep with default settings.
- Returns
- A fnft_nsep_opts_t object with the following options.
localization = fnft_nsep_loc_MIXED
filtering = fnft_nsep_filt_AUTO
max_evals = 20
bounding_box[0] = -FNFT_INF
bounding_box[1] = FNFT_INF
bounding_box[2] = -FNFT_INF
bounding_box[3] = FNFT_INF
normalization_flag = 1
discretization = fnft_nse_discretization_2SPLIT2A
floquet_range = {-1, 1}
floquet_nvals = 2
Dsub = 0
◆ fnft_nsev()
| FNFT_INT fnft_nsev | ( | const FNFT_UINT | D, |
| FNFT_COMPLEX *const | q, | ||
| FNFT_REAL const *const | T, | ||
| const FNFT_UINT | M, | ||
| FNFT_COMPLEX *const | contspec, | ||
| FNFT_REAL const *const | XI, | ||
| FNFT_UINT *const | K_ptr, | ||
| FNFT_COMPLEX *const | bound_states, | ||
| FNFT_COMPLEX *const | normconsts_or_residues, | ||
| const FNFT_INT | kappa, | ||
| fnft_nsev_opts_t * | opts | ||
| ) |
Nonlinear Fourier transform for the nonlinear Schroedinger equation with vanishing boundary conditions. Fast algorithms are used if the discretization supports it.
This routine computes the nonlinear Fourier transform for the nonlinear Schroedinger equation
\[ iq_x + q_{tt} \pm 2q|q|^2=0, \quad q=q(x,t), \]
of Zakharov and Shabat (Soviet. Phys. JTEP 31(1), 1972) for initial conditions with vanishing boundaries
\[ \lim_{t\to \pm \infty }q(x_0,t) = 0 \text{ sufficiently rapidly.} \]
The main references are:
- Wahls and Poor,"Introducing the fast nonlinear Fourier transform," Proc. ICASSP 2013.
- Wahls and Poor, "Fast numerical nonlinear Fourier transforms," IEEE Trans. Inform. Theor. 61(12), 2015.
- Prins and Wahls, " Higher order exponential splittings for the fast non-linear Fourier transform of the KdV equation," Proc. ICASSP 2018, pp. 4524-4528
The routine also utilizes ideas from the following papers:
- Boffetta and Osborne, "Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102(2), 1992.
- Aref, "Control and Detection of Discrete Spectral Amplitudes in Nonlinear Fourier Spectrum," Preprint, arXiv:1605.06328 [math.NA], May 2016.
- Hari and Kschischang, "Bi-Directional Algorithm for Computing Discrete Spectral Amplitudes in the NFT," J. Lightwave Technol. 34(15), 2016.
- Aref et al., "Modulation Over Nonlinear Fourier Spectrum: Continuous and Discrete Spectrum", J. Lightwave Technol. 36(6), 2018.
- Aurentz et al., "Roots of Polynomials: on twisted QR methods for companion matrices and pencils," Preprint, arXiv:1611.02435 [math.NA], Dec. 2016.
- Chimmalgi, Prins and Wahls, "Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators," IEEE Access 7, 2019.
- Prins and Wahls, " Soliton Phase Shift Calculation for the Korteweg–De Vries Equation," IEEE Access, vol. 7, pp. 122914–122930, July 2019.
- Medvedev, Vaseva, Chekhovskoy and Fedoruk, " Exponential fourth order schemes for direct Zakharov-Shabat problem," Optics Express, vol. 28, pp. 20–39, 2020.
The routine supports all discretizations of type fnft_nse_discretization_t. The following discretizations use fast algorithms which have a computational complexity of \( \mathcal{O}(D\log^2 D)\) for \( D\) point continuous spectrum given \( D\) samples:
- fnft_nse_discretization_2SPLIT1A
- fnft_nse_discretization_2SPLIT1B
- fnft_nse_discretization_2SPLIT2A
- fnft_nse_discretization_2SPLIT2B
- fnft_nse_discretization_2SPLIT2S
- fnft_nse_discretization_2SPLIT2_MODAL
- fnft_nse_discretization_2SPLIT3A
- fnft_nse_discretization_2SPLIT3B
- fnft_nse_discretization_2SPLIT3S
- fnft_nse_discretization_2SPLIT4A
- fnft_nse_discretization_2SPLIT4B
- fnft_nse_discretization_2SPLIT5A
- fnft_nse_discretization_2SPLIT5B
- fnft_nse_discretization_2SPLIT6A
- fnft_nse_discretization_2SPLIT6B
- fnft_nse_discretization_2SPLIT7A
- fnft_nse_discretization_2SPLIT7B
- fnft_nse_discretization_2SPLIT8A
- fnft_nse_discretization_2SPLIT8B
- fnft_nse_discretization_4SPLIT4A
- fnft_nse_discretization_4SPLIT4B
The following discretizations use classical algorithms which have a computational complexity of \( \mathcal{O}(D^2)\) for \( D\) point continuous spectrum given \( D\) samples:
- fnft_nse_discretization_BO
- fnft_nse_discretization_CF4_2
- fnft_nse_discretization_CF4_3
- fnft_nse_discretization_CF5_3
- fnft_nse_discretization_CF6_4
- fnft_nse_discretization_ES4
- fnft_nse_discretization_TES4
The accuray of the computed quantities for a given signal depends primarily on the number of samples \( D\) and the numerical method. When the exact spectrum is is know, the accuracy can be quantified by defining a suitable error. The error usually decreases with increasing \( D\) assuming everthing else remains the same. The rate at which the error decreases with increase in \( D\) is known as the order of the method. The orders of the various discretizations can be found at fnft_nse_discretization_t. The orders of the discretizations which use exponential splitting schemes should be the same as their base methods but can deviate when accuracy of the splitting scheme is low. In the following cases the orders of the methods has been observed to be less than expected:
- Focusing case CF4_3, residues have order two instead of four.
- Focusing case CF6_4, residues have order three instead of six.
- Focusing case TES4, residues have order two instead of four.
Application of one step Richardson extrapolation should in theory increase the order by one. However, it can deviate both ways. In case of some smooth signals the order may increase by two instead of one. On the other hand for discontinuous signals it maybe deterimental to apply Richardson extrapolation. In the following cases the orders of the methods has been observed to be less than expected:
- Focusing case CF4_3, residues have order two instead of five.
- Focusing case CF5_3, residues have order five instead of six.
- Focusing case CF6_4, residues have order four instead of seven.
- Focusing case TES4, residues have order two instead of five.
- Parameters
-
[in] D Number of samples [in] q Array of length D, contains samples \( q(t_n)=q(x_0, t_n) \), where \( t_n = T[0] + n(T[1]-T[0])/(D-1) \) and \(n=0,1,\dots,D-1\), of the to-be-transformed signal in ascending order (i.e., \( q(t_0), q(t_1), \dots, q(t_{D-1}) \)) [in] T Array of length 2, contains the position in time of the first and of the last sample. It should be \(T[0]<T[1]\). [in] M Number of points at which the continuous spectrum (aka reflection coefficient) should be computed. [out] contspec Array of length M in which the routine will store the desired samples \( r(\xi_m) \) of the continuous spectrum (aka reflection coefficient) in ascending order, where \( \xi_m = XI[0]+(XI[1]-XI[0])/(M-1) \) and \(m=0,1,\dots,M-1\). Has to be preallocated by the user. If NULL is passed instead, the continuous spectrum will not be computed. By changing the options, it is also possible to compute the values of \( a(\xi) \) and \( b(\xi) \) instead. In that case, twice the amount of memory has to be allocated. [in] XI Array of length 2, contains the position of the first and the last sample of the continuous spectrum. It should be \(XI[0]<XI[1]\). Can also be NULL if contspec==NULL. [in,out] K_ptr Upon entry, *K_ptr should contain the length of the array bound_states. Upon return, *K_ptr contains the number of actually detected bound states. If the length of the array bound_states was not sufficient to store all of the detected bound states, a warning is printed and as many bound states as possible are returned instead. Note that in order to skip the computation of the bound states completely, it is not sufficient to pass *K_ptr==0. Instead, one needs to pass bound_states==NULL. [out] bound_states Array. Upon return, the routine has stored the detected bound states (aka eigenvalues) in the first *K_ptr entries of this array. If NULL is passed instead, the discrete spectrum will not be computed. Has to be preallocated by the user. The user can choose an arbitrary length. Typically, D is a good choice. [out] normconsts_or_residues Array of the same length as bound_states. Upon return, the routine has stored the residues (aka spectral amplitudes) \(\rho_k = b_k\big/ \frac{da(\lambda_k)}{d\lambda}\) in the first *K_ptr entries of this array. By passing a proper opts, it is also possible to store the norming constants (the ' \( b_k \)') or both. Has to be pre-allocated by the user. If NULL is passed instead, the residues will not be computed. [in] kappa =+1 for the focusing nonlinear Schroedinger equation, =-1 for the defocusing one. [in] opts Pointer to a fnft_nsev_opts_t object. The object can be used to modify the behavior of the routine. Use the routine fnft_nsev_default_opts to generate such an object and modify as desired. It is also possible to pass NULL, in which case the routine will use the default options. The user is reponsible to freeing the object after the routine has returned.
- Returns
- FNFT_SUCCESS or one of the FNFT_EC_... error codes defined in fnft_errwarn.h.
◆ fnft_nsev_default_opts()
| fnft_nsev_opts_t fnft_nsev_default_opts | ( | ) |
Creates a new options variable for fnft_nsev with default settings.
- Returns
- A fnft_nsev_opts_t object with the following options.
bound_state_filtering = fnft_nsev_bsfilt_FULL
bound_state_localization = fnft_nsev_bsloc_SUBSAMPLE_AND_REFINE
niter = 10
discspec_type = fnft_nsev_dstype_NORMING_CONSTANTS
contspec_type = fnft_nsev_cstype_REFLECTION_COEFFICIENT
normalization_flag = 1
discretization = fnft_nse_discretization_2SPLIT4B
richardson_extrapolation_flag = 0
◆ fnft_nsev_max_K()
| FNFT_UINT fnft_nsev_max_K | ( | const FNFT_UINT | D, |
| fnft_nsev_opts_t const *const | opts | ||
| ) |
Returns the maximum number of bound states that can be detected by fnft_nsev.
- Parameters
-
[in] D Number of samples that will be passed to fnft_nsev. Should be larger than zero. [in] opts Options that will be passed to fnft_nsev. If NULL is passed, the default options will be used.
- Returns
- Returns the maximum number of bound states or zero on error.
◆ fnft_version()
| FNFT_INT fnft_version | ( | FNFT_UINT *const | major, |
| FNFT_UINT *const | minor, | ||
| FNFT_UINT *const | patch, | ||
| char | suffix[FNFT_VERSION_SUFFIX_MAXLEN+1] | ||
| ) |
Provides the version of the FNFT library.
The version is of the form $major.$minor.$patch$suffix, e.g., "0.2.1-al". The function simply returns the corresponding values defined fnft_config.h.
- Parameters
-
[out] major *major is overwritten with the value of FNFT_VERSION_MAJOR [out] minor *minor is overwritten with the value of FNFT_VERSION_MINOR [out] patch *patch is overwritten with the value of FNFT_VERSION_PATCH [out] suffix suffix is overwritten with the value of FNFT_VERSION_SUFFIX
- Returns
- FNFT_SUCCESS or one of the FNFT_EC_... error codes defined in fnft_errwarn.h.
