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Fast summation
Applications

Direct and fast summation (convolution) More...

Collaboration diagram for Fast summation:

## Modules
| fastsum_matlab
| fastsum_test

## Data Structures
struct | fastsum_plan_
| plan for fast summation algorithm More...

## Macros
#define | X(name) NFFT(name)
| Include header for C99 complex datatype. More...
#define | NF_KUB
#define | EXACT_NEARFIELD (1U<< 0)
| Constant symbols.
#define | NEARFIELD_BOXES (1U<< 1)
#define | STORE_PERMUTATION_X_ALPHA (1U<< 2)
| If this flag is set, and eps_I > 0.0 and NEARFIELD_BOXES is not set, then the vector permutation_x_alpha is stored. More...

## Typedefs
typedef C(* | kernel) (R, int, const R *)
typedef struct fastsum_plan_ | fastsum_plan
| plan for fast summation algorithm

## Functions
static int | max_i (int a, int b)
| max
static R | fak (int n)
| factorial
static R | binom (int n, int m)
| binomial coefficient
static R | BasisPoly (int m, int r, R xx)
| basis polynomial for regularized kernel
C | regkern (kernel k, R xx, int p, const R *param, R a, R b)
| regularized kernel with K_I arbitrary and K_B smooth to zero
static C | regkern1 (kernel k, R xx, int p, const R *param, R a, R b)
| regularized kernel with K_I arbitrary and K_B periodized (used in 1D)
static C | regkern3 (kernel k, R xx, int p, const R *param, R a, R b)
| regularized kernel for even kernels with K_I even and K_B mirrored More...
C | kubintkern (const R x, const C *Add, const int Ad, const R a)
| linear spline interpolation in near field with even kernels More...
static C | kubintkern1 (const R x, const C *Add, const int Ad, const R a)
| cubic spline interpolation in near field with arbitrary kernels
static void | quicksort (int d, int t, R *x, C *alpha, int *permutation_x_alpha, int N)
| quicksort algorithm for source knots and associated coefficients
static void | BuildBox (fastsum_plan *ths)
| initialize box-based search data structures
static C | calc_SearchBox (int d, R *y, R *x, C *alpha, int start, int end_lt, const C *Add, const int Ad, int p, R a, const kernel k, const R *param, const unsigned flags)
| inner computation function for box-based near field correction
static C | SearchBox (R *y, fastsum_plan *ths)
| box-based near field correction
static void | BuildTree (int d, int t, R *x, C *alpha, int *permutation_x_alpha, int N)
| recursive sort of source knots dimension by dimension to get tree structure
static C | SearchTree (const int d, const int t, const R *x, const C *alpha, const R *xmin, const R *xmax, const int N, const kernel k, const R *param, const int Ad, const C *Add, const int p, const unsigned flags)
| fast search in tree of source knots for near field computation
static void | fastsum_precompute_kernel (fastsum_plan *ths)
void | fastsum_init_guru_kernel (fastsum_plan *ths, int d, kernel k, R *param, unsigned flags, int nn, int p, R eps_I, R eps_B)
| initialize node independent part of fast summation plan More...
void | fastsum_init_guru_source_nodes (fastsum_plan *ths, int N_total, int nn_oversampled, int m)
| initialize source nodes dependent part of fast summation plan More...
void | fastsum_init_guru_target_nodes (fastsum_plan *ths, int M_total, int nn_oversampled, int m)
| initialize target nodes dependent part of fast summation plan More...
void | fastsum_init_guru (fastsum_plan *ths, int d, int N_total, int M_total, kernel k, R *param, unsigned flags, int nn, int m, int p, R eps_I, R eps_B)
| initialization of fastsum plan More...
void | fastsum_finalize_source_nodes (fastsum_plan *ths)
| finalization of fastsum plan More...
void | fastsum_finalize_target_nodes (fastsum_plan *ths)
| finalization of fastsum plan More...
void | fastsum_finalize_kernel (fastsum_plan *ths)
| finalization of fastsum plan More...
void | fastsum_finalize (fastsum_plan *ths)
| finalization of fastsum plan More...
void | fastsum_exact (fastsum_plan *ths)
| direct computation of sums More...
void | fastsum_precompute_source_nodes (fastsum_plan *ths)
| precomputation for fastsum More...
void | fastsum_precompute_target_nodes (fastsum_plan *ths)
| precomputation for fastsum More...
void | fastsum_precompute (fastsum_plan *ths)
| precomputation for fastsum More...
void | fastsum_trafo (fastsum_plan *ths)
| fast NFFT-based summation More...
C | gaussian (R x, int der, const R *param)
| K(x)=exp(-x^2/c^2)
C | multiquadric (R x, int der, const R *param)
| K(x)=sqrt(x^2+c^2)
C | inverse_multiquadric (R x, int der, const R *param)
| K(x)=1/sqrt(x^2+c^2)
C | logarithm (R x, int der, const R *param)
| K(x)=log |x|.
C | thinplate_spline (R x, int der, const R *param)
| K(x) = x^2 log |x|.
C | one_over_square (R x, int der, const R *param)
| K(x) = 1/x^2.
C | one_over_modulus (R x, int der, const R *param)
| K(x) = 1/|x|.
C | one_over_x (R x, int der, const R *param)
| K(x) = 1/x.
C | inverse_multiquadric3 (R x, int der, const R *param)
| K(x) = 1/sqrt(x^2+c^2)^3.
C | sinc_kernel (R x, int der, const R *param)
| K(x) = sin(cx)/x.
C | cosc (R x, int der, const R *param)
| K(x) = cos(cx)/x.
C | kcot (R x, int der, const R *param)
| K(x) = cot(cx)
C | one_over_cube (R x, int der, const R *param)
| K(x) = 1/x^3.
C | log_sin (R x, int der, const R *param)
| K(x) = log(|sin(cx)|)
C | laplacian_rbf (R x, int der, const R *param)
| K(x) = exp(-|x|/c)

## Detailed Description

Direct and fast summation (convolution)

Computes the sums

## Macro Definition Documentation

#define X | ( |  | name | ) | NFFT(name)

Include header for C99 complex datatype.

Include header for utils from NFFT3 library. Include header for NFFT3 library.

Definition at line 57 of file fastsum.h.

Referenced by CONCAT(), fpt_init(), fpt_precompute(), fpt_transposed(), main(), nfsft_precompute(), nfsoft_adjoint(), nfsoft_init_guru_advanced(), nfsoft_trafo(), nsfft_adjoint(), nsfft_cp(), nsfft_init(), and nsfft_init_random_nodes_coeffs().

#define STORE_PERMUTATION_X_ALPHA (1U<< 2)

If this flag is set, and eps_I > 0.0 and NEARFIELD_BOXES is not set, then the vector permutation_x_alpha is stored.

Definition at line 79 of file fastsum.h.

Referenced by fastsum_init_guru_source_nodes().

## Function Documentation

static C regkern3 | ( | kernel | k,
|  | R | xx,
|  | int | p,
|  | const R * | param,
|  | R | a,
|  | R | b
| ) |  |
| static

regularized kernel for even kernels with K_I even and K_B mirrored

regularized kernel for even kernels with K_I even and K_B mirrored smooth to K(1/2) (used in dD, d>1)

Definition at line 230 of file fastsum.c.

References BasisPoly().

Referenced by SearchTree().

C kubintkern | ( | const R | x,
|  | const C * | Add,
|  | const int | Ad,
|  | const R | a
| ) |  |

linear spline interpolation in near field with even kernels

cubic spline interpolation in near field with even kernels

Definition at line 318 of file fastsum.c.

Referenced by calc_SearchBox(), and SearchTree().

void fastsum_init_guru_kernel | ( | fastsum_plan * | ths,
|  | int | d,
|  | kernel | k,
|  | R * | param,
|  | unsigned | flags,
|  | int | nn,
|  | int | p,
|  | R | eps_I,
|  | R | eps_B
| ) |  |

initialize node independent part of fast summation plan

Parameters

  ths | The pointer to a fastsum plan.
  d | The dimension of the problem.
  kernel | The kernel function.
  param | The parameters for the kernel function.
  flags | Fastsum flags.
  nn | The expansion degree.
  p | The degree of smoothness.
  eps_I | The inner boundary.
  eps_B | the outer boundary.

Definition at line 779 of file fastsum.c.

References fastsum_plan_::Ad, fastsum_plan_::Add, fastsum_plan_::b, fastsum_plan_::d, fastsum_plan_::eps_B, fastsum_plan_::eps_I, EXACT_NEARFIELD, fastsum_plan_::f_hat, fastsum_plan_::flags, fastsum_plan_::k, fastsum_plan_::kernel_param, max_i(), fastsum_plan_::n, one_over_x(), and fastsum_plan_::p.

Referenced by fastsum_init_guru().

void fastsum_init_guru_source_nodes | ( | fastsum_plan * | ths,
|  | int | N_total,
|  | int | nn_oversampled,
|  | int | m
| ) |  |

initialize source nodes dependent part of fast summation plan

Parameters

  ths | The pointer to a fastsum plan.
  N_total | The number of source knots x.
  nn_oversampled | The oversampled expansion degree for nfft.
  m | The cut-off parameter for the NFFT.

Definition at line 887 of file fastsum.c.

References fastsum_plan_::alpha, fastsum_plan_::d, fastsum_plan_::eps_B, fastsum_plan_::eps_I, fastsum_plan_::f, fastsum_plan_::f_hat, FFT_OUT_OF_PLACE, FFTW_INIT, fastsum_plan_::flags, fastsum_plan_::n, fastsum_plan_::N_total, fastsum_plan_::permutation_x_alpha, PRE_PHI_HUT, PRE_PSI, STORE_PERMUTATION_X_ALPHA, and fastsum_plan_::x.

Referenced by fastsum_init_guru().

void fastsum_init_guru_target_nodes | ( | fastsum_plan * | ths,
|  | int | M_total,
|  | int | nn_oversampled,
|  | int | m
| ) |  |

initialize target nodes dependent part of fast summation plan

Parameters

  ths | The pointer to a fastsum plan.
  M_total | The number of target knots y.
  nn_oversampled | The oversampled expansion degree for nfft.
  m | The cut-off parameter for the NFFT.

Definition at line 955 of file fastsum.c.

References fastsum_plan_::d, fastsum_plan_::f, fastsum_plan_::f_hat, FFT_OUT_OF_PLACE, FFTW_INIT, fastsum_plan_::M_total, fastsum_plan_::n, PRE_PHI_HUT, PRE_PSI, fastsum_plan_::x, and fastsum_plan_::y.

Referenced by fastsum_init_guru().

void fastsum_init_guru | ( | fastsum_plan * | ths,
|  | int | d,
|  | int | N_total,
|  | int | M_total,
|  | kernel | k,
|  | R * | param,
|  | unsigned | flags,
|  | int | nn,
|  | int | m,
|  | int | p,
|  | R | eps_I,
|  | R | eps_B
| ) |  |

initialization of fastsum plan

initialize fast summation plan

Parameters

  ths | The pointer to a fastsum plan.
  d | The dimension of the problem.
  N_total | The number of source knots x.
  M_total | The number of target knots y.
  kernel | The kernel function.
  param | The parameters for the kernel function.
  flags | Fastsum flags.
  nn | The expansion degree.
  m | The cut-off parameter for the NFFT.
  p | The degree of smoothness.
  eps_I | The inner boundary.
  eps_B | the outer boundary.

Definition at line 987 of file fastsum.c.

References fastsum_init_guru_kernel(), fastsum_init_guru_source_nodes(), and fastsum_init_guru_target_nodes().

void fastsum_finalize_source_nodes | ( | fastsum_plan * | ths | ) |

finalization of fastsum plan

finalize source nodes dependent part of plan

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 996 of file fastsum.c.

References fastsum_plan_::alpha, fastsum_plan_::eps_I, fastsum_plan_::flags, fastsum_plan_::permutation_x_alpha, and fastsum_plan_::x.

Referenced by fastsum_finalize().

void fastsum_finalize_target_nodes | ( | fastsum_plan * | ths | ) |

finalization of fastsum plan

finalize target nodes dependent part of plan

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1020 of file fastsum.c.

References fastsum_plan_::f, and fastsum_plan_::y.

Referenced by fastsum_finalize().

void fastsum_finalize_kernel | ( | fastsum_plan * | ths | ) |

finalization of fastsum plan

finalize node independent part of plan

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1029 of file fastsum.c.

References fastsum_plan_::Add, fastsum_plan_::b, fastsum_plan_::eps_I, EXACT_NEARFIELD, fastsum_plan_::f_hat, and fastsum_plan_::flags.

Referenced by fastsum_finalize().

void fastsum_finalize | ( | fastsum_plan * | ths | ) |

finalization of fastsum plan

finalize plan

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1048 of file fastsum.c.

References fastsum_finalize_kernel(), fastsum_finalize_source_nodes(), and fastsum_finalize_target_nodes().

void fastsum_exact | ( | fastsum_plan * | ths | ) |

direct computation of sums

direct summation

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1056 of file fastsum.c.

References fastsum_plan_::alpha, fastsum_plan_::d, fastsum_plan_::f, fastsum_plan_::k, fastsum_plan_::kernel_param, fastsum_plan_::M_total, fastsum_plan_::N_total, fastsum_plan_::x, and fastsum_plan_::y.

void fastsum_precompute_source_nodes | ( | fastsum_plan * | ths | ) |

precomputation for fastsum

sort source nodes, precompute nfft source plan.

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1086 of file fastsum.c.

References fastsum_plan_::alpha, BuildBox(), BuildTree(), fastsum_plan_::d, fastsum_plan_::eps_I, fastsum_plan_::flags, fastsum_plan_::MEASURE_TIME_t, fastsum_plan_::N_total, nfft_elapsed_seconds(), fastsum_plan_::permutation_x_alpha, PRE_FULL_PSI, PRE_LIN_PSI, PRE_PSI, and fastsum_plan_::x.

Referenced by fastsum_precompute().

void fastsum_precompute_target_nodes | ( | fastsum_plan * | ths | ) |

precomputation for fastsum

precompute nfft target plan.

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1141 of file fastsum.c.

References fastsum_plan_::flags, fastsum_plan_::MEASURE_TIME_t, PRE_FULL_PSI, PRE_LIN_PSI, and PRE_PSI.

Referenced by fastsum_precompute().

void fastsum_precompute | ( | fastsum_plan * | ths | ) |

precomputation for fastsum

sort source nodes, precompute nfft plans etc.

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1173 of file fastsum.c.

References fastsum_precompute_source_nodes(), and fastsum_precompute_target_nodes().

void fastsum_trafo | ( | fastsum_plan * | ths | ) |

fast NFFT-based summation

fast NFFT-based summation algorithm

Parameters

  ths | The pointer to a fastsum plan.

Definition at line 1180 of file fastsum.c.

References fastsum_plan_::Ad, fastsum_plan_::Add, fastsum_plan_::alpha, fastsum_plan_::b, fastsum_plan_::d, fastsum_plan_::eps_I, fastsum_plan_::f, fastsum_plan_::f_hat, fastsum_plan_::flags, fastsum_plan_::k, fastsum_plan_::kernel_param, fastsum_plan_::M_total, fastsum_plan_::MEASURE_TIME_t, fastsum_plan_::N_total, nfft_elapsed_seconds(), fastsum_plan_::p, SearchBox(), SearchTree(), fastsum_plan_::x, and fastsum_plan_::y.